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?

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2 \times 3

2 \times 3

2^{34}

2^{34}

a_{i-1}

a_{i-1}

\frac{2}{3}

\frac{2}{3}

\sqrt{2}

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\sum_{i=1}^3

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Easy Math Editor

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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 - 7 years, 6 months ago

Wait, is A) true or false?

Vincent Tandya - 7 years, 6 months ago

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

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 - 7 years, 6 months ago

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

Bob Krueger - 7 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$