SME
Grade 11 Math · Problem type
Add or subtract complex numbers
Use this for “(4 − 2i) − (−1 + 5i)”.
Diagram
z − w = (4 − 2i) − (−1 + 5i) = 5 − 7i: z plus −w.
This is an example. Type your own values to start a new problem.
(
a
+
(
b
)
i)
(
c
+
d
i)
=
p
+
(
q
)
i
C
Real part added
worked out (not typed)
D
Imaginary part added
worked out (not typed)
Formulas
C
=
c
when
s
is
+;
C
=
−
c
when
s
is
−
1 = −1 when − is +; 1 = −(−1) when − is −
D
=
d
when
s
is
+;
D
=
−
d
when
s
is
−
−5 = 5 when − is +; −5 = −5 when − is −
p
=
a
+
C
5 = 4 + 1
q
=
b
+
D
−7 = −2 + (−5)
Assumptions
•
Combine real parts with real parts and imaginary parts with imaginary parts.
•
Subtracting c + di is adding its opposite, −c − di: the picture draws that opposite and adds it.
•
On the plane, adding is joining the arrows end to end.
Step-by-step
WE KNOW
a = 4, b = −2, s: −, c = −1, d = 5
FIND
real part added (C), imaginary part added (D), real part of the answer (p), imaginary part of the answer (q)
Step 1 · Find real part added (C)
Use C = c when s is +; C = −c when s is −
Subtracting: change the sign of the second real part.
C
=
−
c
C
=
−(−1)
C
=
1
Step 2 · Find imaginary part added (D)
Use D = d when s is +; D = −d when s is −
Subtracting: change the sign of the second imaginary part.
D
=
−
d
D
=
−5
Step 3 · Find real part of the answer (p)
Use p = a + C
Real parts together.
p
=
4
+
1
p
=
5
Step 4 · Find imaginary part of the answer (q)
Use q = b + D
Imaginary parts together.
q
=
−2
−
5
(4 − 2i) − (−1 + 5i) = 5 − 7i
q
=
−7
Check
Put the numbers back in. Both sides should match.
1 = −(−1) ✓
−5 = −5 ✓
5 = 4 + 1 ✓
−7 = −2 − 5 ✓
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Complex numbers and quadratic equations with complex solutions
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Complex solutions of a quadratic
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