Mod(ab) = Mod(a)*Mod(b) please help me prove this for all real numbers
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In what context are you using Mod(ab)? Do you mean the absolute value of ab - ie |ab|.
I ask this because the term Mod(x) is used in modular arithmetic which is a tool in elementary number theory
- a field involving integers only. A confusion arises with the term because in number theory, mod(x) is read as modulo x (Here, x is some positive integer), whereas |x| is read as the modulus of x- ie the absolute value of some real number x. I find that it is better to use the |x|, but you might have been taught to use the Mod(x) notation in this regard, which I suppose is fine if the context is clear.
On the assumption that you mean |ab|, note that for all real numbers x
|x| = x, if x >= 0
|x| = -x if x < 0
Also
|-x| = |x| >=0
So |x| >=0 for all real number x. Likewise, |-x| >=0
With this in mind
|a| * |b| = |-a| * |-b| >=0.
Let A = |a| and B = |b|
So A and B are non-negative reals. This takes care of the possibility that either a or b may be negative.
So |a| * |b| = AB >=0
Since we are multiplying the non-negative modulus of the product of a and b
(irrespective of whether either of them is less than zero of not), we must have that
|ab| >=0
But |ab| is the ABSOLUTE value of the product of a and b. That is,
|ab| = AB
Hence
|ab| = |a| * |b|
In other words, the modulus of the product is equal to the product of the moduli
I ask this because the term Mod(x) is used in modular arithmetic which is a tool in elementary number theory
- a field involving integers only. A confusion arises with the term because in number theory, mod(x) is read as modulo x (Here, x is some positive integer), whereas |x| is read as the modulus of x- ie the absolute value of some real number x. I find that it is better to use the |x|, but you might have been taught to use the Mod(x) notation in this regard, which I suppose is fine if the context is clear.
On the assumption that you mean |ab|, note that for all real numbers x
|x| = x, if x >= 0
|x| = -x if x < 0
Also
|-x| = |x| >=0
So |x| >=0 for all real number x. Likewise, |-x| >=0
With this in mind
|a| * |b| = |-a| * |-b| >=0.
Let A = |a| and B = |b|
So A and B are non-negative reals. This takes care of the possibility that either a or b may be negative.
So |a| * |b| = AB >=0
Since we are multiplying the non-negative modulus of the product of a and b
(irrespective of whether either of them is less than zero of not), we must have that
|ab| >=0
But |ab| is the ABSOLUTE value of the product of a and b. That is,
|ab| = AB
Hence
|ab| = |a| * |b|
In other words, the modulus of the product is equal to the product of the moduli