Algebra 2 Complex Numbers: Adding, Subtracting and Multiplying Lesson (Free PDF)

Algebra 2 math lesson: Complex Numbers: Adding, Subtracting and Multiplying – Maths PDF

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This algebra 2 complex numbers lesson for Algebra 2 covers complex numbers: adding, subtracting and multiplying (Common Core N-CN.A.1, N-CN.A.2), with a free PDF to print.

A complex number has the form a + bi, where a and b are real and i is the imaginary unit with i squared equal to -1. Add and subtract by combining real parts and imaginary parts separately. Multiply with the distributive property, then replace each i squared with -1.

Algebra 2 · Common Core State Standards: N-CN.A.1, N-CN.A.2

Key facts in this algebra 2 complex numbers lesson

Key facts

The imaginary unit i satisfies i² = -1, so √(-n) = i√n for n > 0.
A complex number a + bi has real part a and imaginary part b.
Add or subtract real parts and imaginary parts separately.
Multiply by distributing, then replace i² with -1.
Powers of i repeat: i, -1, -i, 1.

What is the imaginary unit i?

No real number squares to a negative, so mathematicians define a new number that does. This number lets us take square roots of negative numbers and solve equations like \(x^{2} + 1 = 0\).

Imaginary unit and complex numbers

The imaginary unit is \(i\), defined by \(i^{2} = -1\), so \(i = \sqrt{-1}\). A complex number has the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. For a positive number \(n\), \(\sqrt{-n} = i\sqrt{n}\).

The complex number 3 + 2i is the point (3, 2): real part on the horizontal axis, imaginary part on the vertical axis.

Every real number is also complex, because \(5 = 5 + 0i\). The number \(-4i\) has real part 0 and imaginary part \(-4\). For example, \(\sqrt{-36} = 6i\) and \(\sqrt{-20} = 2i\sqrt{5}\).

How do the powers of i behave?

Start with \(i^{1} = i\) and \(i^{2} = -1\). Then \(i^{3} = i^{2} \cdot i = -i\) and \(i^{4} = (i^{2})^{2} = 1\). After that the pattern repeats every four powers.

The powers of i repeat in a cycle of length 4.

To find \(i^{n}\), divide \(n\) by 4 and use the remainder. For \(i^{27}\), \(27 = 4 \cdot 6 + 3\), so \(i^{27} = i^{3} = -i\).

How do you add and subtract complex numbers?

Add or subtract the real parts and the imaginary parts separately, just like combining like terms. For \((3 + 2i) + (1 + 4i)\) the real parts give \(3 + 1 = 4\) and the imaginary parts give \(2 + 4 = 6\), so the sum is \(4 + 6i\). On the plane, the sum is the far corner of the parallelogram built from the two numbers.

Adding complex numbers is vector addition: 3 + 2i and 1 + 4i give 4 + 6i.

How do you multiply complex numbers?

Use the distributive property, then replace \(i^{2}\) with \(-1\). Follow these steps for \((2 + 3i)(4 – i)\).

Multiply each term in the first factor by each term in the second.
Replace \(i^{2}\) with \(-1\).
Combine the real parts and the imaginary parts.

Example: \((2 + 3i)(4 – i) = 8 – 2i + 12i – 3i^{2} = 8 + 10i + 3 = 11 + 10i\).

The four partial products 8, -2i, 12i and -3i² add to 11 + 10i.

The set of complex numbers is closed under addition, subtraction, and multiplication, and it keeps the commutative, associative, and distributive properties.

Algebra 2 complex numbers lesson: questions and answers

What is i in math?

The letter i stands for the imaginary unit, defined by i squared equals negative 1. It lets you write square roots of negative numbers, such as the square root of -9 equals 3i.

What is the difference between real and imaginary numbers?

A real number is a point on the ordinary number line. An imaginary number has the form bi with b real and b not zero. A complex number a + bi combines a real part and an imaginary part.

How do you simplify powers of i?

Divide the exponent by 4 and keep the remainder. A remainder of 0, 1, 2, or 3 gives 1, i, -1, or -i. For example, i to the 50th has remainder 2, so it equals -1.

Why does i squared equal -1 when multiplying complex numbers?

Because i squared is defined as -1. After you distribute, every i-squared term becomes a real number, and the final answer is written in a + bi form.

Can a complex number be graphed?

Yes. In the complex plane the horizontal axis shows the real part and the vertical axis shows the imaginary part, so a + bi is plotted at the point (a, b).

How to use this algebra 2 complex numbers lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

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