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Ring-ing in the rational numbers

Recall that a rational number is a number that can be written as \( \frac{a}{b}\), where \(a\) and \(b\) are integers.

We will explore some properties of rational numbers.

1) The sum of 2 rational numbers is always rational.

Proof: Let the 2 numbers be \( \frac{a}{b} \) and \( \frac{c}{d} \), where \(a, b, c, d\) are integers. Then, their sum is \( \frac{ ad+bc}{bd} \), and \( ad+bc\) and \(bd\) are both integers. Hence, this number is rational.

2) The product of 2 irrational numbers does not need to be irrational.

Proof: In the previous post, we showed the \( \sqrt{2} \) is irrational. The product of \( \sqrt{2} \) and \( \sqrt{2} \) is 2, which is rational.

3) The sum of a rational number and an irrational number is always irrational.

Proof: Let the rational number be \( x = \frac{a}{b} \) and the irrational number be \(y\). We will prove this statement by contradiction. Suppose that their sum is rational, of the form \( \frac{ c}{d} \), then we know that \( \frac{a}{b} + y = \frac{c}{d} \), or that \( y = \frac{ c}{d} - \frac{a}{b} = \frac{ cb-ad} { bd} \), which is rational. This contradicts the condition that \(y\) is irrational. Hence the sum is always irrational.

Can you answer the following:

A) What do we know about the product of 2 rational numbers? Is it always rational?

B) What do we know about the sum of 2 irrational numbers? Is it always irrational?

C) What do we know about the product of a rational number and an irrational number? Is it always irrational? [Hint: Be very careful!]


Can someone give me feedback? Is this too hard for Cosines group, or just right? Do you want to see more basic material?

Note by Chung Kevin
4 years ago

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A) Yes, since a/b x c/d = ac/bd which is rational B) No, consider root2 '+ (1-root2) = 1, which is rational C) No, since 0 is a rational number, which when multiplied by anything gives 0, which is rational

Andre Chan - 4 years ago

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Wait, is A) true or false?

Vincent Tandya - 4 years ago

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Sorry, I meant it is. ac/bd is rational.

Andre Chan - 4 years ago

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For CosinesGroup, I think that the material should be slightly more basic, in terms of what a 13-14 year old would typically have access to. To me, this would be on the higher end of Cosinesgroup, or even in Torquegroup.

I liked your "Matchstick puzzles" post, and I think posts similar to that will be appropriate.

Best of Number Theory Staff - 4 years ago

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I found this entertaining and basic enough. It introduces the reader into think about how to formulate basic proofs.

Bob Krueger - 4 years ago

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