Complex numbersEasy
For z₁ = −7 + j and z₂ = 4 + j2, find the real part of the product z₁·z₂.
z₁ = −7 + jz₂ = 4 + j2
Find Re(z₁z₂)
Show hints
- Multiply out the brackets like (a + b)(c + d), giving four terms.
- Replace j² by −1 so that one of the four terms becomes real.
- (−7 + j)(4 + j2) = (−28) + j(−14) + j4 + j²·2
Show the answer and the step-by-step solution
Complex numbers: z = a + jb has real part a and imaginary part b, with j² = −1 (mathematicians write i). Add and subtract part by part; multiply out the brackets and replace j² by −1; divide by multiplying the top and bottom by the conjugate of the denominator, z̄ = a − jb, because z·z̄ = a² + b² is real. On the Argand diagram z is the point (a, b): its distance from the origin is the modulus |z| = √(a² + b²), and the angle from the positive real axis is the argument θ.
Multiply out
(−7 + j)(4 + j2) = (−7)·4 + j·(−7)·2 + j·1·4 + j²·1·2
Use j² = −1
Re = (−28) − 2 = −30
Im = (−14) + 4 = −10
z₁z₂ = −30 − j10
Re(z₁z₂) = −30
Answer
Re(z₁z₂) = −30