Page 38 - 'Blast_Into_Math
P. 38
Blast into Math! ets of nummers: mathematical plaagrounds
Then, 0 can also disguise her friend y in a costume:
(0 in disguise)
(0 in disguise) + y = y.
We will enjoy celebrating Halloween with 0 and 1 and their different costumes throughout this book!
Another fun thing to do in playgrounds is swing on a swing set. You start at the bottom with your feet
on the ground, and if you pump your legs or someone gives you a push, you can swing up high. After
you swing up high, you always swing back down. That’s what makes it fun! Similarly, we’d like to be able
to play an operation and swing up high and then swing back down to the identity.
Definition 3.2.7 Given a set A and a binary operation for which A has an identity, the inverse of an
element a ∈ A is an element b ∈ A so that when we do the operation with a and b the result is the
identity.
For example, the inverse of the number 3 for the operation + is the number −3. This is one way to
define negative numbers. But, these are not in N . So, we need a bigger set, (a bigger playground! ) that
contains N and {0} and all the additive inverses. This set is called Z , and its elements are called integers.
Definition 3.2.8. An integer is either
• a natural number (an element of N);
• the number 0 (the additive identity);
• the additive inverse of an element of N , which for n ∈ N is written −n .
The set of all integers is written Z .
Remark 3.2.9. We use the symbol Z to denote the set of all integers. One way to remember this is that
Zahlen means numbers in German. Even though the symbol comes from German, mathematicians
around the world use the same symbol Z.
Another way to think about additive inverses (or negative numbers) is with the number line. The additive
identity 0 is at the middle of the number line. A positive number is on the right side of 0, and its additive
inverse is equally far from 0 on the left side. So, when you add them together, you end up back in the
middle at 0. You might have learned
x if x> 0
|x| = 0 if x =0
−x if x< 0
38
