Complex Numbers Worksheet Academic | PDF | Complex Number | Numbers - Free Printable

Let’s work through each problem step by step.

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Problem 1:
For the complex number $-10 + 4i$, identify the real number and the imaginary number.

→ A complex number is written as $a + bi$, where:
- $a$ is the real part
- $b$ is the imaginary part (note: it’s just the coefficient, not including the $i$)

So for $-10 + 4i$:
- Real part = $-10$
- Imaginary part = $4$

✔ Final Answer for #1: Real = -10, Imaginary = 4

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Problem 2: Evaluate powers of $i$

Recall:
- $i^1 = i$
- $i^2 = -1$
- $i^3 = i^2 \cdot i = -1 \cdot i = -i$
- $i^4 = (i^2)^2 = (-1)^2 = 1$
- Then it repeats every 4: $i^5 = i$, $i^6 = -1$, etc.

So:

a) $i^2 = -1$
b) $i^3 = -i$
c) $i^4 = 1$
d) $i^9$: Divide 9 by 4 → remainder 1 → so same as $i^1 = i$
e) $i^{14}$: 14 ÷ 4 = 3 remainder 2 → same as $i^2 = -1$

✔ Final Answers for #2:
a) -1
b) -i
c) 1
d) i
e) -1

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Problem 3: Write expressions as complex numbers in standard form ($a + bi$)

We’ll simplify each expression.

a) $(5 + 2i) + (3 - 2i)$
→ Combine real parts: $5 + 3 = 8$
→ Combine imaginary parts: $2i - 2i = 0i$
→ Answer: $8 + 0i$ or just $8$

b) $-i + (7 - 5i) - 3(2 - 3i)$
First distribute the -3:
= $-i + 7 - 5i -6 + 9i$
Now combine real: $7 - 6 = 1$
Imaginary: $-i -5i + 9i = 3i$
→ Answer: $1 + 3i$

c) $(-2 + 4i) + (3 - 9i)$
Real: $-2 + 3 = 1$
Imaginary: $4i - 9i = -5i$
→ Answer: $1 - 5i$

d) $(-2 + 4i) - (3 + 9i)$
Distribute the minus: $-2 + 4i -3 -9i$
Real: $-2 -3 = -5$
Imaginary: $4i - 9i = -5i$
→ Answer: $-5 - 5i$

e) $(5 - 2i) - 2(3 + i)$
Distribute the -2: $5 - 2i -6 -2i$
Real: $5 - 6 = -1$
Imaginary: $-2i -2i = -4i$
→ Answer: $-1 - 4i$

f) $3i(6 - 5i)$
Distribute: $3i \cdot 6 = 18i$, $3i \cdot (-5i) = -15i^2$
But $i^2 = -1$, so $-15(-1) = 15$
→ Total: $15 + 18i$

g) $i(2 + i)$
Distribute: $i \cdot 2 = 2i$, $i \cdot i = i^2 = -1$
→ Total: $-1 + 2i$

h) $(2 + 3i)(1 - 4i)$
Use FOIL:
First: $2 \cdot 1 = 2$
Outer: $2 \cdot (-4i) = -8i$
Inner: $3i \cdot 1 = 3i$
Last: $3i \cdot (-4i) = -12i^2 = -12(-1) = 12$
Combine: Real: $2 + 12 = 14$; Imaginary: $-8i + 3i = -5i$
→ Answer: $14 - 5i$

i) $(-3 + 7i)(1 - 2i)$
FOIL:
First: $-3 \cdot 1 = -3$
Outer: $-3 \cdot (-2i) = 6i$
Inner: $7i \cdot 1 = 7i$
Last: $7i \cdot (-2i) = -14i^2 = 14$
Real: $-3 + 14 = 11$
Imaginary: $6i + 7i = 13i$
→ Answer: $11 + 13i$

j) $(3 - 2i)^2$
This is $(3 - 2i)(3 - 2i)$
FOIL:
First: $3 \cdot 3 = 9$
Outer: $3 \cdot (-2i) = -6i$
Inner: $-2i \cdot 3 = -6i$
Last: $-2i \cdot -2i = 4i^2 = 4(-1) = -4$
Real: $9 - 4 = 5$
Imaginary: $-6i -6i = -12i$
→ Answer: $5 - 12i$

k) $(2i)(1 - 4i)(1 + i)$
Multiply two first: let’s do $(1 - 4i)(1 + i)$
FOIL:
$1 \cdot 1 = 1$
$1 \cdot i = i$
$-4i \cdot 1 = -4i$
$-4i \cdot i = -4i^2 = 4$
Real: $1 + 4 = 5$
Imaginary: $i - 4i = -3i$
So $(1 - 4i)(1 + i) = 5 - 3i$
Now multiply by $2i$:
$2i(5 - 3i) = 10i - 6i^2 = 10i -6(-1) = 10i + 6$
→ Answer: $6 + 10i$

✔ Final Answers for #3:
a) 8
b) 1 + 3i
c) 1 - 5i
d) -5 - 5i
e) -1 - 4i
f) 15 + 18i
g) -1 + 2i
h) 14 - 5i
i) 11 + 13i
j) 5 - 12i
k) 6 + 10i

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Problem 4: Solve each equation

a) $x^2 = -60$
Take square root: $x = \pm \sqrt{-60} = \pm i\sqrt{60}$
Simplify $\sqrt{60} = \sqrt{4 \cdot 15} = 2\sqrt{15}$
→ Answer: $x = \pm 2i\sqrt{15}$

b) $4x^2 + 20 = 0$
Subtract 20: $4x^2 = -20$
Divide by 4: $x^2 = -5$
→ $x = \pm \sqrt{-5} = \pm i\sqrt{5}$

c) $6x^2 + 1 = -5$
Subtract 1: $6x^2 = -6$
Divide by 6: $x^2 = -1$
→ $x = \pm i$

d) $3(x - 1)^2 = -27$
Divide by 3: $(x - 1)^2 = -9$
Take square root: $x - 1 = \pm \sqrt{-9} = \pm 3i$
Add 1: $x = 1 \pm 3i$

e) $(x + 5)^2 + 10 = 2$
Subtract 10: $(x + 5)^2 = -8$
Take square root: $x + 5 = \pm \sqrt{-8} = \pm i\sqrt{8} = \pm 2i\sqrt{2}$
Subtract 5: $x = -5 \pm 2i\sqrt{2}$

f) $5(2x + 8)^2 = -80$
Divide by 5: $(2x + 8)^2 = -16$
Take square root: $2x + 8 = \pm \sqrt{-16} = \pm 4i$
Subtract 8: $2x = -8 \pm 4i$
Divide by 2: $x = -4 \pm 2i$

✔ Final Answers for #4:
a) $x = \pm 2i\sqrt{15}$
b) $x = \pm i\sqrt{5}$
c) $x = \pm i$
d) $x = 1 \pm 3i$
e) $x = -5 \pm 2i\sqrt{2}$
f) $x = -4 \pm 2i$

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Final Answer:
1. Real: -10, Imaginary: 4
2. a) -1, b) -i, c) 1, d) i, e) -1
3. a) 8, b) 1+3i, c) 1-5i, d) -5-5i, e) -1-4i, f) 15+18i, g) -1+2i, h) 14-5i, i) 11+13i, j) 5-12i, k) 6+10i
4. a) ±2i√15, b) ±i√5, c) ±i, d) 1±3i, e) -5±2i√2, f) -4±2i