is 30 an integer 1. For example, 2 , 5 , 0 , − 12 , 244 , − 15 and 8 are all integers. Integer is a math term for a number that is a whole number. The answer to the question: "Is 30 an integer?" Walk through the difference between whole numbers & integers. > How do you check if a value in Java is of an integer type? MATLAB ® has four signed and four unsigned integer classes. Integers are the set of whole numbers and their opposites. He has been teaching from the past 9 years. He provides courses for … These numbers are used to perform various arithmetic operations, like addition, subtraction, multiplication and division.The examples of integers are, 1, 2, 5,8, -9, -12, etc. Is it an integer? An integer.An integer.An integer.An integer. An integer, also called a "round number" or “whole number,” is any positive or negative number that does not include decimal parts or fractions. Quantity A is greater. How to use integer in a sentence. Signed types enable you to work with negative integers as well as positive, but cannot represent as wide a range of numbers as the unsigned types because one bit is used to … $\begingroup$ "For this ratio to not be an integer m and n must be distinct which implies that x is not an integer quantity." 5- (-4)=1 please help i have a test tommorrow and i just dont get what an integer is. Because you can't "count" zero. The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. My only temporary fix was to change the JavaScript that handles the fields, as it was using regex to check if it was and Integer so i changed it to convert the string to int and check if it was NaN Comment Share Find out if 24 is an Integer here! Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. Yes. An even number is any integer (a whole number) that can be divided by 2. You could say that 30/6 is just another name for 5, and then it is an integer. 30 is the integer in this case. Why don't libraries smell like bookstores? integer - any of the natural numbers (positive or negative) or zero; "an integer is a number that is not a fraction" whole number. (30 points) An integer n is prime iff n >= 2 and n's only factors (divisors) are 1 and itself. And 30 is an integer and not a fraction. Suppose x and y are even. A comprehensive database of more than 56 integer quizzes online, test your knowledge with integer quiz questions. What are the three consecutive odd integers with sum of -30? It is a special set of whole numbers comprised of zero, positive numbers and negative numbers and denoted by the letter Z. My only temporary fix was to change the JavaScript that handles the fields, as it was using regex to check if it was and Integer so i changed it to convert the string to int and check if … go, not everyone agrees on a simple thing! Integer Question: Is 24 an Integer? We are simply taking "5 lots of −3", like this: It should be noted that an integer can never be a fraction, a decimal or a per cent. When I as doing my graduation one of my professors Mr. Jaggi told us “Maths student must be polite” thats why I used here ‘may’. sound intelligent. You might be wondering, what is an opposite? python 2.7.4 error: exactly the same as integers. In fact I can't think of any proof of this sentence which doesn't prove the whole theorem along the way. Integer Question: Is 20 an Integer? The opposite of 24 is -24. The number line goes on forever in both directions. Solution for 1. Is a none repeating decimal an integer? Thanks for your help! The number 0 is also considered an integer even though it has neither a positive or negative value. What details make Lochinvar an attractive and romantic figure? Question 256872: The sum of an integer and its square is 30. 30 is the smallest sphenic number, and the smallest of the form 2 × 3 × r, where r is a prime greater than 3. What is the conflict of the story of sinigang? What is an Integer? So they are 1, 2, 3, 4, 5, ... (and so on). What are the steps to the solution? If x is even, then x y is even. An integer is what is more commonly known as a whole number.It may be positive, negative, or the number zero, but it must be whole. In some cases, the definition of whole number will exclude the number zero, or even the set of negative numbers, but this is not as common as the more inclusive use of the term. n is an integer, and k is not an integer. De tekst is beschikbaar onder de licentie Creative Commons Naamsvermelding/Gelijk delen, er kunnen aanvullende voorwaarden van toepassing zijn. The symbol of integers is “ Z “. See more. Examples– -2.4, 3/4, 90.6. Every integer is a rational number. Quantity B is greater. "Natural Numbers" can mean either "Counting Numbers" {1, 2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject. The set of integers is infinite and has no smallest element and no largest element. > is.integer(66L) [1] TRUE > is.integer(66) [1] FALSE Also functions like round will return a declared integer, which is what you are doing with x==round(x). This is shown below. And some people say that zero is NOT a whole number. An integer is a positive or negative whole number. The … PHP does not support unsigned integers. The basic rules of math. For 20 to be an integer, 20 has to be a whole number. How is 30 an even number? Our online integer trivia quizzes can be adapted to suit your requirements for taking some of the top integer quizzes. Furthermore, for 20 to be an integer, you should be able to write 20 without a fractional or decimal component. Definitions. You could try this. (use indirect… It is impossible to answer your question without … 0 0 1 ... An integer is a number that is not a fraction, and decimal numbers are decimal fractions. There are three Proper… Integers The integers are the set of whole numbers and their opposites. Find out if 20 is an Integer here! The farther the number is to the right, the larger the value is. Base Representation; 2: … This is one of those tricky definition questions. Even and Odd Numbers (use indirect… Find the number. It cannot be done. Hello, We have a drafter who's Map 3D 2013 Drafting Settings dialog box is giving him a pop up that says "Please enter an integer from 1 to 30". Solution for 1. Here are some examples: We can compare integers the same way that we compare positive whole numbers. The number 30 as a Roman Numeral XXX The number 30 in various Numeral Systems. Fractions and decimals are not included in the set of integers. please give me an example or two of an integer. I want to know how to test whether an input value is an integer or not. Please note this from the Integer datatype page: "The size of an integer is platform-dependent, although a maximum value of about two billion is the usual value (that's 32 bits signed). Some examples of integers include 1, 3, 4, 8, 99, 108, -43, … Examples of Integers – 1, 6, 15. I have tried using the function isinteger, but I obtain, for example, isinteger(3) = 0.Apparently, any constant is double-precision by Matlab default, and is therefore not recognized as an integer. In the Registry Editor window, navigate to the folder path below depending on your Photoshop version.. Photoshop CC 2018: Navigate to HKEY_CURRENT_USER\SOFTWARE\Adobe\Photoshop\120.0 Photoshop CC 2017: Navigate to HKEY_CURRENT_USER\SOFTWARE\Adobe\Photoshop\110.0 Photoshop CC 2015.5: Navigate to HKEY_CURRENT_USER\SOFTWARE\Adobe\Photoshop\100.0 … For 24 to be an integer, 24 has to be a whole number. A modular type is an integer type with all arithmetic modulo a specified positive modulus; such a type corresponds to an unsigned type with wrap-around semantics. 30 is a whole number that can be written without a fractional component, thus 30 is an integer. It is not only accurate, it makes you In Maths, integers are the numbers which can be positive, negative or zero, but cannot be a fraction. An integer is any number including 0, positive numbers, and negative numbers. But every rational number is not an integer, since for b not equal to 1, if b does not divide a then w don't get an integer. Integer definition, one of the positive or negative numbers 1, 2, 3, etc., or zero. Examples of Integer Multiplication (a) Here is an integer times an integer: 5 xx -3 = -15 Why is this so? But you for sure can say it is a rational number, which is any number that can be expressed as a … Choose from 500 different sets of 30 math integers flashcards on Quizlet. How will understanding of attitudes and predisposition enhance teaching? In the equation 2 + 1/2, the number 2 is the integer and 1/2 is the fraction. Find the integer The sum of an integer and its square is - Answered by a verified Math Tutor or Teacher Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, ... (and so on), (But numbers like ½, 1.1 and 3.5 are not whole numbers.). The number line is used to represent integers. Start by using a number line. Compare -9 and -2. 0 0 1 ... 176 + 125 = 301 Therefore, the lowest positive integer you can add is 125. When did organ music become associated with baseball? An integer (pronounced IN-tuh-jer) is a whole number (not a fractional number) that can be positive, negative, or zero. An integer (from the Latin integer meaning "whole") is colloquially defined as a number that can be written without a fractional component.For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 + 1 / 2, and √ 2 are not. Suppose x and y are even. I want to know how to test whether an input value is an integer or not. Where is telephone area code 035? 4. Consider a function Factor_In_Range (k , n) to return true iff some integer in the range k to n … And everyone agrees on the definition of an integer, so when in doubt say "integer". This is indicated by the arrows. All Rights Reserved. If a a a and b b b are integers such that a ⋅ b = 0 a \cdot b = 0 a ⋅ b = 0, then a = 0 a=0 a = 0 or b = 0 b=0 b = 0. Rational number is a number which can be expressed as a/b where a,b are integers and b is not 0. a) -3 or 10 b) 3 or -10 c) -5 or 6 d) 5 or -6 Help! For 30 to be an integer, 30 has to be a whole number. Ex 6.1, 2 Solve –12x > 30, when x is a natural number –12x > 30 –x > 3012 –x > 2.5 Since x is negative, we multiply both sides by -1 & change the signs (–x) × (-1) < 2.5 × (-1) x < − 2.5 (i) Since x is a natural number, it can be only positive num The base range of a signed integer type includes at least the values of the specified range. is.integer tests if the number is declared as an integer. B. An integer_type_definition defines an integer type; it defines either a signed integer type, or a modular integer type. You need to prove this, as nothing you have said makes it obvious this is true. Learn 30 math integers with free interactive flashcards. But you for sure can say it is a rational number, which is any number that can be expressed as a fraction of two integers.. An octal integer is an integer represented by a string of digits from the set $\{0,1,2,3,4,5,6,7\}$, each representing (in that order) the usual first eight non-negative integers. Integers are like whole numbers, but they also include negative numbers ... but still no fractions allowed! This is one of those tricky definition questions. If a is an odd integer, then a2 +3a+5 is odd. These are all integers (click to mark), and they continue left and right infinitely: Some people (not me) say that whole numbers can also be negative, which makes them So they are 1, 2, 3, 4, 5, ... (and so on). What is an integer? Therefore, these numbers can never be integers: fractions; decimals; percents; Adding and Subtracting Integers. Given an integer number n, return the difference between the product of its digits and the sum of its digits.. He provides courses for Maths and Science at Teachoo. ok i dont get.... is it just considered an integer if the dominate is 1? Counting Numbers are Whole Numbers, but without the zero. Examples of integers are: … If you divide by 2 and there is a remainder left, then the number is odd. The set of integers consists of zero (), the positive natural numbers (1, 2, 3, ...), also called whole numbers or … For example, 3, -10, and 1,025 are all integers, but 2.76 (decimal), 1.5 … Looking at a number line can help you when you need to add or subtract integers. Integers are positive whole numbers and their opposites, including zero. characteristic - the integer part (positive or negative) of the representation of a logarithm; in the expression log 643 = 2.808 the characteristic is 2 Furthermore, for 30 to be an integer, you should be able to write 30 without a fractional or decimal component. If a is an odd integer, then a2 +3a+5 is odd. For any integer a a a, the additive inverse − a-a − a is an integer. (use direct proof) 2. (ii) x is an integer Now, x < 2.5 Since x is an integer, it can be negative but cannot be a decimal So, x can be any negative number less than 3 Hence, the values of x which will satisfy the inequality are = { , 6, 5, 4, 3} Show More The word integer originated from the Latin word “Integer” which means whole. Integers are whole numbers and their negative opposites. Copyright © 2020 Multiply Media, LLC. Integers Integer Classes. Is 30 an integer? When multiplying integers, we can think of multiplying "blocks" of negative numbers. expressed in rational form as 30/1. 2020-09-30 17:50:29 2020-09-30 17:50:29. First of all we should know that “Every whole number is an integer, but every integer may not be a whole number.”. An integer (from the Latin integer meaning "whole") is colloquially defined as a number that can be written without a fractional component.For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 + 1 / 2, and √ 2 are not. Intager. What is the contribution of candido bartolome to gymnastics? You could say that 30/6 is just another name for 5, and then it is an integer. Like this (note: zero isn't positive or negative): For an interesting look at other types of numbers read The Evolution of Numbers, { ... −4, −3, −2, −1, 0, 1, 2, 3, 4, ... }, Adding and Subtracting Positive and Negative Numbers, Negative Integers = { ..., −4, −3, −2, −1 }, Non-Negative Integers = { 0, 1, 2, 3, 4, ... }. Essay On Nature And Environment, How To Get Rid Of Spotted Knapweed, Projection Matrix Songho, Thermomix Recipes Chinese Food, Cherry Ambrosia Salad, Burton Michigan Chamber Of Commerce, Fundamentals Of General, Organic, And Biological Chemistry Pdf, Solar Energy Activities For Elementary Students, " />
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# is 30 an integer

Answer by drk(1908) (Show Source): So if we consider b=1 then we get the set of integers. If x is even, then x y is even. Example 1: Input: n = 234 Output: 15 Explanation: Product of digits = 2 * 3 * 4 = 24 Sum of digits = 2 + 3 + 4 = 9 Result = 24 - 9 = 15 Example 2: Input: n = 4421 Output: 21 Explanation: Product of digits = 4 * 4 * 2 * 1 = 32 Sum of digits = 4 + 4 + 2 + 1 = 11 Result = 32 - 11 = 21 Integer definition is - any of the natural numbers, the negatives of these numbers, or zero. Whats wrong with this code: n = 10 ((n/3)).is_integer() I do not understand why I cannot set n = any number and check if it is an integer or not. So, integers can be negative {−1, −2,−3, −4, ... }, positive {1, 2, 3, 4, ... }, or zero {0}, Integers = { ..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ... }, (But numbers like ½, 1.1 and 3.5 are not integers). (use direct proof) 2. (n)(k+1) A. Integers vs. decimals and fractions An integer, also called a "round number" or “whole number,” is any positive or negative number that does not include decimal parts or fractions. I have tried using the function isinteger, but I obtain, for example, isinteger(3) = 0.Apparently, any constant is double-precision by Matlab default, and is therefore not recognized as an integer. Because you can't \"count\" zero. First of all we should know that “Every whole number is an integer, but every integer may not be a whole number.”. Counting Numbers are Whole Numbers, but without the zero. The opposite of 2 is -2. Integer size can be determined from PHP_INT_SIZE, maximum value from PHP_INT_MAX since PHP 4.4.0 and PHP 5.0.5." Integer Multiplication. when you only want positive integers, say "positive integers". No. When I as doing my graduation one of my professors Mr. Jaggi told us “Maths student must be polite” thats why I used here ‘may’. So there you Fractions, decimals, and percents are out of this basket. Integers are helpful when modeling real life situations. and which one of these would be an integer? Who is the longest reigning WWE Champion of all time? In most programming languages, you can convert a number into an integer … If you can cleanly divide a number by 2, the number is even. The integer 30 is a Composite number. Integer kan verwijzen naar: Integriteit (persoon), ... Deze pagina is voor het laatst bewerkt op 30 jul 2017 om 14:55. Does pumpkin pie need to be refrigerated? Examples of negative integers are -1, -2, -3, and -4. 0 k n k + 2. Properties of the number 30 The integer 30 is an even number. Thus, when x is a natural number, there is no solution. What are the factors of 602? 42 is greater than 30, so 30 is an abundant number. -5 + 5=0 2. 30 has an aliquot sum of 42; the second sphenic number and all sphenic numbers of this form have an aliquot sum 12 greater than themselves. You can declare an integer by putting a L after it. Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. For example, 3, -10, and 1,025 are all integers, but 2.76 (decimal), 1.5 (decimal), and 3 ½ (fraction) are not. For example, is the number -8 a whole number? Integer Rules: A Video Watch this video to better understand the correct procedure for adding, subtracting, multiplying, and dividing positive and negative whole numbers. is YES. Furthermore, for 24 to be an integer, you should be able to write 24 without a fractional or decimal component. 2012-06-30 12:44:11 2012-06-30 12:44:11. The problem with this approach is what you consider to be practically an integer. C. The two quantities are equal. In the Registry Editor window, navigate to the folder path below depending on your Photoshop version.. Photoshop CC 2018: Navigate to HKEY_CURRENT_USER\SOFTWARE\Adobe\Photoshop\120.0 Photoshop CC 2017: Navigate to HKEY_CURRENT_USER\SOFTWARE\Adobe\Photoshop\110.0 … The opposite of -13 is 13. The square of an integer is 30 more than the integer. Examples of positive integers are 1, 2, 3, and 4. He has been teaching from the past 9 years. -----> 1. For example, 2 , 5 , 0 , − 12 , 244 , − 15 and 8 are all integers. Integer is a math term for a number that is a whole number. The answer to the question: "Is 30 an integer?" Walk through the difference between whole numbers & integers. > How do you check if a value in Java is of an integer type? MATLAB ® has four signed and four unsigned integer classes. Integers are the set of whole numbers and their opposites. He has been teaching from the past 9 years. He provides courses for … These numbers are used to perform various arithmetic operations, like addition, subtraction, multiplication and division.The examples of integers are, 1, 2, 5,8, -9, -12, etc. Is it an integer? An integer.An integer.An integer.An integer. An integer, also called a "round number" or “whole number,” is any positive or negative number that does not include decimal parts or fractions. Quantity A is greater. How to use integer in a sentence. Signed types enable you to work with negative integers as well as positive, but cannot represent as wide a range of numbers as the unsigned types because one bit is used to … $\begingroup$ "For this ratio to not be an integer m and n must be distinct which implies that x is not an integer quantity." 5- (-4)=1 please help i have a test tommorrow and i just dont get what an integer is. Because you can't "count" zero. The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. My only temporary fix was to change the JavaScript that handles the fields, as it was using regex to check if it was and Integer so i changed it to convert the string to int and check if it was NaN Comment Share Find out if 24 is an Integer here! Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. Yes. An even number is any integer (a whole number) that can be divided by 2. You could say that 30/6 is just another name for 5, and then it is an integer. 30 is the integer in this case. Why don't libraries smell like bookstores? integer - any of the natural numbers (positive or negative) or zero; "an integer is a number that is not a fraction" whole number. (30 points) An integer n is prime iff n >= 2 and n's only factors (divisors) are 1 and itself. And 30 is an integer and not a fraction. Suppose x and y are even. A comprehensive database of more than 56 integer quizzes online, test your knowledge with integer quiz questions. What are the three consecutive odd integers with sum of -30? It is a special set of whole numbers comprised of zero, positive numbers and negative numbers and denoted by the letter Z. My only temporary fix was to change the JavaScript that handles the fields, as it was using regex to check if it was and Integer so i changed it to convert the string to int and check if … go, not everyone agrees on a simple thing! Integer Question: Is 24 an Integer? We are simply taking "5 lots of −3", like this: It should be noted that an integer can never be a fraction, a decimal or a per cent. When I as doing my graduation one of my professors Mr. Jaggi told us “Maths student must be polite” thats why I used here ‘may’. sound intelligent. You might be wondering, what is an opposite? python 2.7.4 error: exactly the same as integers. In fact I can't think of any proof of this sentence which doesn't prove the whole theorem along the way. Integer Question: Is 20 an Integer? The opposite of 24 is -24. The number line goes on forever in both directions. Solution for 1. Is a none repeating decimal an integer? Thanks for your help! The number 0 is also considered an integer even though it has neither a positive or negative value. What details make Lochinvar an attractive and romantic figure? Question 256872: The sum of an integer and its square is 30. 30 is the smallest sphenic number, and the smallest of the form 2 × 3 × r, where r is a prime greater than 3. What is the conflict of the story of sinigang? What is an Integer? So they are 1, 2, 3, 4, 5, ... (and so on). What are the steps to the solution? If x is even, then x y is even. An integer is what is more commonly known as a whole number.It may be positive, negative, or the number zero, but it must be whole. In some cases, the definition of whole number will exclude the number zero, or even the set of negative numbers, but this is not as common as the more inclusive use of the term. n is an integer, and k is not an integer. De tekst is beschikbaar onder de licentie Creative Commons Naamsvermelding/Gelijk delen, er kunnen aanvullende voorwaarden van toepassing zijn. The symbol of integers is “ Z “. See more. Examples– -2.4, 3/4, 90.6. Every integer is a rational number. Quantity B is greater. "Natural Numbers" can mean either "Counting Numbers" {1, 2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject. The set of integers is infinite and has no smallest element and no largest element. > is.integer(66L) [1] TRUE > is.integer(66) [1] FALSE Also functions like round will return a declared integer, which is what you are doing with x==round(x). This is shown below. And some people say that zero is NOT a whole number. An integer is a positive or negative whole number. The … PHP does not support unsigned integers. The basic rules of math. For 20 to be an integer, 20 has to be a whole number. How is 30 an even number? Our online integer trivia quizzes can be adapted to suit your requirements for taking some of the top integer quizzes. Furthermore, for 20 to be an integer, you should be able to write 20 without a fractional or decimal component. Definitions. You could try this. (use indirect… It is impossible to answer your question without … 0 0 1 ... An integer is a number that is not a fraction, and decimal numbers are decimal fractions. There are three Proper… Integers The integers are the set of whole numbers and their opposites. Find out if 20 is an Integer here! The farther the number is to the right, the larger the value is. Base Representation; 2: … This is one of those tricky definition questions. Even and Odd Numbers (use indirect… Find the number. It cannot be done. Hello, We have a drafter who's Map 3D 2013 Drafting Settings dialog box is giving him a pop up that says "Please enter an integer from 1 to 30". Solution for 1. Here are some examples: We can compare integers the same way that we compare positive whole numbers. The number 30 as a Roman Numeral XXX The number 30 in various Numeral Systems. Fractions and decimals are not included in the set of integers. please give me an example or two of an integer. I want to know how to test whether an input value is an integer or not. Please note this from the Integer datatype page: "The size of an integer is platform-dependent, although a maximum value of about two billion is the usual value (that's 32 bits signed). Some examples of integers include 1, 3, 4, 8, 99, 108, -43, … Examples of Integers – 1, 6, 15. I have tried using the function isinteger, but I obtain, for example, isinteger(3) = 0.Apparently, any constant is double-precision by Matlab default, and is therefore not recognized as an integer. In the Registry Editor window, navigate to the folder path below depending on your Photoshop version.. Photoshop CC 2018: Navigate to HKEY_CURRENT_USER\SOFTWARE\Adobe\Photoshop\120.0 Photoshop CC 2017: Navigate to HKEY_CURRENT_USER\SOFTWARE\Adobe\Photoshop\110.0 Photoshop CC 2015.5: Navigate to HKEY_CURRENT_USER\SOFTWARE\Adobe\Photoshop\100.0 … For 24 to be an integer, 24 has to be a whole number. A modular type is an integer type with all arithmetic modulo a specified positive modulus; such a type corresponds to an unsigned type with wrap-around semantics. 30 is a whole number that can be written without a fractional component, thus 30 is an integer. It is not only accurate, it makes you In Maths, integers are the numbers which can be positive, negative or zero, but cannot be a fraction. An integer is any number including 0, positive numbers, and negative numbers. But every rational number is not an integer, since for b not equal to 1, if b does not divide a then w don't get an integer. Integer definition, one of the positive or negative numbers 1, 2, 3, etc., or zero. Examples of Integer Multiplication (a) Here is an integer times an integer: 5 xx -3 = -15 Why is this so? But you for sure can say it is a rational number, which is any number that can be expressed as a … Choose from 500 different sets of 30 math integers flashcards on Quizlet. How will understanding of attitudes and predisposition enhance teaching? In the equation 2 + 1/2, the number 2 is the integer and 1/2 is the fraction. Find the integer The sum of an integer and its square is - Answered by a verified Math Tutor or Teacher Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, ... (and so on), (But numbers like ½, 1.1 and 3.5 are not whole numbers.). The number line is used to represent integers. Start by using a number line. Compare -9 and -2. 0 0 1 ... 176 + 125 = 301 Therefore, the lowest positive integer you can add is 125. When did organ music become associated with baseball? An integer (pronounced IN-tuh-jer) is a whole number (not a fractional number) that can be positive, negative, or zero. An integer (from the Latin integer meaning "whole") is colloquially defined as a number that can be written without a fractional component.For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 + 1 / 2, and √ 2 are not. Suppose x and y are even. I want to know how to test whether an input value is an integer or not. Where is telephone area code 035? 4. Consider a function Factor_In_Range (k , n) to return true iff some integer in the range k to n … And everyone agrees on the definition of an integer, so when in doubt say "integer". This is indicated by the arrows. All Rights Reserved. If a a a and b b b are integers such that a ⋅ b = 0 a \cdot b = 0 a ⋅ b = 0, then a = 0 a=0 a = 0 or b = 0 b=0 b = 0. Rational number is a number which can be expressed as a/b where a,b are integers and b is not 0. a) -3 or 10 b) 3 or -10 c) -5 or 6 d) 5 or -6 Help! For 30 to be an integer, 30 has to be a whole number. Ex 6.1, 2 Solve –12x > 30, when x is a natural number –12x > 30 –x > 3012 –x > 2.5 Since x is negative, we multiply both sides by -1 & change the signs (–x) × (-1) < 2.5 × (-1) x < − 2.5 (i) Since x is a natural number, it can be only positive num The base range of a signed integer type includes at least the values of the specified range. is.integer tests if the number is declared as an integer. B. An integer_type_definition defines an integer type; it defines either a signed integer type, or a modular integer type. You need to prove this, as nothing you have said makes it obvious this is true. Learn 30 math integers with free interactive flashcards. But you for sure can say it is a rational number, which is any number that can be expressed as a fraction of two integers.. An octal integer is an integer represented by a string of digits from the set $\{0,1,2,3,4,5,6,7\}$, each representing (in that order) the usual first eight non-negative integers. Integers are like whole numbers, but they also include negative numbers ... but still no fractions allowed! This is one of those tricky definition questions. If a is an odd integer, then a2 +3a+5 is odd. These are all integers (click to mark), and they continue left and right infinitely: Some people (not me) say that whole numbers can also be negative, which makes them So they are 1, 2, 3, 4, 5, ... (and so on). What is an integer? Therefore, these numbers can never be integers: fractions; decimals; percents; Adding and Subtracting Integers. Given an integer number n, return the difference between the product of its digits and the sum of its digits.. He provides courses for Maths and Science at Teachoo. ok i dont get.... is it just considered an integer if the dominate is 1? Counting Numbers are Whole Numbers, but without the zero. Examples of integers are: … If you divide by 2 and there is a remainder left, then the number is odd. The set of integers consists of zero (), the positive natural numbers (1, 2, 3, ...), also called whole numbers or … For example, 3, -10, and 1,025 are all integers, but 2.76 (decimal), 1.5 … Looking at a number line can help you when you need to add or subtract integers. Integers are positive whole numbers and their opposites, including zero. characteristic - the integer part (positive or negative) of the representation of a logarithm; in the expression log 643 = 2.808 the characteristic is 2 Furthermore, for 30 to be an integer, you should be able to write 30 without a fractional or decimal component. If a is an odd integer, then a2 +3a+5 is odd. For any integer a a a, the additive inverse − a-a − a is an integer. (use direct proof) 2. (ii) x is an integer Now, x < 2.5 Since x is an integer, it can be negative but cannot be a decimal So, x can be any negative number less than 3 Hence, the values of x which will satisfy the inequality are = { , 6, 5, 4, 3} Show More The word integer originated from the Latin word “Integer” which means whole. Integers are whole numbers and their negative opposites. Copyright © 2020 Multiply Media, LLC. Integers Integer Classes. Is 30 an integer? When multiplying integers, we can think of multiplying "blocks" of negative numbers. expressed in rational form as 30/1. 2020-09-30 17:50:29 2020-09-30 17:50:29. First of all we should know that “Every whole number is an integer, but every integer may not be a whole number.”. An integer (from the Latin integer meaning "whole") is colloquially defined as a number that can be written without a fractional component.For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 + 1 / 2, and √ 2 are not. Intager. What is the contribution of candido bartolome to gymnastics? You could say that 30/6 is just another name for 5, and then it is an integer. Like this (note: zero isn't positive or negative): For an interesting look at other types of numbers read The Evolution of Numbers, { ... −4, −3, −2, −1, 0, 1, 2, 3, 4, ... }, Adding and Subtracting Positive and Negative Numbers, Negative Integers = { ..., −4, −3, −2, −1 }, Non-Negative Integers = { 0, 1, 2, 3, 4, ... }.


# Thread: When theta must be positive and when negative

1. ## When theta must be positive and when negative

I'm grappling with just not seeing when to use negative angles and when to use positive angles when solving trig functions.

If I pick a point on a graph with coordinates (5,-5) the angle formed between the x-axis and the hypotenuse can be either $-45^\circ$ or $315^\circ$ (along with all multiples of $360^\circ$). Yes?

If this is true, then I would think that $\sin(-45^\circ) = \sin(315^\circ) = \dfrac{-\sqrt{2}}{2}$

If I plot the sine function I see that $\sin(-45^\circ) = \dfrac{-\sqrt{2}}{2}$ but my calculator gives me an error when entering $\sin(-45^\circ)$ or $\sin(\dfrac{-\pi}{4})$

Likewise, my current lesson on complex numbers requires that I use the sin and cos functions of $\dfrac{7\pi}{4}$ rather than $\dfrac{-\pi}{4}$

I would appreciate an explanation of when to use a negative reference angle and when to use the positive angle $< 360^\circ$

2. ## Re: When theta must be positive and when negative

Originally Posted by B9766
I'm grappling with just not seeing when to use negative angles and when to use positive angles when solving trig functions.

If I pick a point on a graph with coordinates (5,-5) the angle formed between the x-axis and the hypotenuse can be either $-45^\circ$ or $315^\circ$ (along with all multiples of $360^\circ$). Yes?
If this is true, then I would think that $\sin(-45^\circ) = \sin(315^\circ) = \dfrac{-\sqrt{2}}{2}$
If I plot the sine function I see that $\sin(-45^\circ) = \dfrac{-\sqrt{2}}{2}$ but my calculator gives me an error when entering $\sin(-45^\circ)$ or $\sin(\dfrac{-\pi}{4})$
Likewise, my current lesson on complex numbers requires that I use the sin and cos functions of $\dfrac{7\pi}{4}$ rather than $\dfrac{-\pi}{4}$
I would appreciate an explanation of when to use a negative reference angle and when to use the positive angle $< 360^\circ$
Some older texts & authors still use degrees as well as reference angles which acute. If the terminal side is in quadrant II the reference angle is $180^{\circ}-\theta$;
terminal side is in quadrant III the reference angle is $180^{\circ}+\theta$;terminal side is in quadrant IV the reference angle is $360^{\circ}-\theta$.
In II the $\sin$ is positive, in III the $\tan$ is positive, & in IV the $cos$ is positive.

Look Here. The basic parts of wolframalpha are free to use. The whole site is inexpensive for students & the is a version for small screen devices.

3. ## Re: When theta must be positive and when negative

I don't know why your calculator can't determine sin(-pi/4)). Mine is ok with it, as well as sin(7 pi/4)

Whether to measure angles as clockwise from the x-axis or counterclockwise depends on the application, and may be influenced by the particulars of the problem you are working. As an example, to calculate the y-coordinate of a point on a wheel that is rotating about the origin you would use y=R sin(theta) where theta is positive if the wheel rotates counterclockwise and theta is negative if it's rotating clockwise. This allows for a nice smooth plot of theta and y versus time. As for complex analysis - you are correct that by convention angles are measured clockwise from the real axis, and of course they are given in radians.

4. ## Re: When theta must be positive and when negative

Really, thanks ChipB.

As for the calculator, it was user error. Much to my chagrin, I keep forgetting the TI-84 requires the use of the (-) key rather than the minus key for negative values.

My Trig and Pre-Calc course is a self-study program. I worry sometimes when I don't come up with the same answers as the text.

As an example:

Find Trigonometric form of $5 - 5i$. I came up with $z = 5\sqrt{2} (\cos\dfrac{-\pi}{4}+i \sin\dfrac{-\pi}{4})$

But the text answer was $z = 5\sqrt{2} (\cos\dfrac{7\pi}{4}+i \sin\dfrac{7\pi}{4})$

From your answer it seems either should be accepted as correct. Is that true? Would there normally be a preference?

5. ## Re: When theta must be positive and when negative

Originally Posted by B9766
Really, thanks ChipB.
As for the calculator, it was user error. Much to my chagrin, I keep forgetting the TI-84 requires the use of the (-) key rather than the minus key for negative values.
My Trig and Pre-Calc course is a self-study program. I worry sometimes when I don't come up with the same answers as the text.
As an example:
Find Trigonometric form of $5 - 5i$. I came up with $z = 5\sqrt{2} (\cos\dfrac{-\pi}{4}+i \sin\dfrac{-\pi}{4})$
But the text answer was $z = 5\sqrt{2} (\cos\dfrac{7\pi}{4}+i \sin\dfrac{7\pi}{4})$
From your answer it seems either should be accepted as correct. Is that true? Would there normally be a preference?
The "angle" associated with a complex is known as its argument. So $\displaystyle \arg (5 - 5\bf{i}) = - \frac{\pi }{4}$
Note that I used the negative value. In resent times that has become the preferred notation. However older texts often used the convention that
$\displaystyle 0\le \arg (z) <2\pi$ that seems to be the case with your textbook.
Just note that $2\pi -\frac{\pi}{4}=\frac{7\pi}{4}$ That means that the two are equivalent.

6. ## Re: When theta must be positive and when negative

Thanks for the confirmation, Plato.


# Are polynomials dense in Gaussian Sobolev space?

Let $\mu$ be standard Gaussian measure on $\mathbb{R}^n$, i.e. $d\mu = (2\pi)^{-n/2} e^{-|x|^2/2} dx$, and define the Gaussian Sobolev space $H^1(\mu)$ to be the completion of $C_c^\infty(\mathbb{R}^n)$ under the inner product $$\langle f,g \rangle_{H^1(\mu)} := \int f g\, d\mu + \int \nabla f \cdot \nabla g\, d\mu.$$

It is easy to see that polynomials are in $H^1(\mu)$. Do they form a dense set?

I am quite sure the answer must be yes, but can't find or construct a proof in general. I do have a proof for $n=1$, which I can post if anyone wants. It may be useful to know that the polynomials are dense in $L^2(\mu)$.

Edit: Here is a proof for $n=1$.

It is sufficient to show that any $f \in C^\infty_c(\mathbb{R})$ can be approximated by polynomials. We know polynomials are dense in $L^2(\mu)$, so choose a sequence of polynomials $q_n \to f'$ in $L^2(\mu)$. Set $p_n(x) = \int_0^x q_n(y)\,dy + f(0)$; $p_n$ is also a polynomial. By construction we have $p_n' \to f'$ in $L^2(\mu)$; it remains to show $p_n \to f$ in $L^2(\mu)$. Now we have \begin{align*} \int_0^\infty |p_n(x) - f(x)|^2 e^{-x^2/2} dx &= \int_0^\infty \left(\int_0^x (q_n(y) - f'(y)) dy \right)^2 e^{-x^2/2} dx \\ &\le \int_0^\infty \int_0^x (q_n(y) - f'(y))^2\,dy \,x e^{-x^2/2} dx \\ &= \int_0^\infty (q_n(x) - f'(x))^2 e^{-x^2/2} dx \to 0 \end{align*} where we used Cauchy-Schwarz in the second line and integration by parts in the third. The $\int_{-\infty}^0$ term can be handled the same with appropriate minus signs.

The problem with $n > 1$ is I don't see how to use the fundamental theorem of calculus in the same way.

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Nate, I once needed this result, so I proved it in Dirichlet forms with polynomial domain (Math. Japonica 37 (1992) 1015-1024). There may be better proofs out there, but you could start with this paper.

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Hi Byron, I'm interested as well. I couldn't not find the paper using MathSciNet. Can I get a copy? –  Jonas Teuwen Nov 2 '11 at 21:34
Thanks Byron! I think my library has it, I'll head there right now. The MathSciNet link is ams.org/mathscinet-getitem?mr=1196376, for reference. –  Nate Eldredge Nov 2 '11 at 21:51
@Jonas I have scanned the paper and added it to my "Publications" page. You want Proposition 1.3. I hope my argument holds up to scrutiny! It is not very profound, just a multi-dimensional version of Nate's argument. –  Byron Schmuland Nov 2 '11 at 22:16
@Byron: Thanks. I've got it. –  Jonas Teuwen Nov 2 '11 at 22:16
Thanks a lot for that! –  t.b. Nov 2 '11 at 22:28

Byron's paper, which he linked in his (accepted) answer, has a proof in a more general setting (where the Gaussian measure can be replaced by any measure with exponentially decaying tails). Here is a specialization of it to the Gaussian case, which I wrote up to include in some lecture notes. I guess I was on the right track, the trick was to differentiate more times.

$\newcommand{\R}{\mathbb{R}}$ For continuous $\psi : \R^k \to \R$, let \begin{equation*} I_i \psi(x_1, \dots, x_k) = \int_0^{x_i} \psi(x_1, \dots, x_{i-1}, y, x_{i+1}, \dots, x_k)dy. \end{equation*} By Fubini's theorem, all operators $I_i, 1 \le i \le k$ commute. If $\psi \in L^2(\mu)$ is continuous, then $I_i \psi$ is also continuous, and $\partial_i I_i \psi = \psi$. Moreover, \begin{align*} \int_{0}^\infty |I_i \psi (x_1, \dots, x_k)|^2 e^{-x_i^2/2} dx_i &= \int_{0}^\infty \big\lvert\int_0^{x_i} \psi(\dots, y,\dots)\,dy\big\rvert^2 e^{-x_i^2/2} dx_i \\ &\le \int_0^\infty \int_0^{x_i} |\psi(\dots, y, \dots)|^2 \,dy x_i e^{-x_i^2/2}\,dx_i && \text{Cauchy--Schwarz} \\ &= \int_0^\infty |\psi(\dots, x_i, \dots)|^2 e^{-x_i^2/2} dx_i \end{align*} where in the last line we integrated by parts. We can make the same argument for the integral from $-\infty$ to $0$, adjusting signs as needed, so we have \begin{equation*} \int_\R |I_i \psi(x)|^2 e^{-x_i^2/2} dx_i \le \int_\R |\psi_i(x)|^2 e^{-x_i^2/2} dx_i. \end{equation*} Integrating out the remaining $x_j$ with respect to $e^{-x_j^2/2}$ shows $$||{I_i \psi}||_{L^2(\mu)}^2 \le ||{\psi}||_{L^2(\mu)}^2,$$ i.e. $I_i$ is a contraction on $L^2(\mu)$.

Now for $\phi \in C_c^\infty(\R^k)$, we can approximate $\partial_1 \dots \partial_k \phi$ in $L^2(\mu)$ norm by polynomials $q_n$. If we let $p_n = I_1 \dots I_k q_n$, then $p_n$ is again a polynomial, and $p_n \to I_1 \dots I_k \partial_1 \dots \partial_k \phi = \phi$ in $L^2(\mu)$. Moreover, $\partial_i p_n = I_1 \dots I_{i-1} I_{i+1} \dots I_k q_n \to I_1 \dots I_{i-1} I_{i+1} \dots I_k \partial_1 \dots \partial_k \phi = \partial_i \phi$ in $L^2(\mu)$ also.

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I think I have a different proof. Let $\gamma$ be the Gaussian measure, that is, $\gamma$ is given by the Radon-Nikodym density, $$\mathrm{d}\gamma(x) = \frac{\mathrm{e}^{-x^2}}{\sqrt{\pi}} \mathrm{d}x.$$ Also, consider the Ornstein-Uhlenbeck operator given as, $$L := -\frac12 \Delta + x \cdot \nabla.$$ We can verify that for $u$ and $v$ in $C_{\mathrm{c}}^\infty(\mathbf R^d)$ we have the symmetricity $$\int_{\mathbf{R}^d} u L v \, \gamma(\mathrm{d}x) = \int_{\mathbf{R}^d} \nabla u \cdot \nabla v \, \gamma(\textrm{d}x).$$ The nice thing about this is that the Hermite polynomials are an orthogonal basis for the Gaussian Hilbert space, that is, $L^2(\gamma)$. These can be define using the Rodrigues' formula, that is, $$H_n(x) = (-1)^n \mathrm{e}^{x^2} \partial_x^n \mathrm{e}^{-x^2}.$$ Furthermore, we have, $$L H_n = n H_n.$$ Also, we have, $$H_n' = 2n H_{n - 1}.$$ Proving that the polynomials form a basis for $L^2(\gamma)$ is not hard, just consider the entire function $$F(z) = \int_{-\infty}^\infty \mathrm e^{zx - x^2} \, \frac{\mathrm{d}x}{\sqrt\pi}.$$ Hence, so are the Hermite polynomials as they are linear combinations of the monomials $(x \mapsto x^n)_n$. Higher-order Hermite polynomials are the canonical tensor extensions. So, given an $f$ in $L^2$ we have that $f$ is given in the form $$f = \lim_N f_N = \lim_N \sum_{n = 0}^N a_n \frac{a_n}{\sqrt{n! 2^n}}.$$ And $f_N$ gives the limit of functions that converges to $f$.

Also, recall the orthogonality (after scaling), $$\int_{\mathbf{R}^d} h_n h_m \, \gamma(\mathrm{d}x) = \delta_{nm}.$$ where $$h_n = \frac{H_n}{\sqrt{n! 2^n}}.$$

We can rewrite the inner product on the Sobolev space as $$\langle u, v \rangle_{H^1} = \langle u, (L + 1) v \rangle_{L^2(\gamma)}.$$ We only have to care about the bilinear form $$\mathcal E(u, v) = \int_{\mathbf{R}^d} u L v \, \gamma(\mathrm{d}x).$$ Or so, picking $u = v = f_N - f$ we have after noting that $$g_N := f - f_N = \sum_{n = N + 1}^\infty a_n \frac{H_n}{\sqrt{n! 2^n}},$$ \begin{align} \mathcal E(u, v) &= \int_{\mathbf{R}^d} f_N L f_N \, \gamma(\mathrm{d}x)\\ &\quad + \int_{\mathbf{R}^d} f L f \, \gamma(\mathrm{d}x)\\ &\quad - 2\int_{\mathbf{R}^d} f_N L f \, \gamma(\mathrm{d}x)\\ &= \sum_{n = 0}^\infty n \frac{|a_n|^2}{n! 2^n} - \sum_{n = 0}^N n \frac{|a_n|^2}{n! 2^n}\\ &= \sum_{n = N + 1}^\infty n \frac{|a_n|^2}{n! 2^n}. \end{align}. And as $\sum |a_n|^2$ converges due to the $L^2$ density, so should this.

I hope I did not make any mistakes, just occurred to me while I was biking...

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Kronecker Product (G Dataflow)

Calculates the Kronecker product of two input matrices.

matrix A

The first input matrix.

This input accepts a 2D array of double-precision, floating point numbers or 2D array of complex double-precision, floating point numbers.

Default: Empty array

matrix B

The second input matrix.

This input accepts a 2D array of double-precision, floating point numbers or 2D array of complex double-precision, floating point numbers.

Default: Empty array

error in

Error conditions that occur before this node runs. The node responds to this input according to standard error behavior.

Default: No error

kronecker product

Matrix containing the Kronecker product of the first and second input matrices.

The number of rows in kronecker product is the product of the number of rows in the first and second input matrices. The number of columns in kronecker product is the product of the number of columns in the first and second input matrices.

error out

Error information. The node produces this output according to standard error behavior.

Algorithm for Calculating the Kronecker Product

If A is an n-by-m matrix and B is a k-by-l matrix, the Kronecker product of A and B, C = AB, results in a matrix C with dimensions nk-by-ml. This node calculates the Kronecker product using the following equation.

$C={\left[\begin{array}{cccc}{a}_{11}B& {a}_{12}B& \dots & {a}_{1m}B\\ {a}_{21}B& {a}_{22}B& \dots & {a}_{2m}B\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{n1}B& {a}_{n2}B& \dots & {a}_{nm}B\end{array}\right]}_{nk×ml}$

For example, if

$\begin{array}{cc}A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]& B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]\end{array}$

then

$\begin{array}{ccc}{a}_{11}B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]& {a}_{12}B=\left[\begin{array}{cc}10& 12\\ 14& 16\end{array}\right]& C=\left[\begin{array}{cc}{a}_{11}B& {a}_{12}B\\ {a}_{21}B& {a}_{22}B\end{array}\right]=\left[\begin{array}{cccc}5& 6& 10& 12\\ 7& 8& 14& 16\\ 15& 18& 20& 24\\ 21& 24& 28& 32\end{array}\right]\end{array}$

Where This Node Can Run:

Desktop OS: Windows

FPGA: Not supported


# What is the metric tensor on the n-sphere (hypersphere)?

I am considering the unit sphere (but an extension to one of radius $r$ would be appreciated) centered at the origin. Any coordinate system will do, though the standard angular one (with 1 radial and $n-1$ angular coordinates) would be preferable.

I know that on the 2-sphere we have $ds^2 = d\theta^2+\sin^2(\theta)d\phi^2$ (in spherical coordinates) but I'm not sure how this generalizes to $n$ dimensions.

Added note: If anything can be discovered only about the determinant of the tensor (when presented in matrix form), that would also be quite helpful.

I will define the metric of $$S^{n-1}$$ via pullback of the Euclidean metric on $${\mathbb{R}}^{n}$$.

To start with we take $$n$$-dimension Cartesian co-ordinates: $$(x_1,x_2......x_n)$$. The metric here is $$g_{ij }= \delta_{ij}$$, where $$δ$$ is the Kronecker delta.

We specify the surface patches of $$S^{n-1}$$ by the parametrization $$f$$: $$x_1=r{\cos{\phi_1}},$$

$$x_p=r{\cos{\phi_p}}{\Pi_{m=1}^{p-1}}{\sin{\phi_{m}}},$$

$$x_n=r{\prod_{m=1}^{n-1}}{\sin{\phi_{m}}},$$

Where $$r$$ is the radius of the hypersphere and the angles have the usual range.

We see that the pullback of the Euclidean metric $$g'_{ab} = (f^*g)_{ab}$$ is the metric tensor of the hypersphere. Its components are:

$$g'_{ab} = g_{ij} {\frac{\partial{x_i}}{\partial{\phi_a}}} {\frac{\partial{x_j}}{\partial{\phi_b}}} = {\frac{\partial{x_i}}{\partial{\phi_a}}}{\frac{\partial{x_i}}{\partial{\phi_b}}}$$

We get $$2$$ cases here:

i) $$a>b$$ or $$b>a$$, For these components one obtains a series of terms with alternating signs which vanishes, $$g'_{ab}=0$$ and thus all off-diagonal components of the tensor vanish.

ii) $$a=b$$,

$$g'_{11}=1$$

$$g'_{aa} ={r^2} \prod_{m=1}^{a-1} \sin^2{\phi_{m}}$$
where $$2\leq a\leq {n-1}$$

The determinant is very straightforward to calculate:

$$\det{(g'_{ab})} = {r^2} \prod_{m=1}^{n-1} g'_{mm}$$

Finally, we can write the metric of the hypersphere as:

$$g' = {r^2} \, d\phi_{1}\otimes d\phi_{1} + {r^2} \sum_{a=2}^{n-1} \left( \prod_{m=1}^{a-1} \sin^2{\phi_{m}} \right) d\phi_{a} \otimes d\phi_{a}$$

• Great, thanks for the step-by-step explanation. Although what you've written is fairly clear, you can make it clearer by using the TeX environment, with the $ delimiters, like instead of g'rϕk, write $g'_{r\phi_k}$, which comes out as$g'_{r\phi_k}$. May 27, 2013 at 11:39 • This is a very good answer and very useful too, as it is not that easy to find this piece of information on the net. Proper formatting would make it great. Oct 20, 2014 at 9:12 • @GiuseppeNegro Done. Thanks for the feedback. Nov 10, 2014 at 14:04 • It seems$g'_{11}=r^2$, no? Nov 13, 2018 at 14:17 • Can somebody explain more thoroughly why the off diagonal terms vanish? Apr 16, 2019 at 11:15$\newcommand{\Reals}{\mathbf{R}}$For posterity: Fix$r > 0$, and let$S^{n}(r)$denote the sphere of radius$r$centered at the origin in$\Reals^{n+1}$. Stereographic projection from the north pole$N = (0, \dots, 0, 1)$on the unit sphere$S^{n} = S^{n}(1)$defines a diffeomorphism$\Pi_{N}:S^{n} \setminus \{N\} \to \Reals^{n}given in Cartesian coordinates by \begin{align*} \Pi_{N}(x_{1}, \dots, x_{n}, x_{n+1}) &= \frac{1}{1 - x_{n+1}}(x_{1}, \dots, x_{n}), \\ \Pi_{N}^{-1}(t_{1}, \dots, t_{n}) &= \frac{(2t_{1}, \dots, 2t_{n}, \|t\|^{2} - 1)}{\|t\|^{2} + 1}. \end{align*} In these coordinates, the induced (round) metric on the unit sphere is well-known (and easily checked) to be conformally-Euclidean: $$g(t) = \frac{4 (dt_{1}^{2} + \cdots + dt_{n}^{2})}{(\|t\|^{2} + 1)^{2}}.$$ Stereographic projection from the north pole(0, \dots, 0, r)$of$S^{n}(r)$is given by the scaled mapping$x \mapsto t = r\Pi_{N}(x/r)$, whose inverse is$t \mapsto x = r\Pi_{N}^{-1}(t/r)\$, i.e., \begin{align*} r\Pi_{N}(x_{1}/r, \dots, x_{n}/r, x_{n+1}/r) &= \frac{1}{r - x_{n+1}}(x_{1}, \dots, x_{n}), \\ r\Pi_{N}^{-1}(t_{1}/r, \dots, t_{n}/r) &= \frac{\bigl(2t_{1}, \dots, 2t_{n}, r(\|t/r\|^{2} - 1)\bigr)}{\|t/r\|^{2} + 1}. \end{align*} The induced metric in these coordinates is consequently $$r^{2} g(t/r) = \frac{4 (dt_{1}^{2} + \cdots + dt_{n}^{2})}{(\|t/r\|^{2} + 1)^{2}} = \frac{4r^{4} (dt_{1}^{2} + \cdots + dt_{n}^{2})}{(\|t\|^{2} + r^{2})^{2}}.$$


# A problem in understanding the Intermediate Value Theorem

So, IVT essentially says if a function $$f$$ is continuous over an interval $$[a,b]$$, then the function $$f$$ will take up all the values between $$f(a)$$ and $$f(b)$$ at least once at some point in the interval $$[a,b]$$.

Now, suppose $$f$$ is continuous between $$[3,7]$$ and $$f(3) = 4$$ and $$f(7) = 25$$

According to IVT, $$f(3) = 4 < m < 25 = f(7)$$, i.e., the function takes up all the values between $$f(3)$$ and $$f(7)$$ at least once in the interval. But, according to the below picture, clearly, the function within that interval takes up values different that do not satisfy the above inequality. Please help me fill the gap in my understanding of IVT here.

Please do correct me if I'm mistaken.

This doubt arose from the below KhanAcademy question

• IVT doesn't tell you that ALL values must be between $f(a)$ and $f(b)$ -- it tells you that every value between $f(a)$ and $f(b)$ must be attained. – Nick Peterson Feb 25 at 6:59
• The IVT says that $f$ reaches all values between $f(3)$ and $f(7)$. It doesn't say it can't reach other values as well. – Mathematician 42 Feb 25 at 6:59
• @SathvikR. If it helps: you seem to be interpreting it as: $x\in(a,b)$ implies $f(x)\in (f(a), f(b))$ (assuming $f(a)<f(b)$, etc). But the actual implication is: for all $y\in (f(a), f(b))$, there exists $x\in(a, b)$ so that $f(x)=y$. – Nick Peterson Feb 25 at 7:06
• IVT only says that all values between $4$ and $25$ will be obtained. And in that picture they clearly all are. The IVT says NOTHING at all about values outside $[4,25]$. It doesn't say they are obtained. It doesn't say they aren't obtained. – fleablood Feb 25 at 7:21
• Okay.... it says that every possible output value between $f(3) = 4$ and $f(7) = 25$ will be obtained somewhere in the interval between $3$ and $7$. But it doesn't say that values between $4$ and $25$ are the only values obtained. If I told you that every dog in the Golden Gate Dog shelter has an owner who lives in San Francisco, that does not mean everyone who lives in San Francisco owns a dog in the Golden Gate Dog shelter.... to be continued.... – fleablood Feb 25 at 7:29

The intermediate value theorem tells you that, in that context, for each $$y\in[4,25]$$, there is some $$x\in[3,4]$$ such that $$f(x)=y$$. It does not tell you that if $$x\in[3,4]$$, then $$f(x)\in[4,25]$$. So, there is no contradiction if $$f\bigl([3,4]\bigr)$$ contains values outside $$[4,25]$$.


# Thread: Help with this word problem!

1. ## Help with this word problem!

The value of a new car of a particular make, A, decreases by 22.5% during the first year. In the second year, its value decreases by 12.5% of its value at the beginning of the year. The value of a new car of another make, B, decreases by 25% during the first year. In the second year, its value decreases by 15% of its value at the beginning of the year. A new car of make A costs $48,000 and a new car of make B costs$42,000.

(a) What is the difference between the values of the two cars at the end of the firts year? Express this differnce as a percentage of the value of the car of make A at the end of the first year.

(b) Calculate the difference between the values of the two cars at the end of the second year.

Thankyou!

2. Originally Posted by laser2302

The value of a new car of a particular make, A, decreases by 22.5% during the first year. In the second year, its value decreases by 12.5% of its value at the beginning of the year. The value of a new car of another make, B, decreases by 25% during the first year. In the second year, its value decreases by 15% of its value at the beginning of the year. A new car of make A costs $48,000 and a new car of make B costs$42,000.

(a) What is the difference between the values of the two cars at the end of the firts year? Express this differnce as a percentage of the value of the car of make A at the end of the first year.

(b) Calculate the difference between the values of the two cars at the end of the second year.

Thankyou!
If something decreases by 22.5% means it is worth 77.5% perfect of the original price.
In the beggining the car was worth $x$ after the first year is it $.775x$ after the second year it decreases by 12.5% meaning its value is 87.5% of the new price thus,
$.875(.775x)=.678125x$
Similarily the second car after two years is,
$.85(.75x)=.6375x$ and $.75x$ after one year.

The first problem asks for the difference after the first year.
The first car is worth $.775\times 48,000=37,200$. The second car is worth $.75\times 42,000=31,500$ The difference is,
$37,200-31,500=5,700$

After two years, you have the new values,
$.875\times 37,200=32,250$
$.85\times 31,500=26,775$
Their difference is,
$32,250-26,775= 5475$

3. Originally Posted by ThePerfectHacker
The first problem asks for the difference after the first year.
The first car is worth $.775\times 48,000=37,200$. The second car is worth $.75\times 42,000=31,500$ The difference is,
$37,200-31,500=5,700$
The question asks to write this out in terms of the percentage of the price of car A, so you divide 5,700 by the price of car A

$\frac{5700}{37200}=0.153225806\approx\boxed{15
\text
{percent}}$

4. Hello, laser2302!

Are you sure you can't work this out yourself?

[quote]The value of a new car of a particular make, $A$, decreases by 22.5% during the first year.
In the second year, its value decreases by 12.5% of its value at the beginning of the year.

The value of a new car of another make, $B$, decreases by 25% during the first year.
In the second year, its value decreases by 15% of its value at the beginning of the year.

A new car of make $A$ costs $48,000 and a new car of make $B$ costs$42,000.

(a) What is the difference between the values of the two cars at the end of the first year?
Express this differnce as a percentage of the value of the car of make A at the end of the first year.

(b) Calculate the difference between the values of the two cars at the end of the second year.

During year 1, make $A$ loses $22.5\%\text{ of }\48,000 \:=\:0.225 \times 48,000 \:=\:\10,800$
. . . At the end of year 1, it is worth only: $\48,000 - 10,800 \:= \:\37,200$

During the year 2 $A$ loses $12.5\%\text{ of }\37,200 \:=\:0.125 \times 37,200 \:= \:\4,650$
. . . At the end of year 2, it is worth only: $\37,200 - 4,650\:=\:\32,550$

During year 1, make $B$ loses $25\%\text{ of }\42,000\:=\:0.25 \times 42,000 \:=\:\10,500$
. . . At the end of year 1, it is worth only: $\42,000 - 10,500\:=\:\31,500$

During year 2, $B$ loses $15\%\text{ of }\31,500 \:=\:0.15 \times 31,500\:=\:\4,725$
. . . At the end of year 2, it is worth only: $\31,500 - 4,725\:=\:\26,775$

Now that we've worked all the numbers, we can answer the questions.

(a) The difference in value at the end of year 1 is:
. . . . . $\37,200 - 31,500\:=\:\5,700$ . . .
Boy, that was hard!

The percentage they asked for is: $\frac{5.700}{37,200}\:=\:0.153225806\:\approx\:15. 3\%$

(b) The difference in value at the end of year 2 is:
. . . . . $\32,550 - 26,775\:=\:\5,775$ . . .
I need a nap!

5. Originally Posted by Soroban
Hello, laser2302!

Are you sure you can't work this out yourself?
You know you really shouldn't be so critical, after all, people come here for reassurance and help (or, unfortunately, so they don't have to do the work) and should be treated well.


1. ## Greatest Common Factor

Is there an equation for the greatest common factor of two numbers?

2. Originally Posted by Quick
Is there an equation for the greatest common factor of two numbers?
Equation, not that I know of. Process, yes.

Consider the GCF of 60 and 630.

The prime factorization of 60 is $\displaystyle 2^2 \cdot 3 \cdot 5$.
The prime factorization of 630 is $\displaystyle 2 \cdot 3^2 \cdot 5 \cdot 7$.

The GCF (also known as the Greatest Common Divisor, GCD) will be the number that has a prime factorization that contains the same prime factors as the combination of the lists. In other words,
There is a factor of 2 common to each,
there is a factor of 3 common to each,
there is a factor of 5 common to each.

Thus GCF(60, 630) = 2*3*5 = 30.

If we were talking about GCF(60, 1260) then ($\displaystyle 1260 = 2^2 \cdot 3^2 \cdot 5 \cdot 7$):
There are two factors of 2 common to each,
there is a factor of 3 common to each,
there is a factor of 5 common to each.

Thus GCF(60, 1260) = $\displaystyle 2^2 \cdot 3 \cdot 5$ = 60.

-Dan

PS Now that I think of it, there is a formula, but it isn't anything direct:
Given two numbers x and y, we know that GCF(x, y) = (x*y)/LCM(x, y), where LCM(x, y) is the "Least Common Multiple" of x and y. There is no direct formula I know of to find the LCM either.

3. Originally Posted by topsquark
Equation, not that I know of. Process, yes.

Consider the GCF of 60 and 630.

The prime factorization of 60 is $\displaystyle 2^2 \cdot 3 \cdot 5$.
The prime factorization of 630 is $\displaystyle 2 \cdot 3^2 \cdot 5 \cdot 7$.

The GCF (also known as the Greatest Common Divisor, GCD) will be the number that has a prime factorization that contains the same prime factors as the combination of the lists. In other words,
There is a factor of 2 common to each,
there is a factor of 3 common to each,
there is a factor of 5 common to each.

Thus GCF(60, 630) = 2*3*5 = 30.

If we were talking about GCF(60, 1260) then ($\displaystyle 1260 = 2^2 \cdot 3^2 \cdot 5 \cdot 7$):
There are two factors of 2 common to each,
there is a factor of 3 common to each,
there is a factor of 5 common to each.

Thus GCF(60, 1260) = $\displaystyle 2^2 \cdot 3 \cdot 5$ = 60.

-Dan

PS Now that I think of it, there is a formula, but it isn't anything direct:
Given two numbers x and y, we know that GCF(x, y) = (x*y)/LCM(x, y), where LCM(x, y) is the "Least Common Multiple" of x and y. There is no direct formula I know of to find the LCM either.
In case you were wondering there is a simple process to find the LCM of two numbers as well. Let's take 60 and 630 again.

The prime factorization of 60 is $\displaystyle 2^2 \cdot 3 \cdot 5$.
The prime factorization of 630 is $\displaystyle 2 \cdot 3^2 \cdot 5 \cdot 7$.

The Least Common Multiple will have a prime factorization that is the smallest list of prime factors that completely contains both lists. In other words:
There are two factors of 2,
there are two factors of 3,
There is one factor of 5,
There is one factor of 7.

Thus LCM(60, 630) = $\displaystyle 2^2 \cdot 3^2 \cdot 5 \cdot 7$ = 1260.

(And we can check the equation I gave in the Post Script of the last post:
GCF(60, 630) = (60*630)/LCM(60, 630) = 37800/1260 = 30 as we found in the last post.)

-Dan

4. Originally Posted by topsquark
...
The Least Common Multiple will have a prime factorization that is the smallest list of prime factors that completely contains both lists. ...
Hi,

I taught my pupils an algorithm to calculate the LCM in a tabel: This method works too when you have more than two numbers (for instance when you calculate the sum of fractions). You only have to know the first 20 prime-factors :

5. This seems to be an excellent question for Quicklopedia, my method doth no rely on factorization. Rather on an algorithm (with amazing complexity i.e. really fast) called Euclidean algorithm. I can show it to thee.

6. Originally Posted by ThePerfectHacker
This seems to be an excellent question for Quicklopedia, my method doth no rely on factorization. Rather on an algorithm (with amazing complexity i.e. really fast) called Euclidean algorithm. I can show it to thee.
Actually, for once I found wikipedia gave a good definition (actually, it gave a poor definition for it, but an excellent summary in the "Greatest Common Denominator" section)


# How to find a inverse of a multivariable function?

I have a function $f:\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ defined as:

$$f(x,y) = (3x-y, x-5y)$$

I proved that it's a bijection, now I have to find the inverse function $f^{-1}$.

Because $f$ is a bijection, it has a inverse and this is true:

$$(f^{-1}\circ f)(x,y) = (x,y)$$

$$f^{-1}(3x-y,x-5) = (x,y)$$

I don't know where to go from here. In a one variable function I would do a substitution of the argument of $f^{-1}$ with a variable and express x with that variable, and then just switch places.

I tried to do a substitution like this:

$$3x-y = a$$ $$x-5y = b$$

And then express $x$ and $y$ by $a$ and $b$ , and get this:

$$f^{-1}(x,y) = (\frac{15x-3y}{42}, \frac{x-3y}{14})$$

But I'm not sure if I'm allowed to swap $x$ for $a$, and $y$ for $b$.

Any hint is highly appreciated.

• Note that $f$ is linear, so you could write it as a matrix and then calculate the inverse. – PPR Nov 27 '16 at 19:27
• Recall that when finding the inverse of a bijective function of a singular variable of the form $y=f(x)$ you also "swap the variables." – John Wayland Bales Nov 27 '16 at 19:30
• @JohnWaylandBales Yes, should I do it here as well? And, if so, what sould I swap? – nikol_kok Nov 27 '16 at 19:32
• Your work is absolutely correct as written. You could reduce the first fraction of your final result by a factor of $3$ so that it also had a denominator of $14$. But you swap $x\leftrightarrow a$ and $y\leftrightarrow b$. – John Wayland Bales Nov 27 '16 at 19:38
• @JohnWaylandBales So this is one reliable way of doing this? I mean, I can always swap those two when I'm in a situation like this? – nikol_kok Nov 27 '16 at 19:41

You have a linear function here, given by the matrix $$\begin{bmatrix}3&-1\\1&-5 \end{bmatrix}$$ Can you invert the above?

• @qbert Yes, I know how to find the inverse of that matrix, but I don't know what to do with it :( – nikol_kok Nov 27 '16 at 19:31
• Write it as your answer. Your linear map is given by the above matrix, the inverse map will be given by the inverse of the above matrix ( with respect to the standard basis) – qbert Nov 27 '16 at 19:41
• If you would like to write it in the same form you can apply the inverse matrix to the vector (x,y) – qbert Nov 27 '16 at 19:42
• Don't get me wrong, but I'm not sure what you are saying here. Can you point me to some theory or an explanation of that method? – nikol_kok Nov 27 '16 at 19:50
• @nikol_kok If I understood that correctly, you actually get $f^{-1}(a, b) = (\frac{5a - b}{14}, \frac{a - 3b}{14})$, and you're wondering whether it's OK to just swap $a$ and $b$ for $x$ and $y$? Yes, that works just fine. – Arthur Nov 27 '16 at 19:55

You can split this into two separate functions $u, v:\Bbb R^2\to \Bbb R$ the following way: $$u(x, y) = 3x-y\\ v(x, y) = x-5y$$ and we have $f(x, y) = (u(x, y), v(x, y))$. What we want is $x$ and $y$ expressed in terms of $u$ and $v$, i.e. solve the above set of equations for $x$ and $y$, so that you get two functions $x(u, v), y(u, v)$. Then $f^{-1}(u, v) = (x(u,v), y(u,v))$.

• What set of equations am I supposed to solve? – nikol_kok Nov 27 '16 at 19:47
• @nikol_kok You should solve the equations $$u= 3x-y\\ v= x-5y$$ for $x$ and $y$. This is exactly corresponding to the fact that in order to find the inverse of, say, $g(x) = 5x + 3$, you solve $g = 5x + 3$ for $x$, only in higher dimension. – Arthur Nov 27 '16 at 19:49
• In my question I tried a solution like this. Can you check if it's the same/similar? – nikol_kok Nov 27 '16 at 19:52


# Different solutions of trigonometric equations

Please take a look at this trigonometric equation,
$\cos9x\cos7x = \cos5x\cos3x$

To solve this equation, we can proceed as,

$2\cos9x\cos7x = 2\cos5x\cos3x$
or, $\cos(9x+7x)+\cos(9x-7x) = \cos(5x+3x)+\cos(5x-3x)$
or, $\cos16x+\cos2x = \cos8x+\cos2x$
or, $\cos16x = \cos8x$

From the above situation we can proceed in two ways,

First Way
$\cos16x = \cos8x$
or, $\cos(2\times8x) = \cos8x$
or, $2\cos^28x -1 = \cos8x$
or, $2\cos^28x-\cos8x -1 = 0$
or, $2\cos^28x-2\cos8x+\cos8x -1 = 0$
or, $2\cos8x(\cos8x-1)+(\cos8x -1) = 0$
or, $2\cos8x(\cos8x-1)+(\cos8x -1) = 0$
or, $(\cos8x -1)(2\cos8x+1) = 0$

Either or both of the above factors are zero.

Taking the first one,
$(\cos8x -1) = 0$
or, $\cos8x = 1$
or, $8x = 2n\pi$, where $n$ is an integer, +ve or -ve.
or, $x = {n\pi \over4}$

Taking the second one,
$(2\cos8x+1) = 0$
or, $2\cos8x = -1$
or, $\cos8x = -\frac1 2$
or, $8x = 2n\pi \pm \frac {2\pi} 3$, where $n$ is an integer, +ve or -ve.
or, $x = \frac{n\pi} 4 \pm \frac {\pi} {12}$

Second Way
$\cos16x = \cos8x$
or, $2\sin{8x-16x\over2}\sin{8x+16x\over2} = 0$
or, $2\sin(-4x)\sin{12x} = 0$
or, $-2\sin(4x)\sin{12x} = 0$

Again, either or both of the above factors are zero.

Taking the first one,
$\sin4x = 0$
or, $4x=n\pi$
or, $x=\frac{n\pi}4$

Taking the second one,
$\sin12x = 0$
or, $12x=n\pi$
or, $x=\frac{n\pi}{12}$

Now, as you must have noticed, we are getting two different sets of solutions, $x = \left\{{n\pi \over4}, \frac{n\pi} 4 \pm \frac {\pi} {12}\right\}$ and $x = \left\{{n\pi \over4}, \frac{n\pi}{12}\right\}$. The member $x=\frac{n\pi}4$ is common to both of the sets. Moreover, all these solutions satisfy the equation under consideration.

Could anybody please tell me why is this happening. In addition to the specific answer, some general insight will be most welcome. We have got a number of similar problems in hand. So, unless we can develop some acumen, life may become difficult.

-

If $x\in\{\frac{n\pi}4,\frac{n\pi}4\pm\frac\pi{12}\mid n\in\mathbb Z\}$, then $12x=3n\pi$ or $12x=3n\pi+\pi$ or $12x=3n\pi-\pi$ for some $n\in\mathbb Z$, hence $12x=m\pi$ for some $m\in\mathbb Z$ (namely $m=3n$ or $m=3n+1$ or $m=3n-1$. On the other hand, every $m\in \mathbb Z$ can be written either as $m=3n$ or $m=3n+1$ or $m=3n-1$, depending on the remainder modulo $3$. Therefore $x\in\{\frac{n\pi}4,\frac{n\pi}4\pm\frac\pi{12}\mid n\in\mathbb Z\}\iff \frac{12}\pi x\in\mathbb Z$.
Moreover, if $x=\frac{n\pi}4$ with $n\in \mathbb Z$, then $x=\frac{m\pi}{12}$ with $m:=3n\in\mathbb Z$, hence $\{\frac {n\pi}4\mid n\in\mathbb Z\}\subseteq \{\frac {n\pi}{12}\mid n\in\mathbb Z\}$, which ultimately makes $$\left\{\frac{n\pi}4,\frac{n\pi}4\pm\frac\pi{12}\mid n\in\mathbb Z\right\} = \left\{\frac{n\pi}4,\frac{n\pi}{12}\mid n\in\mathbb Z\right\}=\left\{\frac{n\pi}{12}\mid n\in\mathbb Z\right\}.$$
They aren't actually different sets of numbers. They are both all the numbers of the form $n\pi/12$.
Could you please elaborate a bit, are you suggesting that ${n \pi \over 4} \pm {\pi \over 12}$ and ${n \pi \over 12}$ are the same numbers? –  Masroor Feb 14 '13 at 6:23
$\left\{ \frac{n\pi}{4}, \frac{n\pi}{4} \pm \frac{\pi}{12} \right\} = \left\{ \frac{3n\pi}{12}, \frac{(3n \pm 1)\pi}{12}\right\} = \left\{ \frac{(3n-1)\pi}{12}, \frac{3n\pi}{12}, \frac{(3n+1)\pi}{12} \right\} = \left\{\frac{n\pi}{12}\right\}$, since any integer can be written as one of $3n-1$, $3n$, or $3n+1$. Also, $\left\{\frac{n\pi}{4}, \frac{n\pi}{12}\right\} = \left\{\frac{3n\pi}{12}, \frac{n\pi}{12} \right\} = \left\{\frac{n\pi}{12} \right\}$, because "all integer multiples of $\pi/12$" includes "all integer multiples of $3\pi/12$". –  Blue Feb 14 '13 at 8:12


Perfect squares are numbers that are equal to a number times itself. To simplify this sort of radical, we need to factor the argument (that is, factor whatever is inside the radical symbol) and "take out" one copy of anything that is a square. Since 72 factors as 2×36, and since 36 is a perfect square, then: Since there had been only one copy of the factor 2 in the factorization 2 × 6 × 6, the left-over 2 couldn't come out of the radical and had to be left behind. In other words, we can use the fact that radicals can be manipulated similarly to powers: There are various ways I can approach this simplification. One rule is that you can't leave a square root in the denominator of a fraction. But my steps above show how you can switch back and forth between the different formats (multiplication inside one radical, versus multiplication of two radicals) to help in the simplification process. Solution : √(5/16) = √5 / √16 √(5/16) = √5 / √(4 ⋅ 4) Index of the given radical is 2. Quotient Rule . For the purpose of the examples below, we are assuming that variables in radicals are non-negative, and denominators are nonzero. Simplifying Radicals Coloring Activity. Required fields are marked * Comment. Then simplify the result. A radical expression is composed of three parts: a radical symbol, a radicand, and an index. Determine the index of the radical. This calculator simplifies ANY radical expressions. In this case, the index is two because it is a square root, which … ... Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify. Indeed, we can give a counter example: $$\sqrt{(-3)^2} = \sqrt(9) = 3$$. I used regular formatting for my hand-in answer. Simplifying Radicals Activity. Then, there are negative powers than can be transformed. We can raise numbers to powers other than just 2; we can cube things (being raising things to the third power, or "to the power 3"), raise them to the fourth power (or "to the power 4"), raise them to the 100th power, and so forth. This theorem allows us to use our method of simplifying radicals. How do I do so? (Much like a fungus or a bad house guest.) We are going to be simplifying radicals shortly so we should next define simplified radical form. type (2/ (r3 - 1) + 3/ (r3-2) + 15/ (3-r3)) (1/ (5+r3)). Step 2. In simplifying a radical, try to find the largest square factor of the radicand. When doing your work, use whatever notation works well for you. In reality, what happens is that $$\sqrt{x^2} = |x|$$. For example. Short answer: Yes. Sign up to follow my blog and then send me an email or leave a comment below and I’ll send you the notes or coloring activity for free! root(24)=root(4*6)=root(4)*root(6)=2root(6) 2. simplifying square roots calculator ; t1-83 instructions for algebra ; TI 89 polar math ; simplifying multiplication expressions containing square roots using the ladder method ; integers worksheets free ; free standard grade english past paper questions and answers All that you have to do is simplify the radical like normal and, at the end, multiply the coefficient by any numbers that 'got out' of the square root. All exponents in the radicand must be less than the index. In the first case, we're simplifying to find the one defined value for an expression. "Roots" (or "radicals") are the "opposite" operation of applying exponents; we can "undo" a power with a radical, and we can "undo" a radical with a power. Finance. Step 1 : Decompose the number inside the radical into prime factors. How do we know? Just to have a complete discussion about radicals, we need to define radicals in general, using the following definition: With this definition, we have the following rules: Rule 1.1:    $$\large \displaystyle \sqrt[n]{x^n} = x$$, when $$n$$ is odd. We know that The corresponding of Product Property of Roots says that . Well, simply by using rule 6 of exponents and the definition of radical as a power. Quotient Rule . To simplify a square root: make the number inside the square root as small as possible (but still a whole number): Example: √12 is simpler as 2√3. Then: katex.render("\\sqrt{144\\,} = \\mathbf{\\color{purple}{ 12 }}", typed01);12. Simplify each of the following. In this particular case, the square roots simplify "completely" (that is, down to whole numbers): Simplify: I have three copies of the radical, plus another two copies, giving me— Wait a minute! Another rule is that you can't leave a number under a square root if it has a factor that's a perfect square. What about more difficult radicals? We will start with perhaps the simplest of all examples and then gradually move on to more complicated examples . where a ≥ 0, b > 0 "The square root of a quotient is equal to the quotient of the square roots of the numerator and denominator." Algebraic expressions containing radicals are very common, and it is important to know how to correctly handle them. No radicals appear in the denominator. Step 2 : If you have square root (√), you have to take one term out of the square root for every two same terms multiplied inside the radical. But when we are just simplifying the expression katex.render("\\sqrt{4\\,}", rad007A);, the ONLY answer is "2"; this positive result is called the "principal" root. Product Property of n th Roots. For example . Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify. Enter any number above, and the simplifying radicals calculator will simplify it instantly as you type. ANSWER: This fraction will be in simplified form when the radical is removed from the denominator. So 117 doesn't jump out at me as some type of a perfect square. 1. Question is, do the same rules apply to other radicals (that are not the square root)? We factor, find things that are squares (or, which is the same thing, find factors that occur in pairs), and then we pull out one copy of whatever was squared (or of whatever we'd found a pair of). Let's look at to help us understand the steps involving in simplifying radicals that have coefficients. Some radicals have exact values. It's a little similar to how you would estimate square roots without a calculator. When doing this, it can be helpful to use the fact that we can switch between the multiplication of roots and the root of a multiplication. Subtract the similar radicals, and subtract also the numbers without radical symbols. root(24)=root(4*6)=root(4)*root(6)=2root(6) 2. We use the fact that the product of two radicals is the same as the radical of the product, and vice versa. Physics. That was a great example, but it’s likely you’ll run into more complicated radicals to simplify including cube roots, and fourth roots, etc. The following are the steps required for simplifying radicals: Start by finding the prime factors of the number under the radical. The index is as small as possible. Concretely, we can take the $$y^{-2}$$ in the denominator to the numerator as $$y^2$$. In the same way, we can take the cube root of a number, the fourth root, the 100th root, and so forth. Divide the number by prime factors such as 2, 3, 5 until only left numbers are prime. a square (second) root is written as: katex.render("\\sqrt{\\color{white}{..}\\,}", rad17A); a cube (third) root is written as: katex.render("\\sqrt[{\\scriptstyle 3}]{\\color{white}{..}\\,}", rad16); a fourth root is written as: katex.render("\\sqrt[{\\scriptstyle 4}]{\\color{white}{..}\\,}", rad18); a fifth root is written as: katex.render("\\sqrt[{\\scriptstyle 5}]{\\color{white}{..}\\,}", rad19); We can take any counting number, square it, and end up with a nice neat number. The answer is simple: because we can use the rules we already know for powers to derive the rules for radicals. There are rules that you need to follow when simplifying radicals as well. Indeed, we deal with radicals all the time, especially with $$\sqrt x$$. Simplify the following radicals. I'm ready to evaluate the square root: Yes, I used "times" in my work above. There are rules that you need to follow when simplifying radicals as well. Here is the rule: when a and b are not negative. get rid of parentheses (). Your email address will not be published. The radical sign is the symbol . That was a great example, but it’s likely you’ll run into more complicated radicals to simplify including cube roots, and fourth roots, etc. One rule that applies to radicals is. If you notice a way to factor out a perfect square, it can save you time and effort. Radicals ( or roots ) are the opposite of exponents. How to simplify fraction inside of root? Learn How to Simplify Square Roots. 0. You probably already knew that 122 = 144, so obviously the square root of 144 must be 12. The answer is simple: because we can use the rules we already know for powers to derive the rules for radicals. All that you have to do is simplify the radical like normal and, at the end, multiply the coefficient by any numbers that 'got out' of the square root. Some techniques used are: find the square root of the numerator and denominator separately, reduce the fraction and change to improper fraction. Did you just start learning about radicals (square roots) but you’re struggling with operations? "The square root of a product is equal to the product of the square roots of each factor." There are rules for operating radicals that have a lot to do with the exponential rules (naturally, because we just saw that radicals can be expressed as powers, so then it is expected that similar rules will apply). Concretely, we can take the $$y^{-2}$$ in the denominator to the numerator as $$y^2$$. Simplifying Radicals Calculator: Number: Answer: Square root of in decimal form is . You could put a "times" symbol between the two radicals, but this isn't standard. Simplifying radicals is the process of manipulating a radical expression into a simpler or alternate form. We created a special, thorough section on simplifying radicals in our 30-page digital workbook — the KEY to understanding square root operations that often isn’t explained. Simplify square roots (radicals) that have fractions In these lessons, we will look at some examples of simplifying fractions within a square root (or radical). Components of a Radical Expression . Degrees of Freedom Calculator Paired Samples, Degrees of Freedom Calculator Two Samples. After taking the terms out from radical sign, we have to simplify the fraction. Chemical Reactions Chemical Properties. Perfect Cubes 8 = 2 x 2 x 2 27 = 3 x 3 x 3 64 = 4 x 4 x 4 125 = 5 x 5 x 5. Let's see if we can simplify 5 times the square root of 117. Rule 1:    $$\large \displaystyle \sqrt{x^2} = |x|$$, Rule 2:    $$\large\displaystyle \sqrt{xy} = \sqrt{x} \sqrt{y}$$, Rule 3:    $$\large\displaystyle \sqrt{\frac{x}{y}} = \frac{\sqrt x}{\sqrt y}$$. Often times, you will see (or even your instructor will tell you) that $$\sqrt{x^2} = x$$, with the argument that the "root annihilates the square". In case you're wondering, products of radicals are customarily written as shown above, using "multiplication by juxtaposition", meaning "they're put right next to one another, which we're using to mean that they're multiplied against each other". So let's actually take its prime factorization and see if any of those prime factors show up more than once. The index is as small as possible. Arithmetic Mean Geometric Mean Quadratic Mean Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge. Fraction of a Fraction order of operation: $\pi/2/\pi^2$ 0. Break it down as a product of square roots. In the second case, we're looking for any and all values what will make the original equation true. And for our calculator check…. Determine the index of the radical. 1. root(24) Factor 24 so that one factor is a square number. We can add and subtract like radicals only. You'll usually start with 2, which is the … We will start with perhaps the simplest of all examples and then gradually move on to more complicated examples . (Other roots, such as –2, can be defined using graduate-school topics like "complex analysis" and "branch functions", but you won't need that for years, if ever.). There are five main things you’ll have to do to simplify exponents and radicals. Some techniques used are: find the square root of the numerator and denominator separately, reduce the fraction and change to improper fraction. Radicals (square roots) √4 = 2 √9 = 3 √16 = 4 √25 =5 √36 =6 √49 = 7 √64 =8 √81 =9 √100 =10. If and are real numbers, and is an integer, then. Example 1: to simplify $(\sqrt{2}-1)(\sqrt{2}+1)$ type (r2 - 1)(r2 + 1). Step 1: Find a Perfect Square . Example 1. 2. The radicand contains no fractions. To simplify this radical number, try factoring it out such that one of the factors is a perfect square. Reducing radicals, or imperfect square roots, can be an intimidating prospect. This type of radical is commonly known as the square root. Divide out front and divide under the radicals. You don't want your handwriting to cause the reader to think you mean something other than what you'd intended. Let’s look at some examples of how this can arise. One would be by factoring and then taking two different square roots. Simplify each of the following. URL: https://www.purplemath.com/modules/radicals.htm, Page 1Page 2Page 3Page 4Page 5Page 6Page 7, © 2020 Purplemath. Special care must be taken when simplifying radicals containing variables. By quick inspection, the number 4 is a perfect square that can divide 60. Simplify the following radicals. Rule 1.2:    $$\large \displaystyle \sqrt[n]{x^n} = |x|$$, when $$n$$ is even. For instance, relating cubing and cube-rooting, we have: The "3" in the radical above is called the "index" of the radical (the plural being "indices", pronounced "INN-duh-seez"); the "64" is "the argument of the radical", also called "the radicand". Generally speaking, it is the process of simplifying expressions applied to radicals. Here’s the function defined by the defining formula you see. Simplifying Radicals. Simplifying radical expressions calculator. Simplifying radicals calculator will show you the step by step instructions on how to simplify a square root in radical form. Perfect Cubes 8 = 2 x 2 x 2 27 = 3 x 3 x 3 64 = 4 x 4 x 4 125 = 5 x 5 x 5. You don't have to factor the radicand all the way down to prime numbers when simplifying. In this tutorial we are going to learn how to simplify radicals. (Technically, just the "check mark" part of the symbol is the radical; the line across the top is called the "vinculum".) As soon as you see that you have a pair of factors or a perfect square, and that whatever remains will have nothing that can be pulled out of the radical, you've gone far enough. [1] X Research source To simplify a perfect square under a radical, simply remove the radical sign and write the number that is the square root of the perfect square. The radicand contains no factor (other than 1) which is the nth or greater power of an integer or polynomial. Rule 2:    $$\large\displaystyle \sqrt[n]{xy} = \sqrt[n]{x} \sqrt[n]{y}$$, Rule 3:    $$\large\displaystyle \sqrt[n]{\frac{x}{y}} = \frac{\sqrt[n]{x}}{\sqrt[n]{y}}$$. For instance, 4 is the square of 2, so the square root of 4 contains two copies of the factor 2; thus, we can take a 2 out front, leaving nothing (but an understood 1) inside the radical, which we then drop: Similarly, 49 is the square of 7, so it contains two copies of the factor 7: And 225 is the square of 15, so it contains two copies of the factor 15, so: Note that the value of the simplified radical is positive. Square root, cube root, forth root are all radicals. 2) Product (Multiplication) formula of radicals with equal indices is given by This website uses cookies to ensure you get the best experience. That is, we find anything of which we've got a pair inside the radical, and we move one copy of it out front. Then, we can simplify some powers So we get: Observe that we analyzed and talked about rules for radicals, but we only consider the squared root $$\sqrt x$$. That is, the definition of the square root says that the square root will spit out only the positive root. Generally speaking, it is the process of simplifying expressions applied to radicals. Simplify any radical expressions that are perfect squares. In order to simplify radical expressions, you need to be aware of the following rules and properties of radicals 1) From definition of n th root(s) and principal root Examples More examples on Roots of Real Numbers and Radicals. Here’s how to simplify a radical in six easy steps. So in this case, $$\sqrt{x^2} = -x$$. This website uses cookies to improve your experience. Solved Examples. How to simplify radicals . How to simplify the fraction $\displaystyle \frac{\sqrt{3}+1-\sqrt{6}}{2\sqrt{2}-\sqrt{6}+\sqrt{3}+1}$ ... How do I go about simplifying this complex radical? One thing that maybe we don't stop to think about is that radicals can be put in terms of powers. So … √1700 = √(100 x 17) = 10√17. We'll assume you're ok with this, but you can opt-out if you wish. This is the case when we get $$\sqrt{(-3)^2} = 3$$, because $$|-3| = 3$$. x, y ≥ 0. x, y\ge 0 x,y ≥0 be two non-negative numbers. By using this website, you agree to our Cookie Policy. The expression " katex.render("\\sqrt{9\\,}", rad001); " is read as "root nine", "radical nine", or "the square root of nine". This tucked-in number corresponds to the root that you're taking. Fraction involving Surds. Simple … where a ≥ 0, b > 0 "The square root of a quotient is equal to the quotient of the square roots of the numerator and denominator." Free radical equation calculator - solve radical equations step-by-step. On a side note, let me emphasize that "evaluating" an expression (to find its one value) and "solving" an equation (to find its one or more, or no, solutions) are two very different things. No radicals appear in the denominator. Get your calculator and check if you want: they are both the same value! Cube Roots . To simplify this sort of radical, we need to factor the argument (that is, factor whatever is inside the radical symbol) and "take out" one copy of anything that is a square. Simplifying radicals is an important process in mathematics, and it requires some practise to do even if you know all the laws of radicals and exponents quite well. Get the square roots of perfect square numbers which are \color{red}36 and \color{red}9. The first thing you'll learn to do with square roots is "simplify" terms that add or multiply roots. Reducing radicals, or imperfect square roots, can be an intimidating prospect. Examples. 1. root(24) Factor 24 so that one factor is a square number. Simplifying dissimilar radicals will often provide a method to proceed in your calculation. Since I have two copies of 5, I can take 5 out front. You 're ok with this, but it may  contain '' a square number factor. \cdot. 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