Represent complex numbers on the complex plane in rectangular and polar form and explain why the two forms name the same number

Learning goal: Represent complex numbers, including real and pure imaginary numbers, as points on the complex plane; write a complex number in rectangular form a + bi and in polar form r(cos θ + i sin θ); convert between the forms; and explain why the rectangular and polar forms of a given complex number represent the same number.

Scope note. N.CN.4 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 complex plane. A real number is a point on a number line. A complex number a + bi needs two real numbers, so it is a point in a plane. The complex plane has a horizontal real axis and a vertical imaginary axis, and the number a + bi is plotted at the point with horizontal coordinate a and vertical coordinate b. So 3 + 4i sits 3 units right and 4 units up from the origin. The origin is the number 0. Real numbers, which have imaginary part 0, lie on the real axis: −3 is plotted at (−3, 0). Pure imaginary numbers, which have real part 0, lie on the imaginary axis: 2i is at (0, 2) and −4i is at (0, −4). Every point in the plane is exactly one complex number, and every complex number is exactly one point. A frequent confusion is to plot 2i on the real axis "because 2 is a real number"; the coefficient 2 is the imaginary part, so the point is on the vertical axis.

Rectangular form. The form a + bi is called rectangular form because a and b are the rectangular coordinates of the point. In this form addition and subtraction are easy, since real parts and imaginary parts combine separately.

Polar form. The same point can be located a second way: by its distance r from the origin and the angle θ measured counterclockwise from the positive real axis to the ray from the origin through the point. The distance r is the magnitude |z| = √(a² + b²) from the previous lesson, and the angle θ is called the argument. For a point in the first quadrant, drawing the right triangle with horizontal leg a, vertical leg b and hypotenuse r, right-triangle trigonometry gives a = r cos θ and b = r sin θ. In the other quadrants a or b is negative and the legs have lengths |a| and |b|, but the same two equations still hold, because cosine and sine of any angle are defined so that the point at distance r in direction θ is exactly (r cos θ, r sin θ). Substituting into a + bi gives z = r cos θ + i r sin θ = r(cos θ + i sin θ). This is polar form. The number r is never negative and is unique; r = 0 only for the number 0, which has no defined argument because it points in no direction. The angle θ is not unique: adding or subtracting 360° (or 2π radians) points the same way, so 45° and 405° and −315° all describe the same number. When a single answer is required, an argument in the interval from 0° to 360°, or from −180° to 180°, is usually chosen; say which.

Why the two forms represent the same number. This is the heart of the standard. The two forms are not two different numbers that happen to be related; they are two descriptions of one point. Rectangular form says "go a to the right and b up." Polar form says "turn through θ and go out r." Both instructions land on the same point because a = r cos θ and b = r sin θ are exactly the horizontal and vertical legs of the triangle whose hypotenuse has length r at angle θ. Substituting those legs into a + bi reproduces r(cos θ + i sin θ) term by term. So the equality a + bi = r(cos θ + i sin θ) is an identity about one point, not a coincidence of arithmetic.

Converting rectangular to polar — a worked model inside the lesson. Write 1 + i in polar form. Here a = 1 and b = 1, so r = √(1 + 1) = √2. The point is in the first quadrant, and tan θ = b/a = 1, so θ = 45°. Polar form: √2(cos 45° + i sin 45°). Check by converting back: √2 cos 45° = √2·(√2/2) = 1 and √2 sin 45° = 1, giving 1 + i. Now write −1 + √3 i in polar form: r = √(1 + 3) = 2. The point has negative real part and positive imaginary part, so it is in the second quadrant; the reference angle has tan equal to √3/1, which is 60°, so θ = 180° − 60° = 120°. Polar form: 2(cos 120° + i sin 120°). Check: 2 cos 120° = 2·(−1/2) = −1 and 2 sin 120° = 2·(√3/2) = √3. This particular number returns in the next lesson.

Quadrants are the trap. A calculator's inverse tangent returns an angle between −90° and 90°, which is correct only when the real part is positive. For −1 − i, tan θ = (−1)/(−1) = 1 and a calculator says 45°, but the point is in the third quadrant, so θ = 45° + 180° = 225° (equivalently −135°). Always plot the point, or at least check the signs of a and b, before accepting an angle. For points on the axes, no tangent is needed: 5 has θ = 0°, −3 has θ = 180°, 2i has θ = 90° and −4i has θ = 270° (or −90°).

Converting polar to rectangular. Multiply out. 4(cos 150° + i sin 150°) = 4(−√3/2) + 4(1/2)i = −2√3 + 2i, about −3.46 + 2i. And 2(cos(−60°) + i sin(−60°)) = 2(1/2) + 2(−√3/2)i = 1 − √3 i, about 1 − 1.73i. The magnitude of the result should match r: |1 − √3 i| = √(1 + 3) = 2, as it should.

Real and imaginary numbers in polar form. The real number 5 is 5(cos 0° + i sin 0°); the real number −3 is 3(cos 180° + i sin 180°), with r = 3 rather than −3, because r is a distance; the pure imaginary 2i is 2(cos 90° + i sin 90°). Writing −3 as "−3(cos 0° + i sin 0°)" is a common error; the magnitude must be nonnegative and the sign is carried by the angle.

Angles in degrees or radians. Either unit is fine as long as it is stated. 120° is 2π/3 radians, 45° is π/4, 225° is 5π/4. This unit uses degrees in the text so that the geometry is visible; convert when a problem is set in radians.

Reading the diagram. The graph for this lesson plots five numbers: 3 + 4i with r = 5 and θ ≈ 53.1°, −1 + √3 i with r = 2 and θ = 120°, the pure imaginary −4i with r = 4 and θ = 270°, the real number −3 with r = 3 and θ = 180°, and −3 − 3i with r = 3√2 ≈ 4.24 and θ = 225°. The two drawn segments from the origin are the r values of the first two numbers; their lengths are 5 and 2.

Worked rectangular to polar, second quadrant: Write −2 + 2i in polar form. r = √(4 + 4) = √8 = 2√2. The point has negative real part and positive imaginary part, so it lies in the second quadrant. The reference angle has tangent 2/2 = 1, so it is 45°, and θ = 180° − 45° = 135°. Polar form: 2√2(cos 135° + i sin 135°). Check: 2√2 cos 135° = 2√2·(−√2/2) = −2 and 2√2 sin 135° = 2, giving −2 + 2i.

Worked polar to rectangular: Write 6(cos 210° + i sin 210°) in rectangular form. cos 210° = −√3/2 and sin 210° = −1/2, so the number is 6(−√3/2) + 6(−1/2)i = −3√3 − 3i, about −5.20 − 3i. Check the magnitude: √(27 + 9) = √36 = 6, matching r. The point is in the third quadrant, as an angle of 210° requires.

Worked rectangular to polar with a non-special angle: Write 3 + 4i in polar form. r = √(9 + 16) = 5. First quadrant, tan θ = 4/3, so θ = arctan(4/3) ≈ 53.13°. Polar form: 5(cos 53.13° + i sin 53.13°). Check: 5 cos 53.13° ≈ 3.00 and 5 sin 53.13° ≈ 4.00. Now −3 + 4i: same r = 5, but the point is in the second quadrant, so θ = 180° − 53.13° = 126.87°. Same magnitude, different argument, different number.

Worked misconception: A student writes −1 − i as √2(cos 45° + i sin 45°) because the calculator gave 45° for arctan(1). But √2 cos 45° = 1 and √2 sin 45° = 1, which is 1 + i, the wrong number. The point −1 − i is in the third quadrant, so θ = 225°: √2 cos 225° = −1 and √2 sin 225° = −1. Polar form: √2(cos 225° + i sin 225°).

Plot-and-convert record: On graph paper draw the complex plane with both axes from −5 to 5. Plot 3 + 4i, −1 + √3 i, 2i, −3, −3 − 3i and 4 − 3i, labeling each point with its rectangular form. For each, compute r and θ (in degrees, 0° ≤ θ < 360°), record them in a table, and draw the segment from the origin whose length is r. Beside the table, write the converting equations a = r cos θ and b = r sin θ and show them applied to two of the six numbers.

Independent performance: Choose one complex number in each quadrant, none on an axis. Write each in polar form, showing r, the reference angle and the quadrant adjustment. Then write two numbers in polar form with r = 4 and arguments of your choosing, and convert them to rectangular form. Finish with a paragraph, addressed to someone who has seen only rectangular form, explaining why a + bi and r(cos θ + i sin θ) are the same number, using the right triangle with legs a and b and hypotenuse r. Success criteria: every angle is in the correct quadrant, every conversion checks by converting back, and the paragraph names a = r cos θ and b = r sin θ as the reason the forms agree.