Calculate with a real and an imaginary coordinate

Try

No real number has square −1: the square of a real number is nonnegative. Complex arithmetic introduces a number i satisfying i²=−1. A complex number has the form a+bi, where a and b are real. Think of the two coordinates (a,b) as a way to record the number, not as two separate answers.

Start with (3+2i)+(−1+4i). Combine real terms and imaginary terms to obtain 2+6i. Then subtract the second number instead. The subtraction must act on both of its coordinates, giving 4−2i. Write the parentheses before distributing the minus sign.

Predict

Predict the real and imaginary parts of (3+2i)(−1+4i). There will be four products before like terms are collected. Which one uses i²=−1? A frequent mistake is to treat i² as i or as +1, changing the result.

Here “imaginary part” means the real coefficient b in a+bi. The imaginary term is bi. Thus the imaginary part of 2+6i is 6, not 6i. Real numbers are included in the complex system as numbers with b=0.

Experiment

Use this multiplication ledger and verify each row yourself. The column of unreduced products keeps the mechanism visible.

Pair of terms	Product before reduction	Reduced contribution
3 and −1	−3	−3
3 and 4i	12i	12i
2i and −1	−2i	−2i
2i and 4i	8i²	−8

The total is −11+10i. Check the addition and subtraction from Try with a two-column real/imaginary ledger. Then change only the sign of the second imaginary coordinate and calculate again. Your prediction should identify exactly which products change, rather than starting from an unexplained final formula.

Understand

For real a,b,c,d, distribute (a+bi)(c+di). The four terms are ac, adi, bci and bdi². Replacing i² by −1 and collecting gives (ac−bd)+(ad+bc)i. This derivation explains the multiplication rule for every pair, while the ledger checks one instance. Addition is (a+c)+(b+d)i; subtraction is (a−c)+(b−d)i.

The conjugate of z=a+bi is the number a−bi, written z̄. It keeps the real coordinate and reverses the imaginary coordinate. Multiplying a number by its conjugate gives (a+bi)(a−bi)=a²−abi+abi−b²i²=a²+b². The cross terms cancel and the remaining expression is a nonnegative real number. For z=3+2i the conjugate product is 13, not 5.

This should be distinguished from squaring: (3+2i)²=9+12i−4=5+12i. The conjugate and the original number are different factors unless the imaginary coefficient is zero. A diagram later identifies conjugation as reflection across the real axis; the algebra already explains why the product becomes real.

Complex addition and multiplication retain the real arithmetic laws used in these expansions. We are defining and operating in a larger number system, not claiming that a real variable can be substituted for i.

Apply

Repair a flawed calculation by locating the first invalid step. If someone writes (a+bi)²=a²−b², they have discarded the two mixed products. The full result is a²−b²+2ab i. The mixed part vanishes when a=0 or b=0, not for all complex numbers.

Use multiplication to check any claimed identity. In a written response, record enough intermediate terms that another learner can identify whether a sign error occurred during distribution or when replacing i². A correct final coordinate without this trace does not explain the mechanism.

Continue

Optional depth: conjugation respects addition and multiplication. Addition follows by reversing the summed imaginary coefficient. For products, conjugating (ac−bd)+(ad+bc)i gives the same result as multiplying (a−bi)(c−di). You can prove that equality with four-term expansions.

These actions support N-CN.A.1–2. Next, the conjugate-product identity will produce a reciprocal of every nonzero complex number. This step explains division instead of inventing coordinatewise division, which does not work.