Spivak, Chapter 1 Exercises
1
Spivak Calculus, Chapter 1 Exercises 1. Prove the following: i. If ax = a for some number a 6= 0, then x = 1. Since a 6= 0, there exists a multiplicative inverse a -1 , such that a -1 ax = aa -1 x = 1 ii. x 2 - y 2 = ( x - y)( x + y). x 2 - y 2 = x 2 + xy - xy - y 2 = x( x + y) + (- y)( x + y) = [ x + (- y)]( x + y) = ( x - y)( x + y) iii. If x 2 = y 2 , then x = y or x =- y. | x|= p x 2 1
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Mathematical document with solutions to chapter one exercises of Spivak
Transcript of Spivak, Chapter 1 Exercises
Spivak Calculus, Chapter 1 Exercises
1. Prove the following:
i. If ax = a for some number a 6= 0, then x = 1.
Since a 6= 0, there exists a multiplicative inverse a−1, such that
a−1ax = aa−1
x = 1
ii. x2 − y2 = (x− y)(x+ y).
x2 − y2 = x2 + xy− xy− y2
= x(x+ y)+ (−y)(x+ y)
= [x+ (−y)](x+ y)
= (x− y)(x+ y)
iii. If x2 = y2, then x = y or x =−y.
|x| =√
x2
1
