Page 46 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

Because the integers are closed under addition, multiplication, and subtraction

ad + bc ∈ Z, ad − bc ∈ Z.

This means that x + y and x − y are equal to the product of an integer (respectively ad + bc and
ad − bc ) and the multiplicative inverse of the natural number bd, so by the definition of Q ,

x + y ∈ Q, x − y ∈ Q.

Remark 3.2.13 The letter Q reminds us that the rational numbers consist of quotients of integers.

3.3 The least upper bound property

Although the playground Z is not as big as the playground Q, the integers satisfy an important property
which Q does not. This is known as the least upper bound property. To understand the least upper bound
property, and its mirror-image twin, the greatest lower bound property, we first need to define bounded.

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