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Properties of Complex Number Arithmetic

Properties of Complex Number Arithmetic

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 »  Complex modulus of sum is less than or equal to sum of moduluses

    →  `|z_1 + z_2| <= |z_1| + |z_2|`

Modulus in addition

plain and simple summary

nub

plain and simple summary

nub

dummy

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

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trek

simple steps to build the foundation

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


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Starting on learning "Modulus in Addition". ;; In this page you will learn that the modulus of sum is less than or equal to the sum of modulus.

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| `

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.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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| `

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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Given z1 = a1+i b1 and z2 = a2+i b2, what is the modulus of sum z1 + z2?
multiplied
less than or equal to mod z1 multiplied by mod z2
plus
less than or equal to mod z1 plus mod z2
The answer is 'less than or equal to mod z1 plus mod z2'
mod z1 + z2 less than equal to mod z1 + mod z2 is proven with the representation on complex plane. z1, z2, z1+z2 form a triangle. Length of the sides of the triangle are mod z1, mod z2, mod z1+z2. 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 z1, z2 in complex numbers mod of z1 + z2 less than or equal to mod of z1 + mod of z2
Given z1 = a1+i b1 and z2 = a2+i b2 what is the modulus of difference z1 minus z2?
minus;-
less than or equal mod z1 minus mod z2
plus;+
less than or equal mod z1 + mod z2
The answer is 'less than or equal mod z1 + mod z2'. Subtraction is the inverse of addition. So, modulus of difference z1 minus z2 equals modulus of z1 plus, minus z2. Note : modulus of minus z2 is same as modulus z2.

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