# Proof of Multiplicative Property of Absolute Values

• coverband
In summary, there are multiple proofs for the multiplicative property of absolute values. One approach is to break it into cases, where if a and b are both positive, then |ab| = |a||b|. Another approach is to use the property that |x| = sqrt(x^2), which leads to the proof |ab| = sqrt((ab)^2) = sqrt(a^2 b^2) = sqrt(a^2) sqrt(b^2) = |a||b|. The conversation also touches on the concept of negative absolute values and the confusion it can cause. Overall, the multiplicative property of absolute values can be proven using various methods.
coverband
Hi does anyone know a proof for the multiplicative propery of absolute values

i.e. Prove |ab|=|a||b|

CosbyWest
How about doing exactly what you always do with absolute values: break it into cases.

1) If $a\ge 0$ and $b\ge 0$
Then $ab\ge 0$ so |ab|= ab while |a|= a, |b|= b. ab= (a)(b) so |ab|= |a||b|.

2) If $a\ge 0$ while b< 0
The $ab\le 0$ so |ab|= -ab while |a|= a, |b|= -b. -ab= (a)(-b) so |ab|= |a||b|.

Can you do the other two cases?

Thanks halls!

My book uses the following proof:

$$\left| {ab} \right| = \sqrt {(ab)^2 } = \sqrt {a^2 b^2 } = \sqrt {a^2 } \sqrt {b^2 } = \left| a \right|\left| b \right|$$

Well, if you want to do it the easy way!

I still find |a|=-a when a<0 weird! Surely if a = -a, |-a| = a

When a<0, -a is positive.

coverband said:
I still find |a|=-a when a<0 weird! Surely if a = -a, |-a| = a
Yes that's true. Because if a= -a, then a= 0!

Are you sure that's what you meant to say?

Big-T said:
When a<0, -a is positive.

Yeah I think when you look at the graph of y=|x| it becomes clear (as mud)!

## What is the multiplicative property of absolute values?

The multiplicative property of absolute values states that the absolute value of the product of two numbers is equal to the product of their absolute values.

## How is the multiplicative property of absolute values proven?

The multiplicative property of absolute values can be proven using algebraic manipulations and the definition of absolute value.

## Why is the multiplicative property of absolute values important?

The multiplicative property of absolute values is important because it allows us to simplify mathematical expressions involving absolute values and solve equations involving absolute values more easily.

## Can the multiplicative property of absolute values be extended to more than two numbers?

Yes, the multiplicative property of absolute values can be extended to any number of numbers. The absolute value of the product of multiple numbers is equal to the product of their absolute values.

## Does the multiplicative property of absolute values hold for complex numbers?

Yes, the multiplicative property of absolute values holds for complex numbers. The absolute value of the product of two complex numbers is equal to the product of their absolute values.

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