Answer to Question #3876 in Algebra for junel

Question

If a,b ЄR and b≠0 show that:
a. |a|= √a[sup]2[/sup]
b. |a/b|= |a|/|b|

Answer 1

a. Show that |a|= √a²

Recall that
|a|=a if a>=0
and
|a|=-a otherwise.

Let a>=0. Then
sqrt{a^2} = a = |a|

Suppose a<0. Then a=-b, where b>0, so |a|=b.
On the other hand,
√a² = √(-b)² = √b² = b = |a|

b. Show that |a/b|= |a|/|b|

Consider the following cases:
1) Let a>=0, b>0.
Then |a/b|= a/b = |a|/|b|.

2) Suppose a>=0, b<0.
Then a/b<0, |a|=a, |b|=-b, so
|a/b|= -(a/b) = a/(-b) = |a|/|b|.

3) Suppose a<0, b>0.
Then a/b<0, |a|=-a, |b|=b, so
|a/b|= -(a/b) = (-a)/b = |a|/|b|.

3) Finally, let a<0, b<0.
Then a/b>0, |a|=-a, |b|=-b, so
|a/b|= a/b = (-a)/(-b) = |a|/|b|.