The distance from the origin to the point

Petrolovujhm

Petrolovujhm

Answered question

2022-04-02

The distance from the origin to the point that corresponds to a complex number is known as the modulus of that complex number. There is an interesting relationship involving the moduli of two complex numbers and the modulus of their product. Spread the work around everyone try a few examples and try to figure out what this relationship is.

Answer & Explanation

Laylah Hebert

Laylah Hebert

Beginner2022-04-03Added 15 answers

Step 1
The absolute value ( modulus ) gives the distance from the origin to that point z , where z represents the complex number z=x+iy. The modulus of any complex number can be evaluated by adding the squares of real part and imaginary part of the complex number and by taking the square root of added values.
The modulus of the complex number z=x+iy is calculated by using the formula |z|=x2+y2, where x is the real part of the complex number and y is the imaginary part of the complex number.
Step 2
Consider two complex numbers z1=a+ib and z2=c+id, now let us check the relationship between the moduli of two complex number and modulus of their product.
|z1|=a2+b2 by using the modulus of complex number.
|z2|=c2+d2
Now we can check the modulus of their product
|z1×z2|=|(a+ib)(c+id)|
=|ac+iad+ibc+i2bd|
=ac+iad+ibcbd}
[using i2=1]
=|(acbd)+i(ad+bc)|
Now using the modulus of complex number form
|z1×z2|=(acbd)2+(ad+bc)2
=(ac)2+(bd)22abcd+(ad)2+(bc)2+2abcd
=a2c2+b2d2+b2c2
=c2(a2+b2)+d2(a2+b2)
|z1×z2|=(a2+b2)(c2+d2)
[using the formula (a+b)2=a2+2ab+b2]
Now take product of moduli of two complex numbers z1z2, we get
|z1|×|z2|=a2+b2×c2+d2
[using a×b=a×b]
|z1|×|z2|=(a2+b2)(c2+d2)
Hence the relation between the moduli of two complex numbers and their modulus of product is they are equal.
Hence we have |z1|×|z2|=|z1×z2|
Let us check with an example
z1=1+2i
z2=3+4i
then we get
|z1|=12+22=5
|z2|=32+42
=9+16
|z2|=25=5
simplify the terms by taking square root |z1|×|z2|=55 value
Now consider the modulus of their product
|z1×z2||(1+2i)(3+4i)|
=|3+4i+6i+8i2|
=|3+10i8|
=|5+10i|
(5)2+(10)2
[using i2=1]
=25+100
=125
=5×5×5
|z1×z2|=55
Hence we have |z1||z2|=|z1×z2|

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