COMPLEX NUMBERS · HL
Complex Numbers 5 — Equality
Equating real and imaginary parts to find unknowns.
Section 1 of 2
Equality
Rule
1.Real $=$ Real
2.Img $=$ Img
$2 + 3i + p + qi = 7 + 9i$. Find $p$ and $q$.
$2 + 3i + p + qi = 7 + 9i$
Real: $2 + p = 7 \ \Rightarrow \ p = 5$
Img: $3 + q = 9 \ \Rightarrow \ q = 6$
$a + bi + 5 + 7i = 11 + 19i$. Find $a$ and $b$.
$a + bi + 5 + 7i = 11 + 19i$
Real: $a + 5 = 11$
$a = 6$
Img: $b + 7 = 19$
$b = 12$
Section 2 of 2
Equality with simultaneous equations
$p + qi + 2q + pi = 3 + 2i$. Find $p$ and $q$.
$p + qi + 2q + pi = 3 + 2i$
Group real (R) and imaginary (I):
Real: $p + 2q = 3$
Img: $p + q = 2$
Subtract the equations:
$\begin{aligned} p + 2q &= 3 \\ -p - q &= -2 \\ \hline q &= 1 \end{aligned}$
Back into $p + q = 2$:
$p + 1 = 2$
$p = 1$
SUM
The lot in one box
Equality toolkit
1.Real $=$ Real, Img $=$ Img.
2.Match the parts to get two equations.
3.If both unknowns appear in both parts, solve as simultaneous equations.
End of lesson
Complex Numbers 5 — Equality · HL · Mathslive.ie