Algebraic Form: Standard Form a + bi

The Standard Form of a Complex Number

Whenever you work out an expression with i in it, you already tidy the answer the same way.

Expanding

(1 + i)^2

gives

1 + 2i + i^2

. Nobody leaves it like that:

i^2

is replaced by

-1

, the plain numbers are collected, and what remains is

2i

. The answer ends as a plain number plus some number of

i

's.

Numbers that look quite different all tidy into that one shape.

A subtraction, a sum written backwards, a lone multiple of

i

, an ordinary real number,

i^2

, and a square — each can be rewritten as one real piece plus a real number times

i

.

as first written	tidied	real piece	times i
4 − 5i	4 + (−5)i	4	−5
2i + 3	3 + 2i	3	2
7i	0 + 7i	0	7
5	5 + 0i	5	0
i²	−1 + 0i	−1	0
(1 + i)²	0 + 2i	0	2

What they share: two real numbers, one standing alone and one multiplying i.

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 definition

A complex number written as

a + bi

, with

a

and

b

real, is in algebraic form, also called standard form. The number

a

is its real part and

b

is its imaginary part.

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

Algebraic Form

z = a + bi

Learn more about this formula: Algebraic Form →

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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}

.

So

4 - 5i

has real part

4

and imaginary part

-5

.

Algebraic form is a way of writing a complex number: one expression that carries two real numbers.

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.

Pitfall: Combining the two terms

Writing

3 + 2i

as

5i

adds a real number to an imaginary one and gives the point

(0, 5)

instead of

(3, 2)

. The two terms never merge:

3 + 2i

is already in standard form, with real part

3

and imaginary part

2

.

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

What the name buys: once every number is in this form, questions about them become questions about two real numbers.

Two complex numbers are equal exactly when their real parts match and their imaginary parts match, and sums and conjugates are worked out part by part. None of that has to be re-argued for each new expression: tidy it into

a + bi

first, then read off

a

and

b

. The section on equality of complex numbers is the first use.

The rest of the page covers the notation

\operatorname{Re}(z)

and

\operatorname{Im}(z)

, the real and imaginary parts in detail, equality, the conjugate and its properties, and identities with conjugates.

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}

to

\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)

is

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

is

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

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Notation · Cheat sheet

Complex Number Notation at a Glance

The marks from the notation cards across the complex-number pages: how each is read and what it means. Every group links to the full card on its page.

Basics

z, \; w

z, w: complex numbers

The usual letters for complex numbers, as

x

and

y

are for reals.

a + bi

a plus b i

Real part

a

, imaginary part

b

; both are real numbers.

\mathbb{C}

C, the complex numbers

The set of all

a + bi

; it contains

\mathbb{R}

.

Imaginary numbers

i

i, the imaginary unit

The number with

i^2 = -1

.

\sqrt{-1}

the square root of minus one

Another name for

i

; root rules for reals fail with it.

bi

b i

A pure imaginary number: real part

0

.

Algebraic formThis page

\operatorname{Re}(z), \; \operatorname{Im}(z)

real part, imaginary part

For

z = a + bi

:

a

and

b

.

\operatorname{Im}(z)

is

b

, not

bi

.

a + (-b)i

a plus negative b, i

Written

a - bi

; the imaginary part is

-b

.

z = x + yi

z equals x plus y i

Same form with plane letters: the point

(x, y)

.

Complex conjugate

\bar{z}

z bar, the conjugate

For

z = a + bi

,

\bar{z} = a - bi

: the mirror image across the real axis.

\overline{z_1 + z_2}

the conjugate of the sum

The bar covers the whole expression; it equals

\bar{z}_1 + \bar{z}_2

.

z^*

z star

The same conjugate, as written in physics and engineering.

Absolute value

|z|

the modulus of z

Distance from

0

:

\sqrt{a^2 + b^2}

.

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

mod z squared

Equals

a^2 + b^2

, with no root to take.

|z - c| = R

mod z minus c equals R

The circle of radius

R

around

c

.

Additive inverse

-z

negative z

Both parts change sign:

-a - bi

.

-(a + bi)

minus, quantity a plus b i

The brackets carry the minus to both parts.

z - w

z minus w

Adding the inverse:

z + (-w)

.

Multiplicative inverse

z^{-1}, \; \frac{1}{z}

z inverse; one over z

The number with

z \cdot z^{-1} = 1

; it equals

\frac{\bar{z}}{|z|^2}

.

(3 + 2i)^{-1}

the inverse of three plus two i

Found by multiplying top and bottom by the conjugate:

\frac{3 - 2i}{13}

.

Trigonometric form

r, \; \theta

r and theta, the polar pair

Distance from

0

and angle from the positive real axis.

\arg(z)

the argument of z

Any angle of

z

; there are infinitely many,

2\pi

apart.

\operatorname{Arg}(z)

the principal argument

One chosen angle, usually in

(-\pi, \pi]

; some texts use

[0, 2\pi)

.

\operatorname{cis}\theta

sis theta

Short for

\cos\theta + i\sin\theta

.

Exponential form

e^{i\theta}

e to the i theta

Equals

\operatorname{cis}\theta

(Euler's formula): a point on the unit circle.

re^{i\theta}

r e to the i theta

The whole number

z

: length

r

, angle

\theta

.

e^{i(\theta + 2\pi k)}

theta plus two pi k

The same number for every integer

k

.

De Moivre's theorem

z_k

z sub k, the k-th root

The roots of

z^n = w

, numbered

k = 0, \ldots, n - 1

.

\omega

omega

A primitive root of unity: its powers give all

n

roots of

1

.

R^{1/n}

R to the one over n

The real root of the length; the complex root

w^{1/n}

has

n

values.

Same marks, different job

|z|

The modulus here; for a real number, the absolute value; for a matrix, the determinant.

\bar{z}

The conjugate here; in statistics a bar means a mean, in sets a complement.

z^{-1}

vs

z^*

The reciprocal, against the conjugate: they agree only on the unit circle.

w^{1/n}

n

values for a complex

w

; a single real root for a positive real number.

Complex number symbols table Complex numbers

Complex Number Notation at a Glance·/complex-numbers/basicsLearn Math Class

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:

Algebraic Form

z = a + bi

Learn more about this formula: Algebraic Form →

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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 →

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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:

The Real Part

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

.

z = 2 + 3i: Re(z) = 2 is the horizontal coordinate

The real part is the foot of the point on the horizontal axis, a = 2, and the imaginary part is its height, b = 3; the explorer reads both off the plotted point. Re(z) is a real number, a coordinate, never a complex one. Drag the point and watch a and b update on the complex number explorer.

The imaginary part, treated next, is the other coordinate and just as real a number.

The Imaginary Part

Given a complex number in algebraic form:

Algebraic Form

z = a + bi

Learn more about this formula: Algebraic Form →

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the imaginary part captures the coefficient multiplying

i

:

The Imaginary Part

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:

The Imaginary Part

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 — that is, when

Re(z) = 0

. Zero is included, so

0

is both real and pure imaginary (some textbooks also require

z \neq 0

).

A negative imaginary part

z = 3 − 2i sits three units right of the imaginary axis and two units below the real axis. Its imaginary part is the vertical coordinate with its sign, b = −2 — a real number, not −2i. Dropping the minus sign would put the point at 3 + 2i, above the axis, which is its conjugate. Move z between quadrants on the complex number explorer.

Re(z) = 3 is read off the horizontal axis the same way: the pair (Re z, Im z) is the point's address in the plane.

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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Fact:

i^2 = -1

as a theorem about pairs

In 1837 Hamilton defined a complex number as a pair of reals

(a, b)

, equal only when both entries match, with

(a, b)(c, d) = (ac - bd, ad + bc)

. Then

i = (0, 1)

and

(0, 1)(0, 1) = (-1, 0)

, so

i^2 = -1

is proved, not assumed.

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.

Pitfall: Matching parts before collecting

Solving

x + yi + 3i = 4 + 7i

by matching

x = 4

and

y = 7

leaves out the

3i

: those values give

4 + 10i

. Collected, the left side is

x + (y + 3)i

, so

x = 4

and

y + 3 = 7

, giving

y = 4

.

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

Every complex equation is two real equations in disguise: one for the real parts, one for the imaginary parts.

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.

Pitfall: Negating both parts

Conjugating

-2 + 5i

to

2 - 5i

changes the real part too and gives

-z

, the reflection through the origin. Only the imaginary part changes sign:

\overline{-2 + 5i} = -2 - 5i

.

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

is

3 - 2i

; the conjugate of

-1 - 4i

is

-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.

z and z̄ mirrored in the real axis, |z| = 3.6

The point z and its conjugate z̄ sit at the same distance from the origin, one above the real axis and one below: conjugation keeps the real part and flips the sign of the imaginary part, so it is a reflection in the real axis. The modulus, the length of either arrow, is the same 3.6 for both. Drag z anywhere and watch its mirror image follow on the conjugate and modulus tool.

Conjugation is a reflection, which is why applying it twice returns the original number.

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.

Fact: Algebra cannot tell

i

from

-i

Apart from doing nothing, conjugation is the only rule on

\mathbb{C}

that keeps every real number fixed and respects sums and products. Such a rule must send

i

to a number whose square is

-1

, and only

i

and

-i

qualify.

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.

Classification 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.

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A test without evaluating

The conjugate classifies a complex number

z = a + bi

: compare

\bar{z}

with

z

. The two existence theorems give the first two outcomes.

Existence theorems

\bar{z}

compared with

z

split by what conjugation does to $z$

ℝ

Case 1

\bar{z} = z

Real

conjugation leaves it fixed:

b = 0

, it lives on the real axis

i

Case 2

\bar{z} = -z

Pure imaginary

conjugation negates it:

a = 0

, it lives on the imaginary axis; see pure imaginary numbers

·

Case 3

neither

Off both axes

nonzero real and imaginary parts: the interior of the complex plane

Takeaway

To check whether an expression yields a real result, simply check whether it equals its own conjugate.

Classifying z with its conjugate·/complex-numbers/algebraic-formLearn Math Class

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.

Pitfall: Subtracting the second square

Writing

z \cdot \bar{z} = a^2 - b^2

forgets that

(bi)^2 = -b^2

: for

z = 3 + 2i

it gives

9 - 4 = 5

. The product is

a^2 + b^2

, so

(3 + 2i)(3 - 2i) = 9 + 4 = 13 = |z|^2

.

The product identity is the one used most often in practice, because it turns a complex expression into a single real number. In a division by

c + di

, numerator and denominator are both multiplied by

c - di

, and the new denominator is the product of a number with its own conjugate, the case worked out above with

c

and

d

in place of

a

and

b

. Every quotient in standard form passes through this step.

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

Pairing a number with its conjugate always lands on an axis: the sum and the product are real, the difference pure imaginary.

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)

Common Pitfalls

Six mistakes account for many of the wrong answers in work with the algebraic form. Four are marked where they happen in the sections above; keeping the

i

in the imaginary part and dropping its minus sign are the two that the notation section and The Imaginary Part already point out. Here each is worked through in full, with the wrong and the right working side by side.

Combining the two terms

The real term and the imaginary term of

a + bi

are added together as if they were like terms, so a number such as

3 + 2i

is turned into a single term.

Wrong working:

3 + 2i = (3 + 2)i = 5i

.

Right working:

3

is a real number and

2i

is a real multiple of

i

, so they cannot merge, just as

3 + 2x

does not become

5x

. The number

3 + 2i

is already in standard form, with

Re(z) = 3

and

Im(z) = 2

. The number

5i

has

Re = 0

and

Im = 5

.

Why it happens: in ordinary arithmetic every sum of two numbers ends in a single number, so a sum that stays as two terms looks unfinished.

How to check: plot both.

3 + 2i

is the point

(3, 2)

and

5i

is the point

(0, 5)

, two different points and so two different numbers.

Keeping the i in the imaginary part

The imaginary part of

z = a + bi

is reported as the whole term

bi

instead of the coefficient

b

.

Wrong working:

Im(7 + 4i) = 4i

.

Right working:

Im(z)

is the real number that multiplies

i

. In

7 + 4i

that number is

4

, so

Im(7 + 4i) = 4

, and together with

Re(7 + 4i) = 7

it gives the point

(7, 4)

.

Why it happens: the name "imaginary part" suggests that the answer itself should be imaginary.

How to check:

Re

and

Im

are functions from

\mathbb{C}

to

\mathbb{R}

, so a correct answer never contains

i

. Rebuilding

z = Re(z) + Im(z) \cdot i

must also give back

z

:

7 + 4 \cdot i = 7 + 4i

, while

7 + (4i) \cdot i = 7 - 4 = 3

does not.

Dropping the minus sign

When the imaginary term is subtracted, as in

5 - 3i

, the sign is left behind and only the number after it is reported as the imaginary part.

Wrong working:

5 - 3i

has imaginary part

3

.

Right working: in strict standard form

5 - 3i = 5 + (-3)i

, so

Re(z) = 5

and

Im(z) = -3

.

Why it happens: the minus looks like the operation between the two terms rather than part of the coefficient of

i

.

How to check: plot the point.

Im(z) = -3

puts

5 - 3i

three units below the real axis, at

(5, -3)

. An imaginary part of

3

would put it at

(5, 3)

, which is the conjugate

5 + 3i

.

Matching parts before collecting

In an equation with the unknown

z = x + yi

, the parts are compared before each side has been written as one real part plus one imaginary part, so an extra real or imaginary term is left out of the comparison.

Wrong working:

x + yi + 3i = 4 + 7i

, so

x = 4

and

y = 7

.

Right working: collect the left side first,

x + (y + 3)i = 4 + 7i

. The real parts give

x = 4

and the imaginary parts give

y + 3 = 7

, so

y = 4

.

Why it happens:

x

and

y

look like the real and imaginary parts of the left side, but they are only the parts of

z

, not of the whole expression.

How to check: substitute back.

4 + 4i + 3i = 4 + 7i

is correct, while

x = 4

and

y = 7

give

4 + 7i + 3i = 4 + 10i

.

Negating both parts

The conjugate is formed by changing the sign of every term, the real part included.

Wrong working:

\overline{-2 + 5i} = 2 - 5i

.

Right working: the conjugate keeps the real part and changes only the sign of the imaginary part, so

\overline{-2 + 5i} = -2 - 5i

. The number

2 - 5i

is

-z

, the negative of

z

.

Why it happens: the rule "change the sign" is remembered without the words "of the imaginary part", and changing every sign is what negation does.

How to check: the real part must not change,

Re(\bar{z}) = Re(z)

. On the plane,

\bar{z}

is the mirror image across the real axis, from

(-2, 5)

to

(-2, -5)

, while

2 - 5i

at

(2, -5)

is the reflection through the origin.

Subtracting the second square

In the product

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

, the pattern

(x + y)(x - y) = x^2 - y^2

is applied, but the square of

bi

is taken to be

b^2

.

Wrong working:

(3 + 2i)(3 - 2i) = 3^2 - 2^2 = 9 - 4 = 5

.

Right working:

(3 + 2i)(3 - 2i) = 3^2 - (2i)^2 = 9 - 4i^2 = 9 + 4 = 13

. In general

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

.

Why it happens: the difference of squares is familiar from real algebra, and the

i

inside the second term is lost when that term is squared.

How to check:

z \cdot \bar{z}

must equal

|z|^2

, which is never negative. Here

|3 + 2i| = \sqrt{13}

, so the product must be

13

. For

z = 1 + 2i

the wrong rule even gives

1 - 4 = -3

, while the true product is

1 + 4 = 5

.

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. Polar form is taught on the trigonometric form page.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 →

Can $i^2 = -1$ be proved instead of assumed?

+

Yes, if a complex number is defined as a pair of reals. Hamilton did this in 1837: pairs

(a, b)

are equal only when both entries match, and they multiply by

(a, b)(c, d) = (ac - bd, ad + bc)

. Taking

i = (0, 1)

gives

(0, 1)(0, 1) = (-1, 0)

, which is the real number

-1

. So

i^2 = -1

becomes a calculation, and

a + bi

is shorthand for the pair

(a, b)

.Read more →

What do $\Re(z)$ and $\Im(z)$ mean?

+

They are the real part and the imaginary part, the same as

\operatorname{Re}(z)

and

\operatorname{Im}(z)

, written in Fraktur capitals. The style is the 19th-century German convention and is still standard in physics and older analysis texts. Modern print usually spells the operators out, because the Fraktur letters are hard to read at small sizes. Either way,

\Im(7 + 4i) = 4

, a real number.Read more →

Is there any algebraic difference between $i$ and $-i$ ?

+

No. Both square to

-1

, and conjugation swaps them while keeping every real number fixed and respecting sums and products. So any true statement built from arithmetic and real numbers stays true when

i

is replaced by

-i

throughout. Apart from doing nothing, conjugation is the only rule with these properties. Which square root of

-1

is called

i

is a convention.Read more →

Algebraic Form

Building Numbers from Two Components

Key Terms

The Standard Form of a Complex Number

Real and Imaginary Part Notation

Real and Imaginary Part Notation

The Real Part

The Imaginary Part

Equality of Complex Numbers

The Complex Conjugate

Properties of the Conjugate

Classification Theorems

Useful Identities with Conjugates

Summary: Complete Standard-Form Reference Across Categories

Common Pitfalls

Algebraic Form FAQ

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

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

What is the difference between rectangular and polar form?

When is a complex number purely real or purely imaginary?

Can $i^2 = -1$ be proved instead of assumed?

What do $\Re(z)$ and $\Im(z)$ mean?

Is there any algebraic difference between $i$ and $-i$ ?

Algebraic Form

Building Numbers from Two Components

Key Terms

The Standard Form of a Complex Number

Real and Imaginary Part Notation

Real and Imaginary Part Notation

The Real Part

The Imaginary Part

Equality of Complex Numbers

The Complex Conjugate

Properties of the Conjugate

Classification Theorems

Useful Identities with Conjugates

Summary: Complete Standard-Form Reference Across Categories

Common Pitfalls

Algebraic Form FAQ

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

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

What is the difference between rectangular and polar form?

When is a complex number purely real or purely imaginary?

Can i2=−1i^2 = -1i2=−1 be proved instead of assumed?

What do ℜ(z)\Re(z)ℜ(z) and ℑ(z)\Im(z)ℑ(z) mean?

Is there any algebraic difference between iii and −i-i−i?

Can $i^2 = -1$ be proved instead of assumed?

What do $\Re(z)$ and $\Im(z)$ mean?

Is there any algebraic difference between $i$ and $-i$ ?