







An equation with rational coefficients 
20180526 

From Anagh: If x and y are rational number and
(x+y) + (x  2y)√2 = 2x y + (xy1) √5
Then find the value of x and y. Answered by Penny Nom. 





The decimal form of a fraction 
20180112 

From Tom: Prove that the decimal representation of the quotient of 2 integers must repeat (if it does not terminate). Answered by Penny Nom. 





The square root of 6 is irrational 
20171002 

From John: http://mathcentral.uregina.ca/QQ/database/QQ.09.06/sylvia1.html
In the initial assumption of that proof, root 6 is assumed to be a/b where a and b have no common factors, but why does having a common factor make it irrational? Answered by Penny Nom. 





Five rational numbers between 2 and 1 
20170531 

From Prince: Five rational numbers between 2 and 1 Answered by Penny Nom. 





Rational numbers between 2 and 3 
20170408 

From Medha: Find 12 rational numbers between2 and 3 Answered by Penny Nom. 





Is the square root of 2 plus the square root of 2 irrational? 
20170329 

From haya: how can i prove the the square root of 2 plus the square root of 2 is an irrational number? Answered by Penny Nom. 





A rational number between 4 and 5 
20170103 

From Anshika: Write rational number between 4 and 5 Answered by Penny Nom. 





Why are equivalent fractions equal? Part 2. 
20161122 

From Anandmay: Hi!
I saw one of your answered questions:Why are equivalent fractions equal?
The same question I had for why i searched and ended up entering this wonderful site.
The answer was so much experimental based.I mean,for example,it was explained how
3/4 was equal to 6/8 by dividing a unit into 4 equal parts and taking four of the parts,and then dividing the
same unit into 8 equal parts and then taking 6 of them.The result was that both the taken parts were equal.
(That is,3 out of 4 equal parts were equal to 6 of the 8 equal parts.)
But,this was very much nongeneralized and experimentalbased result(proof).
And we all know that in mathematics,we confirm some particular thing for all the numbers iff we generalize
it.
So,can you explain more clearly 'HOW' the 3 parts of the 4 equal parts and the 6 of the 8 equal parts of the unit
equal(that is the same)?I don't want experimental proof.I want clear proof,and thus please give a generalized
proof for all fractions and their equivalents.
Thanks! Answered by Penny Nom. 





A limit of a rational function 
20160116 

From selvamani: F (x) = x^3+3x^29x2 / x^3x6 and limit x tends to 2, f (x) exist then limit x tends to 2, f (x) is equal to ? How to answer this problem. Answered by Penny Nom. 





Dealing with surds 
20151114 

From Agnes: simplify (1√3)(1÷3+√3) Answered by Penny Nom. 





An example of an irrational number 
20151022 

From Allison: It ask me to find an example of an irrational number less than 5 and I don't
understand what the difference from a rational number and an irrational
number besides the fact that a rational number can be repeated and shown in
a simple fraction and an irrational number can't be written in a simple
fraction.
Can you help me? Answered by Penny Nom. 





2.236... 
20151013 

From Ann: 2.236...
What is the most specific category of numbers does this fall into? Rational or Irrational? Does the .... mean that it repeats? Answered by Harley Weston. 





Square roots 
20150921 

From mariana: I have read various articles on how to find the square root of irrational numbers and every article out there seems to be very confusing.
i read you answer to LUKOW about irrational numbers and i am still quite confused. Say i want to find the square root of 326. i know that it is between 18 and 19 because 18 is the square root of 324 and 19 is the square root of 361 im just very confused about the rest of the process. Please help! ( if possible i would appreciate two examples. thanks) Answered by Penny Nom. 





Is 22/7 is an irrational number? 
20150607 

From Sayak: Is 22/7 is an irrational number Answered by Penny Nom. 





3 over "x" plus 5 
20150223 

From Kenny: 3 over "x" plus 5. I don't know this is either a polynomial and if so how would you define it. Answered by Harley Weston. 





1÷[1√2(order of surd is 4)] 
20140502 

From Anoushka: if t=1÷[1√2(order of surd is 4)] , then t=? Answered by Penny Nom. 





A rational number 
20110517 

From owais: according to definition of irrational number "a number which cannot be
expressed in p/q form is called irrational number"
square root of 2 is a irrational number but if i round off the digit correct
to two decimal places so it becomes 1.41 and we can easily convert
into p/q form so it is a rational number ??? Answered by Penny Nom. 





A rational equation 
20100802 

From Bailey: 3/x
+
x/x+2
______
2/x+2 Answered by Penny Nom. 





1.2731568 
20100616 

From angel: is 1.2731568 a natural number a whole number a irrational number or a inter Answered by Robert Dawson. 





Rational root theorem 
20100414 

From deciree: How could i get the value of x with this equation:
1/13x^3+12x/13=1/2
Thank you! Answered by Harley Weston. 





Rationals 
20091030 

From Jawsh: I don't quite understand the whole idea of rationals.
If you have 3Squareroot 49. Is that equal to 3Squareroot 7^2?
Can you break it down with an example for me please. Answered by Robert Dawson. 





Proof that the root of 27 is irrational 
20091018 

From Scarlet: How do you prove that the square root of 27 is irrational? Answered by Victoria West. 





root 2 is irrational 
20090731 

From prateet: prove that root2 is an irrational number. Answered by Robert Dawson. 





Properties of Inequality 
20090605 

From Nazrul: We know that if a>b then (1/a)<(1/b)
But why 5> 2 implies (1/5)>(1/2) .
Thank you in advance for your help. Answered by Robert J. Dawson. 





Graphing Rational Functions 
20081205 

From Bahra: The question in my homework is: Find all vertical, horizontal, and slant
asymptotes, x and yintercepts, and symetries, and then graph each
function. Check your work with a graphing calculator.
y=3/x+3 meanng: ( y= 3 over x+3) Answered by Janice Cotcher. 





Algebraic fractions 
20081006 

From Kayla: (x^29)/x times (x^34x)/(x^2+5x+6) Answered by Harley Weston. 





Nonterminating, nonrepeating decimals 
20080803 

From Peter: How do you take a random, nonterminating, nonrepeating decimal into a fraction? Answered by Stephen La Rocque. 





Rational numbers 
20080716 

From hinal: list 12 rational numbers which lie between
a) 1 and 0
b) 3 and 3 Answered by Penny Nom. 





How can the distance between two points be irrational? 
20080507 

From Dawn: Yesterday my advanced 7th grade prealgebra students were finding the distance between two points on a coordinate grid using the distance formula. A student asked me how come we or math I guess is able to represent a finite distance with an irrational number. I found this to be a very interesting question. I and the 7 other math teachers on my staff had discussed this and we have a variety of thing to say to the kid, but nothing that amounts to an actual answer. The students have studied what it means for a number to be irrational, and they cannot understand how that can "measure" a line that obviously stops. Got any suggestions? Answered by Stephen La Rocque, Penny Nom, Claude Tardif and Victoria West. 





1/35/6=1/x, solve for x 
20080501 

From Lavada: 1/35/6=1/x How do I solve this problem. Answered by Penny Nom. 





Adding rational functions 
20080429 

From Jonathon: 1/(x+3) + 1/(x^2+5x+6) = Answered by Penny Nom. 





Simplify 5x^5/8 / 15x^2/12 
20080418 

From Kyler: 5x^5/8 / 15x^2/12 Divide the rational expressions. Simplify the results. Answered by Penny Nom. 





Is this an irrational number? 
20080322 

From Tom: Is it possible to build an irrational number with numbers that are just 1's and 2's where only finitely many 2's appear? If so, how do we build it and how do we know it is irrational? Answered by Penny Nom. 





How would I graph: x  2 divided by x^2  4? 
20080319 

From Sean: How would I graph: x  2 divided by x^2  4 using a method in extreme values of
functions or completing the square
x  2/ x^24 Answered by Stephen La Rocque. 





The rational root theorem 
20071209 

From Josh: Prove the rational root theorem (where, in a polynomial equation, all
factors of the constant divided by all factors of the leading coefficient
are possible rational roots of the equation). Answered by Penny Nom. 





What is a rational number between 5.9 and 6? 
20071106 

From cathy: What is the rational number between 5.9 and 6? Answered by Penny Nom. 





An irrational number 
20071029 

From Clara: Find an irrational number between 0.53 (with 53 as repeating)
and 0.54 (with 54 repeating)
I changed each to 53/99 and 54/99 with 1/99 being the difference.
Please help me. Answered by Stephen La Rocque, Penny Nom and Harley Weston. 





Simplifying rational expressions 
20071009 

From Bama: x(x2)+1 divided by x^23x+2 Answered by Stephen La Rocque. 





Subtracting rational expressions 
20071009 

From Pamela: Subtract:
6p – w __ 4c + p
w – 4 w – 4
Answered by Stephen La Rocque. 





Irrational functions 
20071001 

From alicia: i have a question about irrationals functions.
i have been using them quite some time now, but i wonder where they can be found in daily life?
i hope you can help me, Answered by Harley Weston. 





A fraction that cannot be simplified 
20070917 

From Kevin: Make a fraction that cannot be simplified and has 24 as its denominator? Answered by Stephen La Rocque and Chris Langdon. 





The irrationality of PI 
20070807 

From Matthew: I have what I like to think of as a rather interesting question that I can't explain confidently for the life of me.
If we take a circle with a radius of 1 and we calculate the circumference, we can use 2 pi R.
Doing this calculation results in a circumference of 6.28318530717~ which goes on forever.
However, if you were to take a that same circle in the real world, say with radius 1cm and wrap a string around
it, and then measure the string, you don't get 6.28~, you get something like 6.2, a much more finite distance. The length
of the string is not an irrational number, like the math claims it to be. Answered by Claude Tardiff and Stephen La Rocque. 





Simplifying a quartic rational expression using long division 
20070614 

From Megan: x+2/12x^4+17x^3+0x^2+8x40= Answered by Stephen La Rocque and Penny Nom. 





Rational polynomial expressions 
20070609 

From a student: I have a question that continues to lead me to the answer x/(x  1) but according to my math book, the answer should be 1/(x  1).
Who is right? Who is wrong?
Here is the question:
(x^2  6x  27)/(x^2  11x + 18) DIVIDED by (x^3 + 2x^2  3x)/(x^2  2x) Answered by Stephen La Rocque. 





Repeating decimals and rational numbers 
20070531 

From lil: Why are repeating decimals considered rational numbers? Answered by Penny Nom and Gabriel Potter. 





The limit of a rational function 
20070528 

From Imad: 3 _______ 3 _______
lim \/ 1 + x  \/ 1  x
x>0  
x Answered by Penny Nom. 





Positive rational numbers 
20070522 

From Rebecca: What is the definition of positive rationals/positive rational numbers? Answered by Stephen La Rocque. 





Simplifying a rational expression 
20070429 

From Tamika: X^2 + 10X + 24
______________
X + 6
SIMPLIFY EACH RATIONAL EXPRESSION (IF YOU CAN'T SIMPLIFY IT, WRITE ALREADY SIMPLIFIED) Answered by Stephen La Rocque. 





The sum of two rational numbers 
20070404 

From Fathia: Hi,
Could please help me with this question?
Prove that if a and b are rational numbers then a + b is a rational number.
Thank you, Answered by Stephen La Rocque. 





Adding and Subtracting Rational Expressions 
20070321 

From Pricilla: Pricilla
I do not know how to do this problem. Don't know where to start.
[(5)/(2x8)]+[(3x^2+15)/(x^34x^2+5x20)] Answered by Stephen La Rocque. 





Prove that square root of 3 is irrational. 
20061221 

From Humera: Prove that square root of 3 is irrational. Answered by Stephen La Rocque. 





Help me prove that the square root of 6 is irrational 
20060923 

From Sylvia: Could you please help me prove that the square root of 6 is irrational ? Answered by Stephen La Rocque. 





Rational expressions 
20051115 

From Zach: I can solve easy problems such as (x/2)+3=2+(3x/4). That is easy because the Lowest Common Denominator is 4. But what really gets me stuck is a problem like this one.
(6/x2) = ( 21/(x2)(x+2) )+ 1. Answered by Penny Nom. 





2x/x^29  6/x^2+x12 
20050927 

From Emily Ann: 2x/x^{2}9  6/x^{2}+x12 Answered by Penny Nom. 





A rational arithmetic expression 
20050914 

From Vijay: How to simplify following Grade 9 math problem:
(4/9) X (21/32) X (3/14) Answered by Penny Nom. 





Converting rational numbers 
20050807 

From Joe:
I am helping my son with his converting rational
numbers in the form of A/B where A and B are integers and
B not equal to zero
I think I am doing this right but I am not sure so
below are the problems and our answers are beside
them, please let me know if these answerers are correct
Answered by Penny Nom. 





The limit as x approaches 3 of (sqrt(x+1)2)/(x3) 
20050414 

From Abraham: What is the limit as x approaches 3 of
(sqrt(x+1)2)/(x3) Answered by Penny. 





Rational functions 
20050405 

From Nicole: My name is Nicole and I am a teacher at Weyburn Comprehensive School. I am currently teaching both Math B30 and Calculus 30 at the school and I have a question about rational functions. I know that if a rational function (by definition) has common factors in the numerator and the denominator then it is not a rational function (math b30) however in calculus this common factor creates a hole in our graph. Can you explain to me why a common factor or constant does not give us a rational function? Answered by Penny Nom and Leeanne Boehm. 





Rational expressions 
20040924 

From A student: In general, I understand rational expressions except when it comes to solving problems such as:
x+y/2xy  2x/y2x or m4/3m4 + 3m+2/43m
I am confounded by the issue of having to find a common denominator. For example, if I tried to solve these problems by multiplying both denominators they would still be uncommon. Answered by Claude Tardif. 





An equation with rational terms 
20040617 

From Louise: simplify the expression
(a^{2}+n^{2})/2n = a^{2} + (a^{2}n2)/2n Answered by Penny Nom. 





Rational expressions 
20040406 

From Shivani: I am in 11th grade Honors Algebra II. We are currently learning about adding and subtracting rational expressions. I looked at the other questions that were similar to that of mine, but they weren't quite answering my question. I have been working on many problems to practice and see if I can come up with the correct answer, but I can't, I got like 2 out of every 7 right on my school homework. I get the easy ones, where when you have to find the LCD, it's just the other denominator. But, on others you have to really think about it. Those are hard. Here are some of the problems that I cannot get at all: (x2  3x + 1)/(x2  4)  (x2 + 2x +4)/(2  x)  (x  4) /(x  2)
The correct answer is:
(x3 + 4x2 + 7x + 17)/(x2  4) Answered by Penny Nom. 





Polynomials 
20040125 

From Bruce:
A polynomial is defined as
Polynomial functions are functions that have this form:
f(x) = a_{n}x^{n} + a_{n1}x^{n1} + ... + a_{1}x + a_{0}
The value of n must be an nonnegative integer. That is, it must be
whole number; it is equal to zero or a positive integer.
The coefficients, as they are called, are a_{n}, a_{n1},
..., a_{1}, a_{0}. These
are real numbers.
Questions:
 why must n be positive?
 what are some historical facts
about the evolution of the definition?
Answered by Harley Weston. 





Subtracting rational expressions 
20030510 

From Simone: hi, i'm totally lost. i understand that you need to find a lowest common denominator to subtract two fractions (rational expressions) with different denominators. but what if the denominators are "x1" and "x". is x the common denominator? if so what happens to the "1"? do you know of any live online help i can get with the following: 3/(x1)  (12x)/x i've looked through my notes and have no examples that quite match that i can follow to get through it. please help! Answered by Penny Nom. 





Excluded values 
20030222 

From Josh: Why do you think it is necessary to include the "excluded values" when you write your answers to rational expressions? Answered by Penny Nom. 





Rational expressions 
20030122 

From A student:
// = the main fraction line 1.) a+1/a1 + a1/a+1 // a+1/a1  a1/a+1 2.) 2/a(squared)3a+2 + 2/a(squared)a2 // 2/a(squared)1 + 2/ a(squared)+4a+3 Answered by Penny Nom. 





Rational expressions 
20021003 

From Ashley: 1/x(squared) + 5/xy Answered by Penny Nom. 





The square root of 2 
20020305 

From Roger: Does two (2) have a square root or do the numbers just keep going? Are there any other numbers that behave like two when it comes to extracting the square root? Answered by Penny Nom. 





Some irrational numbers 
20010703 

From Kellie:
 Is it possible to write pi as a fraction?
 Is it possible to write the square root of 2 as a fraction.. Explain why?
Answered by Harley Weston. 





Dividing fractions 
20010509 

From Rina: I just wanted to ask if you could help me in math. See I'm having a test soon and its on Dividing Fractions and I just don't get it. My math teacher says that I'll be just fine but I failed my math quiz. I went to ask eric but they could help me so they told me to go to you. So here I am asking you if you could help me. Answered by Penny Nom. 





Adding and subtracting rational expressions 
20010503 

From Donna: Adding and subtracting Rational expressions. I am in grade 10 and I am a student here is an example of the questions: 1/(x+1)  1/(x1) = ? Answered by Penny Nom. 





Cube roots on a calculator 
20010324 

From Will: Hi my name is Will.I have a question about the calculator TI83 Plus or the use of a scientific calculator. It is about using them to turn a cubed root to a decimal. When using that for a rational and irrational number. Rational numbers is a number that terminates or repeates. A irrational number goes on and on and uses ... (like pie). Like when you type the cube root of 8 it gives you 2, and that is a rational number. The squre root of 2 is 1.41421356... that is irrational. So why is it when you put the cube root of 16 in the calculator it says 2.5198421 that is rational it terminates at the ninth digit and my float is set for the tenth? But why when you do it by hand you get the 2 times the cube root of 2 and that is irrational? Why is that? Answered by Judi McDonald. 





The nth place of pi 
20010227 

From Andrew: I have always wondered; is it possible to find the value of an irrational number, such as (phi) at it's nth decimal place??? You would plug the decimal place into the formula and the value would be given at the specified decimal place. When we look at the expression { (sqr 5 + 1)/2 } we are in a sense visualising the number in it's entirety, so the formula may include elements of the above expression in some form. Answered by Claude Tardif. 





More repeating decimals 
20010117 

From Alan: I am neither a math teacher nor a student, but I hope you will consider my question anyway. I recently was discussing repeated decimals with a friend, and went on the web to find out more about a pattern was looking for years ago. In doing so I came upon your correspondence on repeating decimals. . . . Answered by Chris Fisher and Penny Nom. 





Repeating decimal notation 
20001231 

From Clarende Duby: I have seen single repeating decimals for ex. 1/3 = 0.3 with the dot above the 3 to represent the repeating decimal. Then, there is the more common form of the bar over the top of the repeating number or group of numbers (called the period?). Which notation is correct? Answered by Penny Nom. 





Repeating decimals 
20001006 

From Mary O'Sullivan: I understand that with repeating decimals (those with a pattern), the number of digits repeated is put into fraction form with the same number of 9's ex. 0.4444 = 4/9 0.145145145 = 145/999 How can I explain why this is so? Some inquisitive 8th graders are anxious to find out! Answered by Penny Nom and Walter Whiteley. 





Rational Numbers 
20000914 

From Josh Kuhar: How can you tell a number is rational? Answered by Harley Weston. 





Numerator and denominator 
20000618 

From Maureen Beard: What is the origin of the terms numerator and denominater? Answered by Penny Nom. 





The square root of 3 
20000404 

From Mr. William: Prove that root 3 is irrational Answered by Harley Weston. 





Rationals, irrationals and integers 
20000314 

From Erin McKeon: Why does the letter J represent the set of integers, the letter Q represent a set of rational numbers and the letter P represent a set of irrational numbers? What do each of these letters stand for? Answered by Harley Weston. 





Pi 
20000222 

From Shelley Collier: Numbers that can be written in the form p/q where p and q are integers, (q not 0), are known as rational numbers. What you are missing is that p and q must be integers. The fact that Pi is irrational means that you can't have a circle with both the circumference and diameter being integers. In fact you can't even have the circumference and diameter both rational since the quotient of two rationals is again a rational. Answered by Claude Tardif and Penny Nom. 





Irrational algebraic functions 
20000221 

From Bucky Cadena: Here is the multipart problem: Given f(x) = x3* squareroot of x + 4 What does the f(x) intercept equal Find the two values for which f(x) = 5 Find the one value for which f(x) = 3 Answered by Harley Weston. 





The Binomial Theorem for rational exponents 
19990415 

From Angela Evans: The full question is this: Isaac Newton generalized the Binomial Theorem to rational exponents. That is, he derived series of expansions for such expressions as (x+y)^{3} (x+y)^{2/3} (x+y)^{5/6} What did Newton find? What are the first four terms of the series expansions of binomials above? How can this extended Binomial Thrm. be used to aid in calculations? Answered by Penny Nom. 





The square root of two is never supposed to end 
19990306 

From a wondering student: i am algebra II and am in the 9th grade. today we were talking about rational and irrational numbers. When we were talking about square roots my friend and i were talking and we thought of something. if you have a square with sides of length one then the diagonal of the square is the square root of 2. Now the square root of two is never supposed to end. But the diagonal of the square ends so therefore doesn't the square root of 2 end. our math teacher did not really answer our question because it was not in the lesson plan and not to many people would see where we were coming from. the answer is really bugging me and i would like to have your input. Answered by Jack LeSage and Penny Nom. 





Simplifying Radicals 
19990126 

From Mary: I would like to know how to simplify this question:
4 __________________ squareroot7 + squareroot3
I know the answer is (sqrt7  sqrt3) but i would really love to know how to get that answer!! Thanks. Answered by Jack LeSage and Penny Nom. 





Root 17 is Irrational 
19990121 

From John Murdock: If you could help me out with this I would appreciate it. Prove that the square root of 17 is irrational. Answered by Harley Weston. 





Turning Fractions into Decimals 
19990116 

From Lynn: How do you do it step by step Answered by Jack LeSage. 





6.99999... = ? 
19981205 

From Tom: I have had a rather heated arguement with my students. Please settle this for me. Solve <,>, = 6.99999... __ 7 Thank you. Answered by Penny Nom. 





Terminating decimals 
19981116 

From Debra Karr: A college student studying elementary education asked me a question that I could not think of the correct answer. How can you look at a fraction and tell if is a terminating or non terminating decimal? Answered by Jack LeSage and Penny Nom. 





Adding Fractions 
19981015 

From Lindsay: please help me!!!! Answered by Harley Weston. 





Repeating Decimals 
19981001 

From Chris Norton: Could you PLEASE give the formula to find out the number of digits in a repeating decimal before it repeats. I have been trying to get it for weeks from Math sites on the Internet. Can you please, please help me ? Chris Norton Answered by Chris Fisher and Penny Nom. 





Rational and Irrational Numbers 
19980919 

From Ri: I am trying to explain rational & irrational numbers to my niece who is grade 7 and am having difficulties. Could you please explain the difference between rational & irrational numbers. Thank you Ri Answered by Penny Nom. 





Rational Numbers 
19980905 

From Kristin Mckenzie: Iam a secondary student with a math question I don't know how to do. This is the question: Explain whether each decimal number represents a rational number. (a) 0.16 (b) 0.12 (c) 3.125 (d) 0.27 (e) 0,212 112 111 2 .... (f) 0. 457 92 If it wouldn't be a problem i would really appreciate it if you sent back the instructions on how to do the question. My name is Kristin Mckenzie and my return email is lexus1999@hotmail.com Thankyou for you time Answered by Jack LeSage and Harley Weston. 





(2)^x 
19971212 

From Ken Bokinac: During our math class last week we came accross an equation that we could not graph we were wondering if you could help us try and graph it. The question is: y=(2)^x Answered by Harley Weston. 





Division by a fraction. 
19971121 

From Roger Marchegiano: I am interested in a solution to the following extra credit problem for my son in Geometry. We have been unable to produce a satisfactory response: Show a geometric representation of the division of fractions algorithm, (When you divide fractions you multiply the numerator by the reciprocal of the denominator). Answered by Penny Nom. 





A Question About Pi. 
19970708 

From Mike Chan: I have read the section Repeating Decimals in your data base. It mention that 1/17 has at most only 16 repeating digits. But, why does "pi" have an infinite number of digits (and not repeating ). Answered by Penny Nom. 





A proof that e is Irrational. 
19970430 

From Peter Hall and Jenny: We have a little mathematical problem... we need some help proving e is an irrational number! We don't feel very confident in our formulas, so if You have the time to give us a little explanation we would be very grateful!!! Answered by Doug Farenick and Penny Nom. 





Repeating Decimals 
19970124 

From Grant Reed: Is there a way to tell that the repeating decimal for 1/17 has no more than 16 repeating digits? Answered by Penny Nom. 





Ses points de discontinuités est les irrationnelles 
20020108 

From Un eleve: Je voudrait montrer qu'il n'existe pas de fonction de R>R tels que ses points de discontinuités est les irrationnelles, en utilisant la methode de "baire". Answered by Claude Tardif. 

