Exercise on (ir)rational numbers

In summary, the conversation discusses the irrationality of √(2/3) and how it can be proved using the prime factorization of integers. It also addresses the question of whether an irrational number divided by another irrational number is always irrational. The conclusion is that it is not always the case, as shown by the example of √2/√2 = 1. The conversation also provides a proof for why √(p/q) is rational if and only if both p and q are squared numbers.
  • #1
PcumP_Ravenclaw
106
4
Dear all,
I have done question 1 of exercise 2.1 from the book Alan F beardon, Abstract Algebra and Geometry. Please answer some of my doubts.

Q1. a) Show that √(2/3) is irrational. b) Use the prime factorization of integers to show that if √p/q is rational, where p and q are positive integers with no common factors, then p = r^2 and q = s^2 for some integers r and s.

ANS:
a) ##\sqrt{2}## is known to be irrational

## \sqrt{3} ## is irrational because 3 is a prime number and its only factors are 1 and 3??
or it can be proved. p/q is the simplest fraction after cancelling common factors. p and q are integers.

##
\sqrt{3} = \frac{p}{q} \\
3 = \frac{p^2}{q^2} \\
3*q^2 = p^2
##

Please check if my observation below is correct??
Observation:
the multiple is 3 on LHS so if q is odd then p will also be odd. if q is even p will also be even. if both p and q were even numbers then p and q are not in their simplest terms therefore it violates ##\sqrt{3}## being a rational number. because ##\sqrt{}## of any number could be written as p/q but in simplest terms. simplest terms means both cannot be even and both cannot be odd. both can be odd if they are prime numbers e.g 7/3 or 9/5 etc..

In the above case, if q is odd (integer or prime number) p will be a odd number also with a factor 3 (because of the multiple 3) therefore it will not be in its simplest form. Therefore, ##\sqrt{3}## is an irrational number??

√2/3 is irrational because √2 and √3 are irrational! but how to prove that an irrational number divided by another irrational number is also irrational?

b) Please check my answer to the second part of the question? I have written in words can anyone translate to math notation?? Thanks..

##\sqrt{PrimeNumber} ## is always irrational because primes only have factors 1 and themself. Any integer number can be broken down into its prime factors. After cancelling all common primes in numerator and denominator. we are left with non-common primes in numerator and denominator. As square root of prime number is always irrational. the irrational numerator and denominator produce an irrational outcome.

so numerator p and denominator q should be squared numbers to produce integer fractions (after square root operation) which are rational.
 

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  • #2
For part a) an irrational divided by an irrational is not always an irrational. For example, [itex] \frac {\sqrt{2}}{\sqrt{2}}=1 [/itex]. Rather than using evenness or oddness, I would use the fact the if 2 divides [itex]a^2[/itex], then 2 divides a, for a in the integers.

I think you can sharpen up your argument in part b) a little. Just start off with the premise, p and q have no common factors and [itex] \sqrt {\frac {p}{q}} \in \mathbb{Q} [/itex].
 
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1. What are rational and irrational numbers?

Rational numbers are numbers that can be expressed as a ratio of two integers, such as 1/2 or 3/4. Irrational numbers are numbers that cannot be expressed as a ratio of two integers and have an infinite number of non-repeating decimals, such as pi or the square root of 2.

2. What is the importance of understanding rational and irrational numbers in exercise?

Understanding rational and irrational numbers is important in exercise because it helps in accurately measuring and tracking progress. For example, if someone is trying to lose weight and they are keeping track of their weight in decimal form, it is important to know if the number is rational or irrational to avoid inaccuracies in measurement.

3. How do rational and irrational numbers relate to physical activities?

Rational and irrational numbers are used in physical activities to measure distances, time, and other quantities. For example, if someone is running a 5K race, the distance of 5 kilometers can be represented as the rational number 5.0 km or the irrational number 5.000... km, depending on the level of precision needed.

4. Are there any exercises specifically designed to help understand rational and irrational numbers?

Yes, there are exercises and activities that can help improve understanding of rational and irrational numbers. These may include measuring distances with a ruler, estimating the length of irrational numbers on a number line, or converting between rational and irrational numbers in different forms.

5. How can knowing about rational and irrational numbers benefit overall fitness and health?

Knowing about rational and irrational numbers can benefit overall fitness and health by promoting accuracy in tracking progress and setting goals. It can also help in making informed decisions about nutrition and exercise, as some food labels may have irrational numbers such as pi or the golden ratio. Understanding these numbers can also improve problem-solving skills and critical thinking abilities, which are important in making healthy choices.

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