A complex number is a number that can be expressed in the form \( a + bi \), where: - \( a \) and \( b \) are real numbers, - \( i \) is the imaginary unit, defined as \( i = \sqrt{-1} \). In this representation: - \( a \) is called the **real part** of the complex number, - \( b \) is called the **imaginary part** of the complex number.
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A complex number is , where are real numbers and the imaginary unit satisfies . The real part is and the imaginary part is : for they are and , respectively. Equality requires both coordinates to agree. Thus the complex plane gives each complex number a concrete position, while ordinary addition and multiplication become geometric operations.
For and , addition works coordinate by coordinate, whereas multiplication uses :These answers encode different motions: addition by is a translation, while multiplication by is a rotation combined with a uniform change of size. Keeping these pictures distinct prevents a common beginner's confusion.
The complex numbers form a field: every nonzero has a multiplicative inverse, computable using its complex conjugate and modulus. They also supply the missing solutions of a quadratic equation such as , namely . The surprising payoff is that extending arithmetic also makes Euclidean geometry easier to calculate. For a systematic introduction to these definitions, see Jeremy Orloff, MIT 18.04, topic 1, sections 1.3–1.5.
Forms both a:
- division algebra if thought of with complex multiplication as the bilinear map of the algebra
- field