Use conjugates to divide and measure

Try

To divide 7+i by 2+i, find a number q that satisfies q(2+i)=7+i. Guessing by dividing the two coordinates separately would produce 7/2+i, but multiplying that back gives 6+(11/2)i. It fails the defining check.

The conjugate 2−i offers a better route. Multiplying the denominator by its conjugate gives (2+i)(2−i)=5, a nonzero real number. Multiply the numerator and denominator by that same conjugate, so the quotient is unchanged.

Predict

Predict whether the real denominator will always be positive for a nonzero complex denominator. If the denominator is c+di, its conjugate product is c²+d². For real c,d, when could this sum be zero?

Also distinguish modulus from the number itself. The modulus of a+bi is its distance from the origin, √(a²+b²). It is a nonnegative real length. Replacing a complex number by its modulus loses its direction and usually changes the number.

Experiment

Follow the quotient calculation and then perform its independent multiplication check.

Stage	Exact expression	Interpretation
Multiply numerator	(7+i)(2−i)=15−5i	Four terms and i²=−1
Multiply denominator	(2+i)(2−i)=5	Conjugate product
Divide by the real value	(15−5i)/5=3−i	Both coordinates divided by 5
Check	(3−i)(2+i)=7+i	Original numerator recovered

Draw the point (3,−1) or describe its coordinates and squared distance 10. The numerator 7+i has squared modulus 50, and denominator 2+i has squared modulus 5. Their ratio 50/5 matches the quotient’s squared modulus. This check is useful but does not replace multiplication back: length alone cannot determine a complex direction.

Understand

For z=a+bi≠0, the real numbers a,b are not both zero. Thus a²+b²>0. The conjugate product proves that (a−bi)/(a²+b²) is an inverse of z because multiplying it by z gives 1. Division by z means multiplication by this inverse. Zero has no inverse: zero times any number is zero, so it can never produce 1.

The distance formula gives |z|²=a²+b²=z z̄. Conjugation respects multiplication, as a coordinate expansion verifies. Consequently |zw|²=(zw)conjugate(zw)=z z̄ w w̄=|z|²|w|². Both |zw| and |z||w| are nonnegative, so equality of their squares implies |zw|=|z||w|. If w≠0, apply this identity to (z/w)w=z to obtain |z/w|=|z|/|w|. The denominator restriction matters in this length formula too.

For a concrete length calculation, the point −5+12i is 13 units from the origin because 25+144=169. Its conjugate lies at −5−12i, at the same distance. This is why a conjugate product records squared length rather than a signed coordinate.

Apply

Calculate a quotient in three stages: name the nonzero denominator, multiply by its conjugate, and verify by multiplying back. Do not discard the imaginary part just because the new denominator is real. Only that denominator has been made real by the construction.

When the denominator is purely imaginary, the same method still works. For instance, 1/(2i)=−i/2 because (−i/2)(2i)=1. When the denominator is purely real, the formula reduces to familiar division of each numerator coefficient by that real number. These cases belong to one rule.

Continue

Optional depth: if two numbers q₁,q₂ both satisfy qz=1 for z≠0, multiply by the inverse already constructed to obtain q₁=q₂. The reciprocal is unique. Nonzero complex multiplication also has cancellation; the zero exception cannot be removed.

Conjugate division and modulus calculations provide school-entry actions for N-CN.A.3. The next lesson turns the same coordinates into constructions for subtraction, reflection, rotation, distance and midpoint. Keep exact coordinate descriptions alongside sketches so appearance alone is never the evidence.