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(i) Show that for any real numbers a , b , p and q ,
( ap bq ) ( a b )( p q ).
(ii) By taking the square root of the inequality in (i) deduce that
( a p )
( b q )
a b p q
.
Thus AP OA OP .
Summary
-
results on absolute value
Definition When 0 a , a a . When a 0, a a .
Multiplication
ab a
b
·
qn 58
Triangle inequality
qn 61
a b a b
Theorem
qn 63
a b a b .
Historical note
Early steps towards the arithmetical axiomatisation of the number
system were taken by G. Peacock in 1830 and 1845, with his principle
of permanence of forms. Precise arguments assuming the total ordering
of the positive real numbers pervade Euclid's Elements . Thenotation
was introduced by Harriot and first appeared in print in 1631.
Thesymbols and were first used in France about one hundred
years later. Bernoulli's inequality was proved by Jakob Bernoulli in
1689. The arithmetic mean geometric mean inequality appears in
Euclid's Elements (II.5, V.25 and lemma between X.59 and X.60). The
general form of this inequality, for n positive numbers, was established
by C. Maclaurin in 1729. The sequence (1 1/ n ) was first investigated
by L. Euler (1736) who named its limit with his own initial. The
notation for absolutevalue x was due to Weierstrass who used it in
his lectures in Berlin from 1859. It did not appear in print until 1877.
Bolzano (1817) wrote x 1, where we would write x 1. Harnack
(1881) wroteabs m and Jordan (1882) wrotemod m for m .
and
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