Multiplying Complex Numbers by Hand

Complex numbers are numbers of the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit with the defining property i² = −1. This two-part structure extends the one-dimensional number line into a two-dimensional plane, letting you express rotation and oscillation algebraically. Before you attempt any complex number operations, every value must be in standard form: a single real term plus a single imaginary term. No stacked expressions, no unresolved products like 2(3 + 4i) sitting in place of a clean 6 + 8i. Writing both numbers in standard form first is not a formality, it is what makes adding complex numbers, subtracting complex numbers, and especially multiplying complex numbers work without ambiguity. This covers only the arithmetic: addition, subtraction, and multiplication, all done by hand with paper and pencil. Division, powers of i, and polar form each have their own logic and are mentioned here only to say they are separate procedures with separate rules, not extensions of the FOIL method.

Combine Like Terms

Adding complex numbers is the simplest complex number operation because it is exactly like combining like terms in algebra. You add the real parts together, and you add the imaginary parts together. The imaginary unit i is not a variable, but it behaves like one for addition: 3i + 5i = 8i, just as 3x + 5x = 8x. For example, (2 + 3i) + (4 − 5i) = (2 + 4) + (3i − 5i) = 6 − 2i. The result is another complex number in standard form, which means you are done. No multiplication, no i², no simplification beyond combining the two pairs. The only trap is sign errors: when the second number has a negative imaginary part, you are subtracting, not adding. Treat the operation as (a + bi) + (c + di) = (a + c) + (b + d)i, and write out both intermediate sums even if they seem obvious. A skipped step here costs you later when the numbers get messier.

Distribute the Minus Sign

Subtracting complex numbers follows the same logic as addition, but with one critical extra step: distribute the minus sign across the entire second number before you combine anything. For (a + bi) − (c + di), the operation is a + bi − c − di, which rearranges to (a − c) + (b − d)i. The most common failure in all of complex number operations is mishandling this sign. People subtract the real parts correctly, then add the imaginary parts because they forget the minus applies to the d as well. Write it out: (5 + 3i) − (2 + 7i) = 5 + 3i − 2 − 7i = (5 − 2) + (3 − 7)i = 3 − 4i. The second number's imaginary part was +7i, but after distribution it becomes −7i. If you skip the distribution step and write 5 + 3i − 2 + 7i, you get 3 + 10i, which is wrong by 14i.

Apply FOIL, Then Substitute i²

Multiplying complex numbers is where the subject earns its keep, because the procedure is FOIL with a twist: you must remember that i² = −1. For two binomials (a + bi)(c + di), the product is ac + adi + bci + bdi². The First terms give ac. The Outer terms give adi. The Inner terms give bci. The Last terms give bdi², which is bd(−1) = −bd. Combine the two imaginary terms: adi + bci = (ad + bc)i. The final result is (ac − bd) + (ad + bc)i. For example, (2 + 3i)(1 + 4i) = 2 + 8i + 3i + 12i² = 2 + 11i − 12 = −10 + 11i. Notice the real part is 2 − 12, not 2 + 12, because the i² term carries a minus. Failing to substitute i² = −1 is the single most common error in all of complex number arithmetic.

Use the Conjugate for Division Prep

The conjugate of a complex number a + bi is a − bi, formed by flipping the sign of the imaginary part. Multiplying a complex number by its conjugate is a special case of complex number multiplication that always yields a non-negative real number: (a + bi)(a − bi) = a² − (bi)² = a² − b²i² = a² − b²(−1) = a² + b². For example, (3 + 4i)(3 − 4i) = 9 − 12i + 12i − 16i² = 9 + 16 = 25. The cross terms cancel, leaving only the sum of the squares of the real and imaginary coefficients. This operation is the backbone of dividing complex numbers, because multiplying the denominator by its conjugate turns it into a real number, which then divides the numerator normally. If you plan to tackle division later, this is the skill to master now.

This table collects the essential complex number formulas for the operations covered here, plus the one conjugate identity you will reuse constantly. Use it as a quick check while you work; do not memorize the examples, memorize the pattern.

Fix the Three Classic Errors

Three errors account for nearly every wrong answer in manual complex number arithmetic. First, sign errors in subtraction: forgetting to distribute the minus across the entire second number. Fix this by rewriting the problem as addition with the second number's signs flipped. Second, forgetting i² = −1 during multiplication: you do the FOIL correctly, then leave the i² in place. Fix this by making a habit of writing i² = −1 at the top of every multiplication problem. Third, combining real and imaginary parts: adding 3 and 4i to get 7i. Fix this by keeping two separate columns on paper, one for the real coefficients and one for the imaginary coefficients, and never crossing between them. These are not mathematical ignorance; they are mechanical slips, and each has a mechanical remedy.