A point three units right and four units up is the complex number 3+4i. Its distance from zero is five. Multiplying it by i gives −4+3i: the point moves through a quarter-turn without changing its distance. Before treating i as a formal symbol, draw those two vectors and check their perpendicularity with a real dot product.
We identify z=x+iy with the ordered pair (x,y), where x and y are real and i²=−1. Addition is vector addition. Multiplication contains geometry that ordinary coordinatewise multiplication would miss.
Predict
Predict the image of the unit square under multiplication by 2i. Does its area double or quadruple? All lengths double, and the square rotates counterclockwise by a quarter-turn, so area quadruples. A translation z↦z+b changes the location but does not change distances.
Do not confuse a point’s argument, an angle defined modulo 2π, with its modulus, a single nonnegative distance.
Experiment
Complete the missing coordinates before checking the table.
Input z
2iz
Modulus before
Modulus after
1
2i
1
2
i
−2
1
2
1+i
−2+2i
√2
2√2
The table describes exact points, not a sampled proof of the rule for every vector. Draw the edges joining these points to see the orientation and scale together.
Understand
Define the conjugate of z=x+iy by z̄=x−iy and its modulus by |z|=√(x²+y²). Direct multiplication gives zz̄=|z|². Conjugation respects sums and products. Consequently |zw|²=zw z̄w̄=|z|²|w|²; taking nonnegative square roots proves multiplicativity of modulus. A nonzero number has inverse z̄/|z|², so division is defined precisely away from zero.
The triangle inequality is an algebraic theorem as well as a picture. Expand |z+w|² into |z|²+2Re(zw̄)+|w|². For any complex a, Re(a)≤|a|, so the expression is at most (|z|+|w|)². Taking square roots gives |z+w|≤|z|+|w|. Equality requires the two nonzero vectors to point in the same direction; opposing vectors generally produce cancellation.
A disk centered at a with radius r consists of points satisfying |z−a|<r. This definition transfers Euclidean limits and continuity directly to complex numbers. A complex sequence converges exactly when its real and imaginary coordinates converge: each coordinate error is at most the modulus error, while the modulus error is at most the sum of the coordinate errors.
Apply
Repair this claim: “Equal moduli imply equal complex numbers.” The numbers 1 and i disprove it. Equal moduli describe a circle, not a single point. Now compare |z−2| and |z+2|. Squaring both expressions cancels the y² terms and identifies the imaginary axis as their equality set.
Explain why squaring is legitimate here: both sides are nonnegative distances. In later problems, squaring an equation without checking signs can introduce extraneous solutions. The geometric interpretation helps keep that distinction visible.
Continue
Keep a coordinate sketch and an algebraic calculation side by side. The exponential next supplies a systematic way to express rotations. Optional depth: prove that multiplication by x+iy has real matrix with rows (x,−y) and (y,x), and determinant x²+y². This explains both the area factor and why nonzero complex multiplication preserves orientation.
Apply · explain · revise
Practice and reasoning
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Find |(4+2i)−(1−2i)|.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Multiplication by −i rotates a nonzero vector by:
Prove |z+w|≤|z|+|w| without assuming the geometric triangle inequality.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Describe all z with |z−1|=|z+3| and justify the answer.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove ||z|−|w||≤|z−w|.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For nonzero z,w, characterize equality in |z+w|≤|z|+|w|.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
