Algebraic Form

Key Terms

The Standard Form of a Complex Number

Real and Imaginary Part Notation

The Real Part

The Imaginary Part

Equality of Complex Numbers

The Complex Conjugate

Properties of the Conjugate

Existence Theorems

Useful Identities with Conjugates

Summary: Complete Standard-Form Reference Across Categories

Algebraic Form FAQ

Building Numbers from Two Components

Once the imaginary unit

i

enters our mathematical toolkit, we need a systematic way to combine it with real numbers. The algebraic form provides this structure — a standardized notation that expresses every complex number as the sum of a real part and an imaginary part. This representation serves as the foundation for arithmetic, comparison, and manipulation throughout complex analysis.

Key Terms

Algebraic Form— the standard representation

z = a + bi

Real Part— the component

a

z = a + bi

Imaginary Part— the coefficient

b

z = a + bi

Complex Number— any number expressible in algebraic form

Complex Conjugate— obtained by negating the imaginary part

See All Complex Numbers Definitions →

The Standard Form of a Complex Number

Every complex number can be written in a single canonical format:

Algebraic Form

z = a + bi

Learn more about this formula: Algebraic Form →

Browse all complex numbers formulas →

Here

a

and

b

are real numbers and

i

is the imaginary unit satisfying

i^2 = -1

. This representation, called the standard form or algebraic form, provides a uniform way to express any element of

\mathbb{C}

.

The structure mirrors a binomial with two distinct terms. The first term

a

stands alone as an ordinary real number. The second term

bi

combines a real coefficient

b

with the imaginary unit. Addition joins them into a single mathematical object. The number

3 + 2i

contains real term

3

and imaginary term

2i

. The number

-1 + 7i

has real term

-1

and imaginary term

7i

. Even

4 - 5i

fits this pattern when read as

4 + (-5)i

.

The real number

a

is called the real part of

z

. The real number

b

— not

bi

, just

b

— is called the imaginary part. This terminology causes confusion at first, since the "imaginary part" is itself a real number. The name indicates which component of

z

the value describes, not the nature of

b

as a quantity.

The imaginary unit

i

serves a structural role: it separates and distinguishes the two components. Without

i

, we could not tell where the real part ends and the imaginary part begins. The expression

a + bi

packages two independent real values into one object, with

i

acting as the marker that identifies which piece is which. This packaging allows complex numbers to encode two-dimensional information — a feature that becomes central when we explore the geometric representation.

Special cases deserve mention. When

b = 0

, the imaginary term vanishes and

z = a + 0i = a

reduces to a pure real number. When

a = 0

, the real term vanishes and

z = 0 + bi = bi

becomes a pure imaginary number. When both

a = 0

and

b = 0

, we obtain

z = 0

, the complex number zero. Every real number and every pure imaginary number fits within the algebraic form as a special case — the complex numbers contain all previous number systems as subsets.

Condition on a, b	Form reduces to	Category	Example
b = 0	a + 0i = a	real number	5 = 5 + 0i
a = 0, b ≠ 0	0 + bi = bi	pure imaginary	7i = 0 + 7i
a = 0 and b = 0	0 + 0i = 0	zero — the only number both real and pure imaginary	0
a ≠ 0 and b ≠ 0	a + bi	general complex number	3 + 2i

Real and Imaginary Part Notation

Notation

Real and Imaginary Part Notation

The extraction marks live here: the operator pair that splits a complex number into its two real coordinates, the strict spelling that keeps signs honest, and the letters that turn one complex unknown into two real ones.

z

a + bi

and

\mathbb{C}

come from complex numbers basics,

i

from the imaginary unit; the conjugate bar

\bar{z}

of The Complex Conjugate below is owned by its dedicated page.

Re(z)

Im(z)

the real part of z; the imaginary part of z

The extraction pair:

Re(z) = a

and

Im(z) = b

for

z = a + bi

— both functions from

\mathbb{C}

\mathbb{R}

, so the output never carries an

i

. Careful print sets the operators upright,

\operatorname{Re}(z)

rather than italic, via \operatorname{Re} — the LaTeX reference has the ready-made commands.

CasesOperator style drops the parentheses —

\operatorname{Re}\, z

is common in analysis. The pair acts as coordinates:

z = 3 - 2i

lands at

(Re(z), Im(z)) = (3, -2)

; the conjugate extraction identities live in The Real Part and The Imaginary Part below.

Also written

\Re(z)

and

\Im(z)

— Fraktur capitals, the 19th-century German convention still standard in physics and older analysis texts; hard to read at small sizes, which is why modern print spells the operators out.

Do not confuse

Im(z) = bi

. The commonest error in the topic — the function returns the bare coefficient:

Im(7 + 4i)

4

, a real number, never

4i

Same glyph elsewhere

\operatorname{Im}

is also the image of a linear map —

\operatorname{Im}(T)

in image and kernel — an unrelated job on an unrelated object; what follows the mark (complex number vs transformation) decides.

a + (-b)i

a plus negative b, i

The strict spelling of a subtraction:

4 - 5i

4 + (-5)i

. Rewriting makes the coefficient of

i

visible before anything is extracted or conjugated — the general form's sign convention applied under pressure.

CasesNeeded exactly where signs are harvested: reading off

Im(5 - 3i) = -3

, building

\bar{z}

, matching coefficients in Equality of Complex Numbers below.

Do not confuseA detachable minus. Reporting

Im(5 - 3i) = 3

drops the sign the notation had already absorbed into the coefficient — the whole point of the strict form is that the sign travels with

b

z = x + yi

z equals x plus y i

The unknowns convention: when

z

itself is what you solve for, its parts take the unknown letters —

x, y

real by declaration. One complex unknown becomes two real ones, the engine behind Equality of Complex Numbers below.

CasesAnalysis extends the split to outputs:

w = u + iv

for values

w = f(z)

, keeping input parts

(x, y)

and output parts

(u, v)

in separate alphabets.

Do not confuseFree letters. The method works only because

x, y \in \mathbb{R}

is declared — treat them as complex and one equation stops splitting into two.

All complex number symbols →↑ Complex Numbers

z	Strict standard form a + bi	Re(z)	Im(z)	Common mistake to avoid
7 + 4i	7 + 4i	7	4	—
5 − 3i	5 + (−3)i	5	−3	reporting Im(z) = 3 (the negative sign belongs to Im, not to i)
4i	0 + 4i	0	4	reporting Im(z) = 4i (the i is a marker; the part is just the coefficient)
−2	−2 + 0i	−2	0	—
0	0 + 0i	0	0	—

The Real Part

The real part of a complex number extracts its horizontal component. For any number written in algebraic form:

z = a + bi

the real part is the value

a

, denoted:

Real and Imaginary Parts

\operatorname{Re}(z) = a, \qquad \operatorname{Im}(z) = b \quad \text{for} \;\; z = a + bi

Learn more about this formula: Real and Imaginary Parts →

Browse all complex numbers formulas →

This function accepts a complex number as input and returns a real number as output. No matter how elaborate the original expression,

Re(z)

always belongs to the real number line.

On the complex plane, the real part measures horizontal displacement from the origin — movement along the real axis. The number

z = 5 - 3i

sits at coordinates

(5, -3)

, so

Re(z) = 5

gives the horizontal position.

A useful identity connects the real part to the conjugate:

Re(z) = \frac{z + \bar{z}}{2}

Adding a number to its conjugate cancels the imaginary terms, leaving twice the real part. This identity often simplifies calculations where extracting the real component matters.

Real numbers are characterized by having no imaginary contribution:

z

is real if and only if

Re(z) = z

. Equivalently, real numbers satisfy

Im(z) = 0

The Imaginary Part

Given a complex number in algebraic form:

z = a + bi

the imaginary part captures the coefficient multiplying

i

Im(z) = b

Students frequently misunderstand this definition. The output is

b

— a real number — not the product

bi

. When

z = 7 + 4i

, the imaginary part equals

4

, an ordinary real value. The word "imaginary" in the name refers to the role this component plays within

z

, not to any special nature of

b

itself.

Geometrically,

Im(z)

specifies how far the point lies above or below the horizontal axis in the complex plane. Consider

z = 5 - 3i

, located at

(5, -3)

. Here

Im(z) = -3

, indicating three units below the real axis. Dropping the minus sign would place the point incorrectly.

The conjugate provides an algebraic route to isolate this component:

Im(z) = \frac{z - \bar{z}}{2i}

Taking the difference

z - \bar{z}

eliminates the real terms entirely, leaving only the imaginary contribution.

A complex number qualifies as pure imaginary precisely when its real part vanishes while the number itself remains nonzero — that is, when

Re(z) = 0

and

z \neq 0

Equality of Complex Numbers

When do two complex numbers count as the same? The answer demands more than casual inspection — equality in

\mathbb{C}

requires matching both components independently. Two complex numbers

z_1 = a + bi

and

z_2 = c + di

are equal if and only if

a = c

and

b = d

. The real parts must match, and the imaginary parts must match. Neither condition alone suffices.

Formally:

Equality of Complex Numbers

a + bi = c + di \iff a = c \;\text{ and }\; b = d

Learn more about this formula: Equality of Complex Numbers →

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This principle seems obvious yet carries profound implications. The equation

a + bi = c + di

splits into two separate real equations:

a = c

from comparing real parts, and

b = d

from comparing imaginary parts. A single complex equation yields two real constraints. This doubling of information proves invaluable for solving problems.

Consider the equation

z^2 = 3 + 4i

where we seek

z = x + yi

with real unknowns

x

and

y

. Expanding:

(x + yi)^2 = x^2 + 2xyi + y^2i^2 = (x^2 - y^2) + 2xyi

. For this to equal

3 + 4i

, we need

x^2 - y^2 = 3

and

2xy = 4

. The original complex equation has transformed into a system of two real equations in two real unknowns — a problem with familiar solution techniques.

This method extends broadly. Whenever a complex equation appears, equating real and imaginary parts separately converts the problem into real arithmetic. The strategy applies to polynomial equations, functional equations, and identities. What looks like one equation is actually two, and exploiting this duality simplifies countless calculations throughout complex analysis.

The equality criterion also explains why complex numbers cannot be ordered. With real numbers, we compare single values and declare one larger or smaller. Complex numbers carry two independent values, and no consistent rule determines whether

3 + 2i

should rank above or below

1 + 5i

. We can compare their moduli, but the numbers themselves resist ordering.

Complex equation (with z = x + yi)	Real-part equation	Imaginary-part equation
a + bi = c + di (general rule)	a = c	b = d
(x + yi)² = 3 + 4i	x² − y² = 3	2xy = 4
(x + yi) + i = 2	x = 2	y + 1 = 0
(x + yi)(1 + i) = 5	x − y = 5	x + y = 0

The Complex Conjugate

Every complex number

z = a + bi

has a companion called its conjugate, written

\bar{z}

and defined as

\bar{z} = a - bi

. The operation preserves the real part while negating the imaginary part — only the sign in front of

bi

changes.

The geometric meaning becomes clear on the complex plane. If

z

sits at point

(a, b)

, then

\bar{z}

sits at

(a, -b)

. The two points share identical horizontal position but opposite vertical positions. Drawing both reveals mirror images reflected across the real axis. The conjugate of

3 + 2i

3 - 2i

; the conjugate of

-1 - 4i

-1 + 4i

; the conjugate of

5

(a real number) is simply

5

.

Alternative notation uses

z^*

instead of

\bar{z}

, particularly in physics and engineering contexts. Both symbols denote the same operation. This text favors the overline notation

\bar{z}

as standard in pure mathematics.

The conjugate appears throughout complex analysis — in division, in computing the modulus, in classifying numbers as real or pure imaginary, and in polynomial theory. A dedicated section explores the conjugate comprehensively: its algebraic properties, geometric interpretations, and applications to equations and simplification.

Properties of the Conjugate

The conjugate operation obeys algebraic laws that make it compatible with standard arithmetic. These properties transform complex calculations by allowing conjugation to pass through sums, products, and quotients in predictable ways.

Taking the conjugate twice returns the original number:

\overline{\bar{z}} = z

. Reflection across the real axis, performed twice, brings every point back to its starting location. This involution property guarantees that conjugation is reversible.

Conjugation distributes over addition:

\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}

. The conjugate of a sum equals the sum of the conjugates. The same holds for subtraction. This linearity means we can conjugate term by term when facing complicated expressions.

Conjugation also distributes over multiplication:

\overline{z_1 \cdot z_2} = \bar{z_1} \cdot \bar{z_2}

. The conjugate of a product equals the product of the conjugates. Extending to division:

\overline{z_1 / z_2} = \bar{z_1} / \bar{z_2}

, valid when

z_2 \neq 0

. Powers follow naturally:

\overline{z^n} = (\bar{z})^n

for any integer

n

.

These rules prove essential when simplifying expressions, verifying identities, and solving equations. Rather than computing a complicated product and then conjugating, we can conjugate each factor first and multiply afterward — often a simpler path. The dedicated conjugate page provides complete proofs and extended applications of these properties.

Existence Theorems

The conjugate provides algebraic tests for classifying complex numbers. Two theorems characterize when a complex number belongs to special subcategories — the real numbers and the pure imaginary numbers.

The first theorem states: a complex number

z

is real if and only if

z = \bar{z}

. To verify, let

z = a + bi

. If

z

is real, then

b = 0

, so

z = a

and

\bar{z} = a

, giving

z = \bar{z}

. Conversely, if

z = \bar{z}

, then

a + bi = a - bi

, which forces

bi = -bi

, meaning

2bi = 0

and thus

b = 0

. The number must be real. This theorem lets us test whether an expression yields a real result without fully evaluating it — simply check if it equals its own conjugate.

The second theorem states: a complex number

z

is pure imaginary if and only if

\bar{z} = -z

. Again let

z = a + bi

. If

z

is pure imaginary, then

a = 0

, so

z = bi

and

\bar{z} = -bi = -z

. Conversely, if

\bar{z} = -z

, then

a - bi = -a - bi

, forcing

a = -a

and therefore

a = 0

. The number has no real part and qualifies as pure imaginary.

Both theorems connect the conjugate operation to fundamental classification questions. A number lives on the real axis precisely when conjugation leaves it fixed. A number lives on the imaginary axis precisely when conjugation negates it. Every other complex number — those with nonzero real and imaginary parts — satisfies neither condition, lying off both axes in the interior of the complex plane.

Useful Identities with Conjugates

Three identities involving a complex number and its conjugate appear constantly in applications. Each reveals structural information about how the real and imaginary parts interact.

The sum of a number and its conjugate yields twice the real part:

z + \bar{z} = (a + bi) + (a - bi) = 2a

. The imaginary terms cancel completely, leaving a purely real result. This identity extracts the real part through arithmetic:

Re(z) = \frac{z + \bar{z}}{2}

. Whenever a calculation produces

z + \bar{z}

, we know immediately the answer is real without further investigation.

The difference between a number and its conjugate yields twice the imaginary part times

i

z - \bar{z} = (a + bi) - (a - bi) = 2bi

. The real terms cancel, leaving a purely imaginary result. This identity extracts the imaginary part:

Im(z) = \frac{z - \bar{z}}{2i}

. Recognizing

z - \bar{z}

signals that the result lies on the imaginary axis.

The product of a number and its conjugate yields the sum of squares of the components:

z \cdot \bar{z} = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2i^2 = a^2 + b^2

. This result is always real and always non-negative. It equals zero only when both

a = 0

and

b = 0

, meaning

z = 0

. This identity connects directly to the modulus: since

|z|^2 = a^2 + b^2

, we have

z \cdot \bar{z} = |z|^2

. The product also enables division — multiplying numerator and denominator by the conjugate of the denominator produces a real denominator

|z|^2

, converting the quotient to standard form.

Additional identities and applications appear in the complex conjugate dedicated page.

Summary: Complete Standard-Form Reference Across Categories

The algebraic form

z = a + bi

packages many facets of a single complex number — its real part, its imaginary part, its conjugate, its category, and its position in the complex plane. The table below collects all of these for a representative selection of complex numbers, providing a one-glance reference for what each piece of the standard form encodes.

z	Re(z)	Im(z)	z̄	Category	Plane location
3 + 2i	3	2	3 − 2i	general complex	(3, 2) — Q1
−1 − 4i	−1	−4	−1 + 4i	general complex	(−1, −4) — Q3
5 − 3i	5	−3	5 + 3i	general complex	(5, −3) — Q4
5	5	0	5	real (b = 0)	(5, 0) — on real axis
7i	0	7	−7i	pure imaginary (a = 0)	(0, 7) — on imaginary axis
0	0	0	0	zero — both real and pure imaginary	origin (0, 0)

Algebraic Form FAQ

Why is the imaginary part of a complex number a real number?

Because the imaginary part names the coefficient, not the term. In 3 + 4i the imaginary part is 4, a plain real number, while 4i is the imaginary term. The distinction matters whenever a formula asks for Im(z), since supplying 4i instead of 4 makes every subsequent step wrong.Read more →

How do you solve an equation by equating real and imaginary parts?

Two complex numbers are equal only when both components match, so a single complex equation splits into two real equations. Collect everything into the form a + bi on each side, set the real parts equal, set the imaginary parts equal, and solve the resulting pair simultaneously.Read more →

What is the difference between rectangular and polar form?

Rectangular form gives the horizontal and vertical components as a + bi. Polar form gives distance from the origin and angle from the positive real axis instead. They describe the same point using different coordinates, and each suits different work: rectangular for adding, polar for multiplying and taking powers.Read more →

When is a complex number purely real or purely imaginary?

It is purely real when its imaginary part is zero, leaving a point on the horizontal axis. It is purely imaginary when its real part is zero, leaving a point on the vertical axis. Zero satisfies both conditions at once, which places it at the origin where the two axes meet.Read more →