Nonreal Numbers — Definition, Examples & Table
Nonreal Numbers
The complex numbers that are not real. That is, the complex numbers with a nontrivial imaginary part.
For example, 3 + 2i is nonreal, 2i is nonreal, but 3 is real.
Key Formula
z=a+biwhere b=0
Where:
- z = A nonreal complex number
- a = The real part, which can be any real number (including 0)
- b = The imaginary part, which must be nonzero for the number to be nonreal
- i = The imaginary unit, defined by i² = −1
Worked Example
Problem: Classify each of the following numbers as real or nonreal: 5, −3 + 4i, 7i, 0, √2 − i.
Step 1: Write each number in the standard complex form a + bi and identify the imaginary part b.
5=5+0i,−3+4i,7i=0+7i,0=0+0i,2−i=2+(−1)i
Step 2: A number is nonreal when b ≠ 0, and real when b = 0.
Step 3: Check each value of b: For 5, b = 0 (real). For −3 + 4i, b = 4 (nonreal). For 7i, b = 7 (nonreal). For 0, b = 0 (real). For √2 − i, b = −1 (nonreal).
Answer: Real: 5 and 0. Nonreal: −3 + 4i, 7i, and √2 − i.
Frequently Asked Questions
Is a purely imaginary number (like 2i) considered nonreal?
Yes. A purely imaginary number has the form 0 + bi where b ≠ 0. Because its imaginary part is nonzero, it is nonreal. Every purely imaginary number is nonreal, but not every nonreal number is purely imaginary — for example, 3 + 2i has both a real and an imaginary part.
Are nonreal numbers the same as imaginary numbers?
The terms are sometimes used interchangeably in casual speech, but they can mean different things. "Imaginary number" often refers specifically to numbers of the form bi (purely imaginary), while "nonreal number" includes any complex number with b ≠ 0, such as 3 + 2i. To avoid confusion, use "nonreal" when you mean any complex number that is not on the real number line.
Nonreal numbers vs. Purely imaginary numbers
Every purely imaginary number (like 5i) is nonreal, but not every nonreal number is purely imaginary. A nonreal number can have a nonzero real part as well — for example, 3 + 2i is nonreal but not purely imaginary. Purely imaginary numbers are a subset of nonreal numbers, which are in turn a subset of all complex numbers.
Why It Matters
Nonreal numbers arise naturally when you solve polynomial equations that have no real solutions, such as x2+1=0. The Fundamental Theorem of Algebra guarantees that every polynomial of degree n has exactly n complex roots (counted with multiplicity), and for polynomials with real coefficients, the nonreal roots always come in conjugate pairs like a+bi and a−bi. Understanding which numbers are nonreal helps you interpret discriminants, factor polynomials completely, and work with signals and rotations in physics and engineering.
Common Mistakes
Mistake: Thinking that 0 is nonreal because it "isn't positive or negative."
Correction: Zero is a real number — it equals 0 + 0i, so its imaginary part is 0. A number is nonreal only when its imaginary part is nonzero.
Mistake: Assuming a number like 4 + 0i is nonreal because it is written in complex form.
Correction: The form a + bi with b = 0 is just a real number. Writing 4 as 4 + 0i does not make it nonreal; what matters is whether b ≠ 0.
Related Terms
- Complex Numbers — The full set containing both real and nonreal numbers
- Real Numbers — Complex numbers with imaginary part equal to zero
- Imaginary Part — The coefficient b that determines if a number is nonreal
- Imaginary Numbers — Purely imaginary numbers are a subset of nonreal numbers
- Imaginary Unit — Defined as i² = −1, the building block of nonreal numbers
- Complex Conjugate — Nonreal roots of real polynomials come in conjugate pairs
- Nontrivial — Nonreal means having a nontrivial imaginary part