Complex numbers · Eqora guide
Complex numbers in math: square or squared modulus?
Separate a complex product from a real distance. Use conjugates to explain squared modulus, division and an independent check.
For z = 3 + 4i, you calculate z² = −7 + 24i. Another calculation gives 3² + 4² = 25. Are these competing answers to the same question? No: the first is the square of z, while the second is its squared modulus, |z|². Complex numbers in math make the distinction essential. Multiplying a number by itself and measuring its distance from zero are different operations, even when both expressions contain a square.
This guide develops the distinction using algebra and the complex plane. You will identify real and imaginary parts, expand a square, form a conjugate, divide by a nonzero complex number and check a distance from a shifted center. Everything is worked out on paper without an app or solver. We use the imaginary unit i with i² = −1, real coefficients, and ordinary Cartesian coordinates with the same scale on both axes.
Eqora publishes this explanation as a learning aid, not an exam substitute or a guarantee of correctness or grades. Check the question, notation, assumptions and result. The pool photographs suggest reflection, paired objects and separation. They are conceptual illustrations, not scale drawings of the complex plane; no ring size, tile count or water reflection encodes our numerical values. The formulas and independent checks provide the mathematical justification.
Complex numbers in math have two real coordinates
Write z = a + bi, where a and b are real. The real part Re(z) is a; the imaginary part Im(z) is b, not bi. For 3 + 4i, those parts are three and four. For 2 − 3i, they are two and negative three. Calling b an imaginary part does not make b itself imaginary. It is a real coefficient telling you how much of the unit i is present.
Represent a + bi by the point (a, b). The horizontal axis records the real part, and the vertical axis records the imaginary coefficient. The origin represents zero. A negative b places the point below the horizontal axis, not at a negative distance from the origin. Our geometry-diagram guide explains why coordinates and stated relationships are more reliable than measuring an illustrative picture that may not be drawn to scale.
Squaring keeps the cross term and uses i² = −1
The square z² means z multiplied by z. Expand both factors: (a + bi)² = a² + 2abi + b²i². Since i² = −1, this becomes (a² − b²) + 2abi. The minus sign belongs to the real part because the two factors i multiply to negative one. The middle term does not disappear. It comes from the two products between one real term and one imaginary term.
For z = 3 + 4i, the expansion is 9 + 24i + 16i² = −7 + 24i. Writing 9 + 16 = 25 skips the cross term and replaces i² by positive one. That does not square this complex number. You can verify by distributing the two original factors separately: nine, twelve i, twelve i and negative sixteen. Group real contributions together and imaginary contributions together only after preserving their signs.
For z = i alone, z² = −1, although its distance from zero is one. This small example immediately rejects the claim that every complex square is nonnegative. Nonnegativity applies to real squares and to squared moduli, not to arbitrary complex squares. The usual real ordering is not extended to all complex numbers while preserving the ordinary ordered-field rules; a nonreal product is not a positive or negative real distance.
The modulus is a distance, not another complex point
The modulus |a + bi| is √(a² + b²). It is the Euclidean distance from (a, b) to the origin. The right triangle formed by the coordinate projections explains the sum of squares: each leg length is the magnitude of its coordinate, and squaring removes the sign. For 3 + 4i, |z| = √(9 + 16) = 5. The output is a nonnegative real number, not a new pair of coordinates.
Squaring this distance gives |z|² = a² + b² = 25. Keep the bars while writing the square. The expressions z² and |z|² ask different questions; removing the bars changes the operation. Also distinguish |z| from |z|²: five is the distance, twenty-five its square. If a problem asks for a length, stop at the square root rather than reporting the squared distance because it looks simpler.
The same modulus can belong to many numbers. Both 3 + 4i and −3 + 4i lie five units from the origin, but their real parts differ. A fixed positive modulus describes a circle, not one unique point. Zero is the only complex number with modulus zero. Coordinate distances require the same scale on both axes; stretching the displayed axes differently changes the picture’s appearance, not the modulus formula.
Conjugation reflects the imaginary coordinate
The conjugate of z = a + bi is z̄ = a − bi. Keep a unchanged and reverse the sign of b. Thus 3 + 4i has conjugate 3 − 4i, while 2 − 3i has conjugate 2 + 3i. Applying conjugation twice recovers the original number. Geometrically, this is reflection across the real axis, so the distance from the origin stays the same even though a nonzero imaginary coordinate changes sign.
Conjugation is not the same as taking the opposite. The opposite of 3 + 4i is −3 − 4i, which reverses both coefficients. Nor is conjugation automatically a reciprocal. Ohio State’s Ximera conjugate lesson derives the relationship between conjugation, squared modulus and multiplicative inverse. The operation is useful because paired imaginary terms can cancel in particular expressions; it does not mean every expression involving a conjugate becomes real.
Add z and z̄: (a + bi) + (a − bi) = 2a. Subtract them: z − z̄ = 2bi. These two calculations offer a quick sign check and show which part each operation isolates. Real numbers satisfy z̄ = z, because b = 0. Pure imaginary numbers satisfy z̄ = −z. Do not infer either relation for a general number with both coefficients nonzero.
A conjugate pair produces the squared modulus
Multiply different factors this time: z z̄ = (a + bi)(a − bi). The cross terms −abi and abi cancel, leaving a² − b²i² = a² + b². Therefore z z̄ = |z|². The second factor must be the conjugate for this identity. In z², the two cross terms have the same sign instead and combine into 2abi. That single change of factor explains why the results differ.
For 3 + 4i, z z̄ = (3 + 4i)(3 − 4i) = 25, while z² = −7 + 24i. But the modulus of z² is √(49 + 576) = 25. So |z²| = |z|², even though z² is not |z|². This is a useful comparison of notation: placing bars around the product converts the result into a distance. It does not erase the imaginary part of the original product itself.
The two squares agree when z is real. For a nonreal z, they do not: equality of imaginary parts would require 2ab = 0, and equality of real parts would require a² − b² = a² + b², forcing b = 0. This shows why a coincidence in a real-only example is not a general complex-number rule. Check both parts before declaring two complex expressions equal.
Divide by changing both numerator and denominator
To divide by a nonzero complex number c + di, multiply the whole fraction by (c − di)/(c − di). This factor equals one, so the quotient’s value is unchanged. The new denominator is c² + d², a positive real number because c and d are not both zero. Conjugating only the denominator would change the fraction instead of creating an equivalent expression.
For (3 + 4i)/(1 − 2i), the denominator’s conjugate is 1 + 2i. The numerator becomes (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i² = −5 + 10i. The denominator becomes (1 − 2i)(1 + 2i) = 5. Divide both numerator terms by five: the result is −1 + 2i. Notice that the conjugate belongs to the denominator, not whichever number is easiest to recognize.
Verify without repeating the same fraction manipulation: (−1 + 2i)(1 − 2i) = −1 + 2i + 2i − 4i² = 3 + 4i. Multiplying the proposed quotient by the original denominator recovers the original numerator. Our independent answer-checking guide emphasizes this return to the starting relation. A decimal approximation may look plausible; an exact reverse multiplication checks the real and imaginary coefficients separately.
An inverse needs a nonzero squared modulus
Set the numerator to one. For z ≠ 0, 1/z = z̄/|z|². With z = 3 + 4i, this gives (3 − 4i)/25 = 3/25 − (4/25)i. The conjugate alone would produce a product of twenty-five with z, not one. Division by the squared modulus provides the normalization needed for a multiplicative inverse. Parentheses around the numerator make clear that both coefficients are divided by twenty-five.
At zero, |z|² = 0 and the expression is undefined. Do not cancel this denominator or treat it as a limiting instruction supplied by the problem. No complex number multiplied by zero gives one. For a nonzero real z, the same formula reduces to the usual real reciprocal. Naming the permitted domain first prevents a correct symbolic identity from being applied at a forbidden value.
A shifted center changes the number inside the bars
The distance from z to another point c is |z − c|, not generally |z| − |c|. Let z = 3 + 4i and c = 1 + i. Subtract corresponding coefficients: z − c = 2 + 3i. The distance is √(2² + 3²) = √13. Subtracting the two distances from the origin, 5 − √2, would measure a different quantity and loses the directions of the two points.
An equation |z − c| = r describes a circle centered at c when r > 0, a single point when r = 0, and no points when r < 0. Writing z = a + bi converts |z − (1 + i)| = 2 into (a − 1)² + (b − 1)² = 4. The center is (1, 1); the radius is two. Four is the squared radius, not the radius itself.
These are distances in the mathematical coordinate model. If coordinates represent physical quantities, the interpretation and compatible units must be stated separately. A pool reflection cannot determine the coordinate signs, scale or center. Use the subtraction and the modulus formula, then test a known point: z = 3 + i lies two units from 1 + i. This checks the center and radius without trusting the sketch’s apparent size.
Transfer the distinction to a new number
Try w = 2 − 3i without consulting the earlier numerical results. Find w², w̄, |w|, |w|², 1/w and w/(1 + i). Predict which outputs must be real and which may remain complex. Work through the operations before reading the checks. The sign of the imaginary coefficient changes from our main example, so copying its pattern of plus signs will not demonstrate understanding.
The square is w² = 4 − 12i + 9i² = −5 − 12i. The conjugate is 2 + 3i, and w w̄ = 13, giving |w| = √13 and |w|² = 13. The inverse is (2 + 3i)/13. Finally, multiplying the quotient’s numerator by 1 − i gives −1 − 5i, while its denominator is two: w/(1 + i) = −1/2 − (5/2)i.
Check that (−1/2 − (5/2)i)(1 + i) recovers 2 − 3i and that the modulus of −5 − 12i is thirteen. Explain the different roles of thirteen, √13 and the complex square. If they became interchangeable in your notes, restore the bars and factor labels. Then choose another number and repeat the comparison. Understanding means being able to justify the operation, not merely recognizing a familiar answer.
