squares <- c(i = (1i)^2, minus_i = (-1i)^2)
squares
#> i minus_i
#> -1+0i -1+0iComplex Numbers
1 Complex numbers
DefDefinition 1 (Imaginary unit)
Definition 1 (Imaginary unit) The imaginary unit \(i\) is a number whose square is \(-1\) [1]:
\[i^2 \stackrel{\text{def}}{=}-1.\]
Apart from that one rule, \(i\) obeys the usual rules of arithmetic:
- sums and products with \(i\) are commutative;
- sums and products with \(i\) are associative;
- multiplication distributes over addition.
Axler [2], Definition 1.1, p. 2 makes this idea rigorous: it defines a complex number as an ordered pair \((a, b)\) of real numbers, written \(a + bi\), defines addition and multiplication of such pairs, writes \(0 + 1i\) as \(i\), and leaves it to the reader to verify that \(i^2 = -1\).
ExmExample 1 (Powers of the imaginary unit)
Example 1 (Powers of the imaginary unit) The powers of \(i\) repeat in a cycle of four:
\[ \begin{aligned} i^3 &= i^2 \cdot i && \text{(split off one factor of } i \text{)} \\ &= (-1) \cdot i && \text{(}\href{#def-imaginary-unit}{\text{Definition~1}}\text{)} \\ &= -i && \text{(multiply)} \end{aligned} \]
and
\[ \begin{aligned} i^4 &= i^2 \cdot i^2 && \text{(split the power into two squares)} \\ &= (-1) \cdot(-1) && \text{(}\href{#def-imaginary-unit}{\text{Definition~1}}\text{, twice)} \\ &= 1 && \text{(multiply)} \end{aligned} \]
so the cycle starts again:
\[ \begin{aligned} i^5 &= i^4 \cdot i && \text{(split off one factor of } i \text{)} \\ &= 1 \cdot i && \text{(} i^4 = 1 \text{)} \\ &= i && \text{(multiply)} \end{aligned} \]
and the powers run \(i, -1, -i, 1, i, \ldots\).
Example 2 (No real number squares to \(-1\)) The imaginary unit is not a real number, because the square of every real number \(x\) is at least \(0\):
- if \(x \ge 0\), then \(x^2 = x \cdot x\) is a product of two nonnegative numbers, so \(x^2 \ge 0\);
- if \(x < 0\), then \(x^2 = x \cdot x\) is a product of two negative numbers, so \(x^2 > 0\).
For instance, \(3^2 = 9\) and \((-3)^2 = 9\); neither is \(-1\). So the equation \(x^2 = -1\) has no real solution, and Definition 1 introduces a new number to solve it.
Theorem 1 (The number \(-i\) also squares to \(-1\)) \[(-i)^2 = -1.\]
ProofProof
Proof. \[ \begin{aligned} (-i)^2 &= (-1)^2\,i^2 && \text{(} -i = (-1) \cdot i \text{, and } (ab)^2 = a^2 b^2 \text{ because products commute and associate)} \\ &= 1 \cdot(-1) && \text{(} (-1)^2 = 1 \text{ and }\href{#def-imaginary-unit}{\text{Definition~1}}\text{)} \\ &= -1 && \text{(multiply)} \end{aligned} \]
Example 3 (Squaring \(i\) and \(-i\) in R) R has the imaginary unit as 1i. The code below squares it and its negative.
Both squares have real part -1 and imaginary part 0, as Definition 1 and Theorem 1 say.
Remark 1 (The rule does not tell \(i\) from \(-i\)). Because \(-i\) obeys the same rule as \(i\) (Theorem 1), the rule \(i^2 = -1\) does not single one of them out. The symbol \(i\) names one of these two square roots of \(-1\), chosen once and for all.
DefDefinition 2 (Complex number)
Definition 2 (Complex number) A complex number [3] is a number of the form
\[z = a + b\,i,\]
where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit (Definition 1). The set of all complex numbers is \(\mathbb{C}\stackrel{\text{def}}{=}\{a + b\,i : a, b \in \mathbb{R}\}\).
DefDefinition 3 (Real part)
Definition 3 (Real part) The real part of a complex number \(z = a + b\,i\) (Definition 2) is \(\operatorname{Re} z \stackrel{\text{def}}{=}a\).
DefDefinition 4 (Imaginary part)
Definition 4 (Imaginary part) The imaginary part of a complex number \(z = a + b\,i\) (Definition 2) is \(\operatorname{Im} z \stackrel{\text{def}}{=}b\).
DefDefinition 5 (Equal complex numbers)
Definition 5 (Equal complex numbers) Two complex numbers (Definition 2) are equal exactly when their real parts (Definition 3) are equal and their imaginary parts (Definition 4) are equal.
Axler [2], Definition 1.1, p. 2 defines \(\mathbb{C}\) as the set of ordered pairs \((a, b)\) of real numbers, written \(a + bi\), and Axler [2], Definition 4.1, p. 120 defines the real and imaginary parts.
ExmExample 4 (Real and imaginary parts)
Example 4 (Real and imaginary parts) For \(z = 2 - 5\,i\), the real part is \(\operatorname{Re} z = 2\) and the imaginary part is \(\operatorname{Im} z = -5\): the imaginary part is the real number \(-5\), not \(-5\,i\).
The numbers \(1 + i\) and \(1 - i\) have the same real part, \(1\), but different imaginary parts, \(1\) and \(-1\), so they are different complex numbers.
The real number \(7\) is the complex number \(7 + 0\,i\), with real part \(7\) and imaginary part \(0\).
LemTheorem 2 (Every real number is a complex number)
Theorem 2 (Every real number is a complex number) A complex number \(z\) is a real number exactly when \(\operatorname{Im} z = 0\). In particular, every real number \(a\) is the complex number \(a + 0\,i\), with \(\operatorname{Re} a = a\) and \(\operatorname{Im} a = 0\), so \(\mathbb{R}\) is a subset of \(\mathbb{C}\).
ProofProof
Proof. Let \(a\) be a real number. Since \(0\,i = 0\),
\[ \begin{aligned} a + 0\,i &= a + 0 && \text{(} 0\,i = 0 \text{)} \\ &= a && \text{(}\href{algebra-sums.qmd#thm-add-ident}{\text{Theorem~4 in Convexity, Infimum and Sums}}\text{)} \end{aligned} \]
so \(a\) is the complex number \(a + 0\,i\), whose imaginary part is \(0\). The real part of \(a + 0\,i\) is \(a\), and by the equality rule in Definition 5 no other pair of real and imaginary parts gives the same number.
Conversely, if \(z = a + b\,i\) has \(b = \operatorname{Im} z = 0\), then \(z = a + 0\,i = a\) by the same calculation, so \(z\) is real.
ExmExample 5 (Which complex numbers are real)
Example 5 (Which complex numbers are real) R stores complex numbers as a real part and an imaginary part. The code below converts the real number \(7\) to a complex number and reads the parts of three complex numbers.
z <- c(as.complex(7), 2 - 5i, 3 + 0i)
data.frame(z = z, re = Re(z), im = Im(z), is_real = Im(z) == 0)The real number \(7\) becomes the complex number 7+0i, with imaginary part 0. Of the 3 numbers, 2 have imaginary part \(0\), and by Theorem 2 these are the real ones. The number \(2 - 5\,i\) is not real, because its imaginary part is -5.
LemTheorem 3 (Adding and multiplying complex numbers)
Theorem 3 (Adding and multiplying complex numbers) For real numbers \(a, b, c, d\):
\[(a + b\,i) + (c + d\,i) = (a + c) + (b + d)\,i\]
and
\[(a + b\,i)(c + d\,i) = (ac - bd) + (ad + bc)\,i.\]
ProofProof
Proof. Sum.
\[ \begin{aligned} (a + b\,i) + (c + d\,i) &= a + c + b\,i + d\,i && \text{(rearrange the terms)} \\ &= (a + c) + (b + d)\,i && \text{(factor out } i \text{)} \end{aligned} \]
Product.
\[ \begin{aligned} (a + b\,i)(c + d\,i) &= ac + ad\,i + bc\,i + bd\,i^2 && \text{(distribute)} \\ &= ac + ad\,i + bc\,i - bd && \text{(}\href{#def-imaginary-unit}{\text{Definition~1}}\text{)} \\ &= (ac - bd) + (ad + bc)\,i && \text{(group the real terms and the terms with } i \text{)} \end{aligned} \]
Axler [2], Definition 1.1, p. 2 takes these two formulas as the definitions of addition and multiplication.
ExmExample 6 (Adding and multiplying two complex numbers)
Example 6 (Adding and multiplying two complex numbers) Let \(w = 1 + 2\,i\) and \(z = 3 - i\). Their sum is
\[ \begin{aligned} w + z &= (1 + 3) + (2 + (-1))\,i && \text{(}\href{#thm-complex-arithmetic}{\text{Theorem~3}}\text{, sum)} \\ &= 4 + i && \text{(add)} \end{aligned} \]
and their product, multiplying out directly, is
\[ \begin{aligned} w z &= 1 \cdot 3 + 1 \cdot(-i) + 2\,i \cdot 3 + 2\,i \cdot(-i) && \text{(distribute)} \\ &= 3 - i + 6\,i - 2\,i^2 && \text{(multiply)} \\ &= 3 - i + 6\,i + 2 && \text{(}\href{#def-imaginary-unit}{\text{Definition~1}}\text{)} \\ &= (3 + 2) + (-1 + 6)\,i && \text{(group the real terms and the terms with } i \text{)} \\ &= 5 + 5\,i && \text{(add)} \end{aligned} \]
The product formula in Theorem 3 gives the same answer: with \(a = 1\), \(b = 2\), \(c = 3\) and \(d = -1\),
\[ \begin{aligned} ac - bd &= 3 - (-2) \\ &= 5 \end{aligned} \]
and
\[ \begin{aligned} ad + bc &= -1 + 6 \\ &= 5. \end{aligned} \]
DefDefinition 6 (Complex conjugate)
Definition 6 (Complex conjugate) The complex conjugate [4] of a complex number \(z = a + b\,i\) (Definition 2) is
\[\overline{z} \stackrel{\text{def}}{=}a - b\,i.\]
Axler [2], Definition 4.2, p. 120 gives the same definition, as \(\overline{z} = \operatorname{Re} z - (\operatorname{Im} z)\,i\).
ExmExample 7 (Complex conjugates)
Example 7 (Complex conjugates)
\(\overline{3 + 4\,i} = 3 - 4\,i\).
\[ \begin{aligned} \overline{-2\,i} &= \overline{0 + (-2)\,i} \\ &= 0 - (-2)\,i \\ &= 2\,i. \end{aligned} \]
-
\[ \begin{aligned} \overline{5} &= \overline{5 + 0\,i} \\ &= 5 - 0\,i \\ &= 5: \end{aligned} \]
a real number is its own complex conjugate. A complex number with a nonzero imaginary part, such as \(3 + 4\,i\), is not.
DefDefinition 7 (Absolute value of a complex number, modulus)
Definition 7 (Absolute value of a complex number, modulus) The absolute value, or modulus [3], of a complex number \(z = a + b\,i\) (Definition 2) is
\[\mathopen{}\left|z\right|\mathclose{} \stackrel{\text{def}}{=}\sqrt{a^2 + b^2}.\]
Axler [2], Definition 4.2, p. 120 gives the same definition.
ExmExample 8 (Absolute values of complex numbers)
Example 8 (Absolute values of complex numbers)
\[ \begin{aligned} \mathopen{}\left|3 + 4\,i\right|\mathclose{} &= \sqrt{3^2 + 4^2} \\ &= \sqrt{25} \\ &= 5. \end{aligned} \]
\[ \begin{aligned} \mathopen{}\left|-2\,i\right|\mathclose{} &= \sqrt{0^2 + (-2)^2} \\ &= \sqrt{4} \\ &= 2. \end{aligned} \]
-
For a real number \(a = a + 0\,i\),
\[ \begin{aligned} \mathopen{}\left|a\right|\mathclose{} &= \sqrt{a^2 + 0^2} \\ &= \sqrt{a^2}, \end{aligned} \]
which is the usual absolute value of \(a\); for instance,
\[ \begin{aligned} \mathopen{}\left|-3\right|\mathclose{} &= \sqrt{9} \\ &= 3. \end{aligned} \]
LemTheorem 4 (A complex number times its conjugate)
Theorem 4 (A complex number times its conjugate) For every complex number \(z\),
\[z\,\overline{z} = \mathopen{}\left|z\right|\mathclose{}^2.\]
In particular, \(z\,\overline{z}\) is a nonnegative real number.
ProofProof
Proof. Write \(z = a + b\,i\) with \(a, b\) real. Then
\[ \begin{aligned} z\,\overline{z} &= (a + b\,i)(a - b\,i) && \text{(}\href{#def-complex-conjugate}{\text{Definition~6}}\text{)} \\ &= a^2 - ab\,i + ab\,i - b^2\,i^2 && \text{(distribute)} \\ &= a^2 - b^2\,i^2 && \text{(cancel } ab\,i \text{)} \\ &= a^2 + b^2 && \text{(}\href{#def-imaginary-unit}{\text{Definition~1}}\text{)} \\ &= \mathopen{}\left|z\right|\mathclose{}^2 && \text{(}\href{#def-complex-modulus}{\text{Definition~7}}\text{)} \end{aligned} \]
Axler [2], result 4.4, p. 121 lists this identity among the properties of complex numbers.
Example 9 (Multiplying \(3 + 4\,i\) by its conjugate) \[ \begin{aligned} &(3 + 4\,i)(3 - 4\,i) \\ &= 9 - 12\,i + 12\,i - 16\,i^2 && \text{(distribute)} \\ &= 9 - 16\,i^2 && \text{(cancel } 12\,i \text{)} \\ &= 9 + 16 && \text{(}\href{#def-imaginary-unit}{\text{Definition~1}}\text{)} \\ &= 25 && \text{(add)} \end{aligned} \]
which is \(\mathopen{}\left|3 + 4\,i\right|\mathclose{}^2 = 5^2\) from Example 8, as Theorem 4 says.
References
[1]
Wikipedia contributors, “Imaginary unit — Wikipedia, the free encyclopedia.” 2026. Available: https://en.wikipedia.org/wiki/Imaginary_unit
[2]
S. Axler, Linear algebra done right, 4th ed. in Undergraduate texts in mathematics. Springer, 2024. doi: 10.1007/978-3-031-41026-0.
[3]
Wikipedia contributors, “Complex number — Wikipedia, the free encyclopedia.” 2026. Available: https://en.wikipedia.org/wiki/Complex_number
[4]
Wikipedia contributors, “Complex conjugate — Wikipedia, the free encyclopedia.” 2026. Available: https://en.wikipedia.org/wiki/Complex_conjugate