complex number: meaning, definition, pronunciation and examples
C2Formal/Academic/Technical
Quick answer
What does “complex number” mean?
A number consisting of a real part and an imaginary part (a multiple of i, where i = √-1), expressed as a + bi.
Audio
Pronunciation
Definition
Meaning and Definition
A number consisting of a real part and an imaginary part (a multiple of i, where i = √-1), expressed as a + bi.
In mathematics, an element of the complex number system used to extend the real numbers to solve equations with no real solutions; also used abstractly to describe multifaceted systems with interacting components in non-mathematical contexts (e.g., 'a complex number of factors').
Dialectal Variation
British vs American Usage
Differences
No significant difference in the core mathematical term. Pronunciation and vowel emphasis may differ slightly (see IPA). The non-mathematical extended use is equally rare in both varieties.
Connotations
Solely mathematical and precise. Carries connotations of advanced mathematics, engineering, and physics.
Frequency
Extremely low frequency in general language but high frequency in advanced STEM (Science, Technology, Engineering, Mathematics) contexts.
Grammar
How to Use “complex number” in a Sentence
N + of + complex number(s)complex number + V (e.g., represents, has)V + complex number (e.g., calculate, plot)Adj + complex number (e.g., arbitrary, given, conjugate)Vocabulary
Collocations
Examples
Examples of “complex number” in a Sentence
adjective
British English
- The complex-number solution was more elegant.
- We need a complex-number approach.
American English
- The complex-number solution was more elegant.
- We need a complex-number approach.
Usage
Meaning in Context
Business
Virtually never used.
Academic
Core term in pure and applied mathematics, physics, electrical engineering, and related fields.
Everyday
Extremely rare, only when explaining advanced concepts in simple terms.
Technical
Fundamental term. Used in signal processing, control theory, quantum mechanics, and complex analysis.
Vocabulary
Synonyms of “complex number”
Weak
Vocabulary
Antonyms of “complex number”
Watch out
Common Mistakes When Using “complex number”
- Using 'complicated number' instead of the fixed term 'complex number'.
- Treating 'i' as a variable instead of the defined imaginary unit (√-1).
- Forgetting that complex numbers are typically written as a + bi, not a + ib.
FAQ
Frequently Asked Questions
No. 'Complex number' is a fixed, precise mathematical term. 'Complicated number' is a vague descriptive phrase and is not used in technical contexts.
Yes. A real number is a special case of a complex number where the imaginary part (bi) is zero (e.g., 5 + 0i).
They are fundamental in electrical engineering (AC circuits, signal processing), quantum physics, fluid dynamics, and in solving equations that have no real-number solutions.
The symbol 'i' represents the imaginary unit, defined by the property i² = -1. It is not a variable but a constant.
A number consisting of a real part and an imaginary part (a multiple of i, where i = √-1), expressed as a + bi.
Complex number is usually formal/academic/technical in register.
Complex number: in British English it is pronounced /ˌkɒm.pleks ˈnʌm.bər/, and in American English it is pronounced /ˌkɑːm.pleks ˈnʌm.bɚ/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think of a 'complex' apartment building: it has two addresses — a real street number (the real part 'a') and an imaginary unit number (the imaginary part 'bi'). Together they form the full 'complex number'.
Conceptual Metaphor
A TWO-DIMENSIONAL POINT (real part = x-coordinate, imaginary part = y-coordinate). A HYBRID ENTITY (combining two different types of information into one entity).
Practice
Quiz
What is the standard form for writing a complex number?