Once the imaginary uniti 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.
The Standard Form of a Complex Number
Every complex number can be written in a single canonical format: z=a+bi, where a and b are real numbers and i is the imaginary unit satisfying i2=−1. This representation, called the standard form or algebraic form, provides a uniform way to express any element of 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.
Real and Imaginary Parts: Formal Notation
Mathematics benefits from precise notation, and the components of a complex number receive dedicated symbols. For any z=a+bi, we write Re(z)=a to denote the real part and Im(z)=b to denote the imaginary part. These functions extract the individual coordinates from the combined expression.
The notation Re(z) reads as "the real part of z." Applied to z=7+4i, we obtain Re(7+4i)=7. The function strips away the imaginary term and returns what remains. Similarly, Im(z) reads as "the imaginary part of z," and Im(7+4i)=4. Note again: the output is 4, not 4i. The function extracts the coefficient, not the entire imaginary term.
Geometric interpretation reinforces the meaning. On the complex plane, the real part measures horizontal displacement from the origin — movement along the real axis. The imaginary part measures vertical displacement — movement along the imaginary axis. The number z=3−2i sits at coordinates (3,−2): three units right and two units down. Here Re(z)=3 gives the horizontal coordinate and Im(z)=−2 gives the vertical coordinate.
Negative imaginary parts require careful handling. Consider z=5−3i. Writing this in strict standard form yields z=5+(−3)i, revealing that the coefficient of i is −3. Therefore Re(z)=5 and Im(z)=−3. The negative sign belongs to the imaginary part; it does not float separately. Students who report Im(5−3i)=3 have dropped the sign and produced an incorrect answer.
Both Re and Im are functions from C to R. They accept a complex number as input and return a real number as output. No complex number emerges from either function — even when applied to the most elaborate complex expression, the result always belongs to the real number line. This property makes Re and Im essential tools for decomposing complex problems into real-valued components that standard techniques can address.
Equality of Complex Numbers
When do two complex numbers count as the same? The answer demands more than casual inspection — equality in C requires matching both components independently. Two complex numbers z1=a+bi and z2=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.
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 z2=3+4i where we seek z=x+yi with real unknowns x and y. Expanding: (x+yi)2=x2+2xyi+y2i2=(x2−y2)+2xyi. For this to equal 3+4i, we need x2−y2=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.
The Complex Conjugate
Every complex number z=a+bi has a companion called its conjugate, written zˉ and defined as zˉ=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 zˉ 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 zˉ, particularly in physics and engineering contexts. Both symbols denote the same operation. This text favors the overline notation zˉ 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: zˉ=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: z1+z2=z1ˉ+z2ˉ. 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: z1⋅z2=z1ˉ⋅z2ˉ. The conjugate of a product equals the product of the conjugates. Extending to division: z1/z2=z1ˉ/z2ˉ, valid when z2=0. Powers follow naturally: zn=(zˉ)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=zˉ. To verify, let z=a+bi. If z is real, then b=0, so z=a and zˉ=a, giving z=zˉ. Conversely, if z=zˉ, 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 zˉ=−z. Again let z=a+bi. If z is pure imaginary, then a=0, so z=bi and zˉ=−bi=−z. Conversely, if zˉ=−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+zˉ=(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)=2z+zˉ. Whenever a calculation produces z+zˉ, 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−zˉ=(a+bi)−(a−bi)=2bi. The real terms cancel, leaving a purely imaginary result. This identity extracts the imaginary part: Im(z)=2iz−zˉ. Recognizing z−zˉ 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⋅zˉ=(a+bi)(a−bi)=a2−(bi)2=a2−b2i2=a2+b2. 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=a2+b2, we have z⋅zˉ=∣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.