Complex numbers: adding, subtracting, multiplying and dividing

Complex numbers extend arithmetic to a second dimension. An imaginary unit (i) satisfies (i^2=-1), and a number such as (5-3i) combines a real part of 5 with an imaginary part of -3. Students in senior secondary mathematics classes across Australia, including NSW and Victoria, use this notation when exploring algebra, geometry, electrical models and rotations.

The most useful mental picture is a coordinate grid. If east-west movement is the real part and north-south movement is the imaginary part, (6+4i) describes a point 6 units east and 4 units north. A location 3 blocks east and 2 blocks north of Parliament House in Canberra could therefore be labelled (3+2i). Multiplying by (i) then turns that vector through 90 degrees.

For complex numbers, adding, subtracting, multiplying and dividing follow a set of consistent algebraic rules. Addition and subtraction work part by part, while multiplication and division require care with (i^2=-1). Australian classrooms usually write a decimal point, so (1.5+i) means one and a half, not one comma five. On a calculator, check that complex mode is active and the imaginary key uses (i).

Real numbers remain part of the same system: (42+0i) is simply 42. At a Woolworths checkout, an A$42.60 trolley and an A$13.75 shop at Coles differ by A$28.85; in complex-number notation, that difference is (28.85+0i). The Sydney-to-Melbourne road journey of roughly 880 kilometres can likewise start as (880+0i).

Reading and representing complex numbers

Write every complex number in standard form (a+bi), where (a) and (b) are real numbers. The sign before (i) belongs to the imaginary part, so (7-2i) has real part 7 and imaginary part -2. A zero may be shown explicitly, as in (0-6i), because keeping the two parts separate prevents many later errors.

The conjugate of (a+bi) is (a-bi). It is especially important in division because ((a+bi)(a-bi)=a^2+b^2), a real number. Another useful measure is the modulus, (|a+bi|=\sqrt{a^2+b^2}); for (3+4i), the modulus is 5, matching the 3-4-5 right triangle on the coordinate grid.

Adding and subtracting in standard form

Add corresponding parts: ((a+bi)+(c+di)=(a+c)+(b+d)i). Subtraction follows the same pattern: ((a+bi)-(c+di)=(a-c)+(b-d)i). For example, ((3+2i)+(1-4i)=4-2i), because (3+1=4) and (2+(-4)=-2).

To subtract, reverse the signs inside the second bracket before combining. Thus ((3+2i)-(1-4i)) becomes (3+2i-1+4i=2+6i). A common check is to confirm that adding the difference back to the original number restores the starting value.

Multiplying complex numbers

Multiplication uses the distributive law and the rule (i^2=-1). If (z_1=a+bi) and (z_2=c+di), then (z_1z_2=(ac-bd)+(ad+bc)i). The FOIL expansion gives the same result: multiply the real parts, subtract the product of the imaginary parts, then combine the two cross terms as the imaginary part.

For example, ((3+2i)(1-4i)=3-12i+2i-8i^2). Since (-8i^2=8), the answer is (11-10i). Check the structure before calculating: the real part comes from (3\times1-2\times(-4)=11), and the imaginary part comes from (3\times(-4)+2\times1=-10).

Multiplication is commutative, so changing the order should not change the answer. A useful special case is ((a+bi)(a-bi)=a^2+b^2); this also shows why conjugates appear in division. When one factor is (i), the result rotates the other factor by 90 degrees, a fact used in geometry and engineering.

Dividing complex numbers

To divide (z_1=a+bi) by (z_2=c+di), multiply the numerator and denominator by the conjugate (c-di):

[ \frac{a+bi}{c+di}

\frac{(a+bi)(c-di)}{c^2+d^2}. ]

The denominator is now real and non-zero whenever (c) and (d) are not both zero. Simplify the numerator, then divide both its real and imaginary parts by (c^2+d^2).

For (\frac{3+2i}{1-i}), multiply by (\frac{1+i}{1+i}). The numerator becomes ((3+2i)(1+i)=3+3i+2i+2i^2=1+5i), and the denominator becomes ((1-i)(1+i)=1-i^2=2). Hence the quotient is (\frac12+\frac52i).

Do not cancel an (i) as though it were an ordinary factor, and never divide by zero. Leaving a quotient as a fraction is valid, but standard form is usually clearer for checking. In polar form, division also means dividing the moduli and subtracting the arguments, which can be efficient for powers and repeated rotations.

Quick rules for checking complex-number work

The first list covers the operations that work directly on matching parts:

The second list covers products and quotients:

A reliable final check depends on the operation: estimate the modulus after multiplication, or multiply the quotient by the divisor after division. Your next step is to write (2+3i), (1-i), (4+0i) and (-2i) on paper, then calculate one addition, one subtraction, one product and one quotient using these rules.