COMPLEX NUMBERS · HL
Complex Numbers 3 — Division and Argand Diagram
Dividing complex numbers and plotting on the Argand diagram.
Section 1 of 2
Division
Rule
1.Multiply above and below by the conjugate of the bottom.
Simplify (i) $\dfrac{2 + 3i}{5 + 6i}$
Bottom $5 + 6i \ \Rightarrow$ Conjugate $5 - 6i$
$\dfrac{2 + 3i}{5 + 6i} \cdot \dfrac{5 - 6i}{5 - 6i}$
Top:
$(2 + 3i)(5 - 6i)$
$2(5 - 6i) + 3i(5 - 6i)$
$10 - 12i + 15i - 18i^2$
$10 + 3i + 18 = 28 + 3i$
Bottom:
$(5 + 6i)(5 - 6i)$
$5(5 - 6i) + 6i(5 - 6i)$
$25 - 30i + 30i - 36i^2$
$25 + 36 = 61$
$\dfrac{28 + 3i}{61}$
Simplify $\dfrac{3 - 2i}{7 + 3i}$
Bottom is $7 + 3i$. Conjugate is $7 - 3i$.
$\dfrac{3 - 2i}{7 + 3i} \cdot \dfrac{7 - 3i}{7 - 3i}$ (Top · Top, Bottom · Bottom)
Top:
$(3 - 2i)(7 - 3i)$
$3(7 - 3i) - 2i(7 - 3i)$
$21 - 9i - 14i + 6i^2$
$21 - 23i - 6 = 15 - 23i$
Source flag: pages 2–3 of this PDF scanned blank, so the bottom and final answer for this one are missing. See chat note.
Section 2 of 2
Argand Diagram
$z = 3 + 2i, \qquad w = -4 - i$
Plot on Argand diagram
$z = -2 + 3i, \qquad w = 4 - 2i$
Plotting an Argand diagram
1.This is the same as the line where we have $x$ and $y$ axes, but now we have real and imaginary instead.
2.Real is first: go left ($-$) or right ($+$).
3.Imaginary second: up ($+$) or down ($-$).
SUM
The lot in one box
Division and Argand toolkit
1.Division: multiply above and below by the conjugate of the bottom.
2.Argand: real along the horizontal axis, imaginary up the vertical axis.
3.Real first (left/right), imaginary second (up/down).
End of lesson
Complex Numbers 3 — Division and Argand Diagram · HL · Mathslive.ie