Complex Number

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Complex Number: meaning, method and solved example (Algebra 2) – Maths PDF

In short: A complex number has the form a + bi, combining a real part a and an imaginary part b with the imaginary unit i, where i² = −1.

Algebra 2Numbers and OperationsN-CN.A.1

What is Complex Number?

A complex number bundles two ordinary numbers into one: \(a + bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i^2 = -1\). Every real number is a special case with \(b = 0\), and every pure imaginary number has \(a = 0\).

Treat \(i\) like a variable and replace \(i^2\) by \(-1\). Addition combines like parts: \((a + bi) + (c + di) = (a + c) + (b + d)i\). On the complex plane, \(a + bi\) is the point \((a, b)\), with the horizontal axis real and the vertical axis imaginary.

How to spot it

  • A quadratic equation has solutions of the form \(2 \pm 3i\).
  • A problem says to write the answer in \(a + bi\) form.
  • Two expressions with \(i\) must be added, subtracted, multiplied or divided.
  • A point on a plane is labeled with a real axis and an imaginary axis.

The method

  1. To add or subtract, combine real parts together and imaginary parts together.
  2. To multiply, expand with FOIL as with binomials, then replace each \(i^2\) with \(-1\).
  3. Collect the real terms and the imaginary terms to write the result as \(a + bi\).
  4. To plot, place the real part along the horizontal axis and the imaginary part along the vertical axis.
z = 3 + 2iw = 4 – 5iz + w = 7 – 3irealimaginary
Each number a + bi is the point (a, b); adding two of them adds their coordinates.

Complex Number: solved example

Problem

Let \(z = 3 + 2i\) and \(w = 4 – 5i\). Find \(z + w\) and \(zw\).

Solution

Sum: \((3 + 4) + (2 – 5)i = 7 – 3i\).

Product: \((3 + 2i)(4 – 5i) = 12 – 15i + 8i – 10i^2 = 12 – 7i + 10 = 22 – 7i\), since \(-10i^2 = +10\). The results are \(7 – 3i\) and \(22 – 7i\).

Try it

Question 1

Compute \((5 – 3i) + (-2 + 9i)\).

Show the answer

Real parts: \(5 – 2 = 3\). Imaginary parts: \(-3 + 9 = 6\). The sum is \(3 + 6i\).

Question 2

Name the real part and the imaginary part of \(7 – 4i\).

Show the answer

The real part is 7 and the imaginary part is \(-4\), the coefficient of \(i\).

Complex Number vs. Real Numbers

Real numbers sit on a single number line, and none of them squares to a negative. A complex number lives on a plane with two directions. Every real number such as 6 is also complex, written \(6 + 0i\), but a number like \(6 + 2i\) is not real because its imaginary part is not zero.

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