Imaginary Numbers (Intro)

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Understand why imaginary numbers were introduced.
  • Interpret \(i\) as the square root of \(-1\).
  • Rewrite negative square roots using \(i\).
  • Distinguish between real, imaginary, and complex numbers.

Key Ideas

Imaginary numbers arise when we try to take the square root of a negative number.

Since no real number satisfies
\[ x^2 = -1, \]
we define a new number:

\[ i = \sqrt{-1}. \]

This leads to:

  • \(\sqrt{-9} = 3i\)
  • \(\sqrt{-25} = 5i\)
  • \(\sqrt{-a} = i\sqrt{a}\) for any positive \(a\)

A pure imaginary number looks like \(bi\), where \(b\) is real.

A complex number has the form
\[ a + bi, \] where \(a\) and \(b\) are real.

Common Problem Types

Rewriting Square Roots of Negatives

Convert negative square roots into multiples of \(i\).

Example:
\(\sqrt{-12} = i\sqrt{12} = 2i\sqrt{3}\).

Distinguishing Real vs. Imaginary vs. Complex

Classify numbers.

Example:

  • \(7\) → real
  • \(4i\) → imaginary
  • \(3 - 2i\) → complex

Using the Definition of \(i\)

Use \(i^2 = -1\) to simplify expressions.

Example:
\(\sqrt{-1} = i\), \(\sqrt{-16} = 4i\).

Converting Real–Imaginary Forms

Rewrite expressions like \(\sqrt{-a}\) into \(i\sqrt{a}\).

Example:
\(\sqrt{-50} = 5i\sqrt{2}\)

Strategies

  • Always rewrite \(\sqrt{-a}\) as \(i\sqrt{a}\).
  • Factor the negative first:
    \[ \sqrt{-a} = \sqrt{-1}\sqrt{a} = i\sqrt{a}. \]
  • Check whether the expression simplifies (perfect squares).
  • Recognize “complex number” includes all real numbers too.

Worked Examples

Example 1 — Rewrite a negative square root

\[ \sqrt{-18} = i\sqrt{18} = 3i\sqrt{2}. \]

Example 2 — Classify a number

Number: \(12 - 7i\)

  • Real part: \(12\)
  • Imaginary part: \(-7i\)

This is complex.

Example 3 — Evaluate

\[ \sqrt{-49} = 7i. \]

WarningCommon Mistakes
  • Forgetting to pull out the negative: rewrite \(\sqrt{-a}\) as \(i\sqrt{a}\).
  • Thinking imaginary numbers “don’t exist”—they extend the number system.
  • Mixing up real vs. imaginary vs. complex categories.
  • Simplifying incorrectly: \(\sqrt{-ab} \neq \sqrt{-a}\sqrt{-b}\).

Practice Problems

  1. Rewrite: \(\sqrt{-20}\).
  2. Determine the type (real, imaginary, complex):
    1. \(5i\)
    2. \(-3\)
    3. \(7 + 4i\)
  3. Simplify: \(\sqrt{-64}\).
  4. Rewrite: \(\sqrt{-72}\) in simplest radical form.

1.
\[ \sqrt{-20} = i\sqrt{20} = 2i\sqrt{5}. \]

2a. \(5i\): imaginary
2b. \(-3\): real
2c. \(7 + 4i\): complex

3.
\[ \sqrt{-64} = 8i. \]

4.
\[ \sqrt{-72} = i\sqrt{72} = 6i\sqrt{2}. \]

Summary

  • Imaginary numbers arise from the definition \(i = \sqrt{-1}\).
  • Rewrite \(\sqrt{-a}\) as \(i\sqrt{a}\).
  • Pure imaginary numbers look like \(bi\).
  • Complex numbers look like \(a + bi\).
  • Always factor out \(\sqrt{-1} = i\).
  • Real numbers are a subset of complex numbers.
  • Check for perfect squares before simplifying.