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mathsComplex NumbersProperties of Complex Number Arithmetic

### Modulus in addition

Modulus of sum is less than or equal to the sum of modulus.

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Given z_1 = a_1+ib_1 and z_2 = a_2+ib_2, what is the modulus of sum |z_1 + z_2|?

• <=|z_1| xx |z_2|
• <=|z_1| + |z_2|
• <=|z_1| + |z_2|

The answer is '<=|z_1| + |z_2|'.

|z_1 + z_2| <= |z_1| + |z_2| is proven with the representation on complex plane. z_1, z_2, z_1+z_2 form a triangle. Length of the sides of the triangle are |z_1|, |z_2|, |z_1+z_2|. Sum of any two sides of the triangle is greater than the third side.

•  Modulus of sum is less than or equal to the sum of modulus.

Modulus of Sum: For complex numbers z_1, z_2 in CC
|z_1 + z_2| <= |z_1| + |z_2|

Solved Exercise Problem:

Given z_1 = a_1+ib_1 and z_2 = a_2+ib_2 what is the modulus of difference |z_1 - z_2|?

• <=|z_1| - |z_2|
• <=|z_1| + |z_2|
• <=|z_1| + |z_2|

The answer is '<=|z_1| + |z_2|'. Subtraction is the inverse of addition. So, |z_1 - z_2| = |z_1 + (-z_2)|. <= |z_1|+|-z_2| <= |z_1| + |z_2|. Note : |-z_2| = |z_2|

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