math-2017:N.CN.3
math-2017:N.CN.3 Find the conjugate of a complex number; use conjugates to find magnitudes and quotients of complex numbers.
Learning goal: Find the conjugate of a complex number, use the product of a number and its conjugate to find magnitudes, and use conjugates to write quotients of complex numbers in the form a + bi.
Scope note. N.CN.3 is marked (+) in Ohio's Learning Standards for mathematics, which Ohio uses to indicate a standard "for students who are planning on taking advanced courses". It is taught here in full, as extension content for students on that path.
The conjugate. The conjugate of the complex number z = a + bi is the number a − bi: same real part, opposite imaginary part. It is written z̄ (read "z bar"). The conjugate of 3 + 4i is 3 − 4i; the conjugate of −2 − 5i is −2 + 5i; the conjugate of 7i is −7i; the conjugate of the real number 6 is 6 itself, because 6 = 6 + 0i and changing the sign of 0 does nothing. Conjugating twice returns the original number. Only the sign of the imaginary part changes; a student who writes the conjugate of −2 − 5i as 2 + 5i has changed both signs and produced the opposite of the number, which is a different thing.
The key product. Multiply a number by its conjugate: (a + bi)(a − bi) = a² − abi + abi − b²i² = a² + b². The two middle terms cancel because they are opposites, and −b²i² = +b². So z·z̄ = a² + b², a real number that is never negative and is 0 only when z = 0. This is the fact that makes everything else in the lesson work. For z = 3 + 4i: (3 + 4i)(3 − 4i) = 9 + 16 = 25.
Magnitude. The magnitude (also called the modulus or absolute value) of z = a + bi is |z| = √(a² + b²), the nonnegative square root of the conjugate product. So |z|² = z·z̄. For 3 + 4i the magnitude is √25 = 5. For −2 + 2i it is √(4 + 4) = √8 = 2√2, about 2.83. For the real number −7 it is √49 = 7, agreeing with the ordinary absolute value. For the pure imaginary number −7i it is also 7. For 1 − i√3 it is √(1 + 3) = 2. Two things to notice: a magnitude is always a real number that is 0 or positive, and a number and its conjugate have the same magnitude, because a² + (−b)² = a² + b². In the next lessons the magnitude becomes the distance from 0 to the number in the complex plane; for now it is defined by the formula.
A misconception about magnitude. |3 + 4i| is not 3 + 4 = 7 and not 3² + 4² = 25. The magnitude is the square root of the sum of the squares, exactly like the length of the hypotenuse of a right triangle with legs 3 and 4. Forgetting the square root is the most common slip; checking that |3 + 4i| = 5 rather than 25 is a good habit, since 5² = 25 = z·z̄.
Division: the problem. What does (2 + i)/(3 − 4i) mean, and how is it written in the form a + bi? Dividing by a real number is easy: (2 + i)/5 = 2/5 + (1/5)i. Dividing by a number with a nonzero imaginary part is not obviously possible, because the standard form has no i in a denominator. The conjugate solves this: multiplying numerator and denominator by the conjugate of the denominator makes the denominator the real number a² + b², after which division by a real number is just dividing both parts.
A worked model inside the lesson. Write (2 + i)/(3 − 4i) in the form a + bi. The conjugate of the denominator is 3 + 4i. Multiply numerator and denominator by it: [(2 + i)(3 + 4i)] / [(3 − 4i)(3 + 4i)]. Denominator: 9 + 16 = 25. Numerator: 2·3 + 2·4i + i·3 + i·4i = 6 + 8i + 3i + 4i² = 6 + 11i − 4 = 2 + 11i. So the quotient is (2 + 11i)/25 = 2/25 + (11/25)i, that is, 0.08 + 0.44i. Check by multiplying back: (0.08 + 0.44i)(3 − 4i) = 0.24 − 0.32i + 1.32i − 1.76i² = 0.24 + 1.00i + 1.76 = 2 + i, which is the numerator we started with. The check is the reason the method is trustworthy: a quotient q = z/w is correct exactly when q·w = z.
Why multiplying top and bottom is allowed. Multiplying numerator and denominator by the same nonzero number is multiplying the fraction by 1, so the value does not change. Multiplying by the conjugate specifically is what makes the new denominator real, because w·w̄ = |w|² is real. Any nonzero complex number w has a nonzero magnitude, so division by any nonzero complex number is possible; the only forbidden division is by 0, as with real numbers.
A second example, with a clean answer. (4 + 2i)/(1 − i). Multiply by (1 + i)/(1 + i): denominator 1 + 1 = 2; numerator (4 + 2i)(1 + i) = 4 + 4i + 2i + 2i² = 4 + 6i − 2 = 2 + 6i. Quotient: (2 + 6i)/2 = 1 + 3i. Check: (1 + 3i)(1 − i) = 1 − i + 3i − 3i² = 1 + 2i + 3 = 4 + 2i.
The reciprocal of i. 1/i: the conjugate of i is −i, so 1/i = (1·(−i))/(i·(−i)) = −i/(−i²) = −i/1 = −i. Check: i·(−i) = −i² = 1. So 1/i = −i, a fact that surprises many students and is worth remembering.
Magnitudes and products. Because |z|² = z·z̄ and the conjugate of a product is the product of the conjugates, |zw| = |z|·|w|. For instance, |(3 + 4i)(1 − 2i)| should equal 5·√5. Computing the product gives 3 − 6i + 4i − 8i² = 11 − 2i, whose magnitude is √(121 + 4) = √125 = 5√5, as predicted. Likewise |z/w| = |z|/|w| for w ≠ 0. These rules let magnitudes be found without expanding, which is useful when the product is messy.
Worked conjugate and magnitude: For z = −5 + 12i, find z̄, z·z̄ and |z|. The conjugate is −5 − 12i. The product is (−5 + 12i)(−5 − 12i) = 25 + 60i − 60i − 144i² = 25 + 144 = 169. The magnitude is |z| = √169 = 13. Check the relationship: |z|² = 169 = z·z̄. Also |z̄| = √(25 + 144) = 13, the same magnitude.
Worked quotient: Write (7 − i)/(2 + 3i) in the form a + bi. Multiply by (2 − 3i)/(2 − 3i). Denominator: 4 + 9 = 13. Numerator: (7 − i)(2 − 3i) = 14 − 21i − 2i + 3i² = 14 − 23i − 3 = 11 − 23i. Quotient: 11/13 − (23/13)i, about 0.846 − 1.769i. Check: (11/13 − 23i/13)(2 + 3i) = 22/13 + 33i/13 − 46i/13 − 69i²/13 = (22 + 69)/13 + (33 − 46)i/13 = 91/13 − 13i/13 = 7 − i.
Worked division by a pure imaginary number: Write (3 + 2i)/(5i) in the form a + bi. The conjugate of 5i is −5i. Numerator: (3 + 2i)(−5i) = −15i − 10i² = 10 − 15i. Denominator: (5i)(−5i) = −25i² = 25. Quotient: 10/25 − (15/25)i = 2/5 − (3/5)i. Check: (2/5 − 3i/5)(5i) = 2i − 3i² = 3 + 2i.
Worked magnitude of a quotient without dividing: Find |(5 − 5i)/(2 + i)|. Using |z/w| = |z|/|w|: |5 − 5i| = √50 = 5√2 and |2 + i| = √5, so the magnitude is 5√2/√5 = 5√(2/5) = √10, about 3.16. Confirm by dividing: (5 − 5i)(2 − i)/5 = (10 − 5i − 10i + 5i²)/5 = (5 − 15i)/5 = 1 − 3i, whose magnitude is √(1 + 9) = √10.
Conjugate table: Make a table with columns z, z̄, z·z̄, |z|. Fill it for 3 + 4i, −2 + 2i, −7, −7i, 1 − i√3, 5 + 12i and 1 + i. Show the conjugate product as a full expansion in the first three rows, then use a² + b² directly. Beside the table, write one sentence explaining why z·z̄ can never be negative and one explaining why |z| = |z̄|.
Independent performance: Choose a nonzero complex number w = c + di with both c and d nonzero, and a second number z. Write z/w in the form a + bi by multiplying numerator and denominator by w̄, showing the denominator become the real number c² + d². Then check by multiplying your quotient by w and confirming that z comes back. Finally, compute |z|, |w| and |z/w| separately and confirm that |z/w| = |z|/|w|. Success criteria: the denominator is explicitly shown to be real, the multiply-back check reproduces z exactly, and the magnitude relationship holds to the accuracy of your rounding.
An adult may read the prompts aloud. Answer independently with words, objects, drawings or a demonstration before opening the answer guide. Award 1 point for a correct result or claim and 1 point for a matching explanation or demonstration; where a question asks only for a result (a corrected sentence, a spelling, a count), a fully correct result earns both points, and the answer guide shows what a complete response contains. Record prompting separately. The total (two points per question) is local practice feedback, not an Ohio proficiency level.
Write the conjugate of −4 + 9i and of 6, and explain why a real number is its own conjugate.
The conjugate of −4 + 9i is −4 − 9i. The conjugate of 6 is 6, because 6 = 6 + 0i and changing the sign of the imaginary part 0 leaves 0.
Find (8 − 6i)(8 + 6i) and use it to state |8 − 6i|.
The product is 64 − 36i² = 64 + 36 = 100. Since z·z̄ = |z|², the magnitude is |8 − 6i| = √100 = 10.
A student says |5 + 12i| = 17. Explain the error and give the correct magnitude.
The student added the parts. The magnitude is √(5² + 12²) = √(25 + 144) = √169 = 13, the square root of the sum of the squares.
Write (1 + i)/(1 − i) in the form a + bi and check.
Multiply by (1 + i)/(1 + i): numerator (1 + i)² = 1 + 2i + i² = 2i; denominator 1 + 1 = 2. Quotient: 2i/2 = i. Check: i(1 − i) = i − i² = 1 + i.
Write 10/(3 + i) in the form a + bi.
Multiply by (3 − i)/(3 − i): numerator 10(3 − i) = 30 − 10i; denominator 9 + 1 = 10. Quotient: 3 − i. Check: (3 − i)(3 + i) = 9 + 1 = 10.
Explain why multiplying the numerator and denominator of z/w by w̄ does not change the value of the quotient, and why it is useful.
w̄/w̄ equals 1 for any nonzero w, so the quotient is multiplied by 1. It is useful because w·w̄ = |w|² is a real number, and dividing by a real number just divides each part.
Find |(3 + 4i)(1 − 2i)| in two ways.
By the product rule: |3 + 4i|·|1 − 2i| = 5·√5 = 5√5. By expanding: (3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i² = 11 − 2i, with magnitude √(121 + 4) = √125 = 5√5. Both agree.
Show that 1/i = −i and check by multiplication.
The conjugate of i is −i: 1/i = (−i)/(i·(−i)) = −i/(−i²) = −i/1 = −i. Check: i·(−i) = −i² = 1, so −i is the reciprocal of i.