266 PROPERTIES OF THE INTEGERS [CHAP. 11
Absolute Value
Theabsolute valueof an integera. written|a|, is formally defined by|a|={
a ifa≥ 0
−a ifa< 0Accordingly,|a|>0 except whena=0. Geometrically speaking,|a|may be viewed as the distance between
the pointsaand 0 on the number lineR. Also,|a−b|=|b−a|may be viewed as the distance between the
pointsaandb. For example:
(a)|− 3 |= 3 ;| 7 |= 7 ;|− 13 |= 13 ; (b)| 2 − 7 |=|− 5 |= 5 ;| 7 − 2 |=| 5 |= 5Some properties of the absolute value function follow. (Problems 11.6 and 11.7 prove (iii) and (iv).)
Proposition 11.4: Letaandbbe any integers. Then:
(i) |a|≥ 0 ,and|a|=0iffa= 0 (iv) |a±b|≤|a|+|b|
(ii) −|a|≤a≤|a| (v) ||a|−|b||≤|a±b|
(iii) |ab|=|a||b|11.3Mathematical Induction
The principle of mathematical induction stated below essentially asserts that the positive integersNbegin
with the number 1 and the rest are obtained by successively adding 1. That is, we begin with 1, then 2= 1 +1,
then 3= 2 +1, then 4= 3 +1, and so on. The principle makes precise the vague phrase “and so on.”
Principle of Mathematical Induction:LetSbe a set of positive integers with the following two properties:
(i) 1 belongs toS.(ii) Ifkbelongs toS, thenk+1 belongs toS.ThenSis the set of all positive integers.
We shall not prove this principle. On the contrary, when the setNof positive integers (natural numbers) is
developed axiomatically, this principle is given as one of the axioms.
There is an equivalent form of the above principle which is usually used when proving theorems:
Principle of Mathematical Induction:LetPbe aproposition defined on the integersn≥1 such that:
(i) P( 1 )is true.(ii) P(k+ 1 )is true wheneverP(k)is true.ThenPis true for every integern≥1.
EXAMPLE 11.1
(a) LetPbe the proposition that the sum of the firstnodd numbers isn^2 ; that is:P (n): 1 + 3 + 5 +···+( 2 n− 1 )=n^2(Thenth odd number is 2n−1 and the next odd number is 2n+1.)
Clearly,P (n)is true forn=1; that is:
P( 1 ): 1 = 12