Example Questions in Arithmetic
1. Why is 5/sqrt(5) equal to sqrt(5)?
2. which is correct: 8 * 3 means 8 three times, or 8 * 3
means 3 eight times?

Question # 1

Why is 5/sqrt(5) equal to sqrt(5)?

sqrt(5) is that number which, when multiplied by itself,
gives 5.

This means that sqrt(5) * sqrt(5) = 5

sqrt(5) * sqrt(5) = 5

Dividing equals by equals gives equals

( sqrt(5) * sqrt(5) ) / sqrt(5) = 5/sqrt(5)

On the left hand side of the equation, one of
the sqrt(5) 's in the numerator cancels the sqrt(5) in the
denomerator.

sqrt(5) = 5 / sqrt(5)

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2. which is correct: 8 * 3 means 8 three times, or 8 * 3
means 3 eight times?

Here is how I would show that the two ways 8 * 3 and 3 * 8
mean exactly the same thing.

Take 24 identical square cards

You can make 8 rows of three or 3 rows of eight.

If you make 8 rows of three, you can view the 8 rows of three
from the perpendicular side, and see 3 rows of 8.

How you view the cards does not change what they are.

This shows that 8 * 3 is the same as 3 * 8.

On the question does 8 * 3 mean 8 three times or 3 eight
times.

Take your choice. Either one will work consistently.

Occasionally a mathematician will choose one of the two
possible definitions as "The Definition".
I never bother to remember which way was chosen. I am
dyslexic with respect to such definitions.

I shall give an intuitive argument for each of the two ways.

First my argument for saying that 8 * 3 means 8 three times.

8 * 3 means 8 times 3 means 8 replicated 3 times means
8 + 8 + 8.

Second my argument for saying that 8 * 3 means 3 eight times.

8 * 3 means 8 threes means 8 replications of 3 means
3 + 3 + 3 + 3 +3 + 3 + 3 +3.

Which you choose is an arbitrary convention.

It is a mathematical property of 3*8 that the choice must be
an arbitrary convention.

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