By Julie Rowlett
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Additional resources for Blast into Math - A Fun and Rigorous Introduction to Pure Mathematics
We also need rules for playing games in the playground of rational numbers. 1. Let x and y be rational numbers, and a and c be elements of Z , and b and d elements of N such that c y= . d a x= , b Then xy = ac . bd If b = d , then x+y = a+c , b x−y = a−c . b and 2. The multiplicative inverse of an integer z ∈ Z with z < 0 is −1 , −z and x= a −a = . b −b 3. If x = 0 and a > 0 , then the multiplicative inverse of x is b . a 42 Blast into Math! Sets of numbers: mathematical playground If a < 0 , then −a ∈ N , and the multiplicative inverse of x is − −b b b = = .
The product of −x and −y is equal to the product of xy . That is (−x) ∗ (−y) = xy. 39 Blast into Math! Sets of numbers: mathematical playground According to these rules, the integers are closed under addition, subtraction, and multiplication. The integers Z are a bigger playground than N because they contain additive inverses, which is like having an addition swingset. Starting at the additive identity 0 , we can swing over to an element of N like 3 , and then we can get a push over to its additive inverse −3 , add them together and end up back where we started at 0 .
You may have seen the rational numbers defined differently like in the following proposition. As long as we can prove that two definitions are equivalent, then they have the same meaning, and so mathematically it is equivalent no matter which definition we choose to use. 11 (Rational Proposition) Any rational number can be written as p q where p ∈ Z and q ∈ N . Proof: If a rational number is an integer, then that integer is the “ p ” in the theorem, and since p = p, 1 the multiplicative identity 1 is the “ q ” in the theorem.