Page 36 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

The preceding proposition shows that we can play the games of addition and multiplication without
leaving the playground of natural numbers! Playgrounds often have merry-go-rounds. If we jump on
a merry-go-round and give it a push, it will spin bringing us back to where we started. Similarly, if we
have a natural number like 5, and we want to play the game of multiplication starting with 5 so that we
end up back at 5 again, can we do that in our playground? Is there some y ∈ N such that

5 ∗ y =5 ?

I bet you know the answer. That’s right, y =1.

Definition 3.2.6 Given a binary operation on a set A = ∅ , an identity for that operation is an element
I ∈ A so that, for any a ∈ A, if we perform the operation with I and a, the result is a.

The multiplicative identity is 1, and 1 ∈ N. What about the additive identity? Is there some number
z such that

x + z = x ?

Yes, the additive identity is 0.

A fun mathematical trick is to disguise the additive or multiplicative identity. You can think of the additive
and multiplicative identities 0 and 1 as playful numbers who enjoy celebrating Halloween by disguising

themselves in different costumes. They are so playful, that they disguise their friends in costumes too!
The multiplicative identity often disguises himself in the “costume”

x
1= forany x =0.
x

Since

1 ∗ y = y ∀y,

1 can also disguise his friend y in a costume: once 1 is wearing his costume we then put y into the
costume

(1 in disguise)
(1 in disguise) ∗ y = y.

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