For any two complex numbers z1 and z2, prove that Re (z1z2) = Re z1 Re z2-Im z1 Im z2.
Solution and Explanation
Let z1=x1+iy1 and z2=x2+iy2
∴ z1z2=(x1+iy1) (x2+iy2)
= x1(x2+iy2)+iy1(x2+iy2)
= x1x2+ix1y2+iy1x2+i2y1y2
= x1x2+ix1y2+iy1x2--y1y2 [i2=-1]
⇒ (x1x2-y1y2)+i(x1y2+y1x2)
⇒ Re(z1z2)=Re z1Re Z2-Imz1 ImZ2
Hence, proved.
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Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.