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The Imaginary Unit i

  • Grade 11
  • Mathematics
  • N-CN.A.1
  • Number & Quantity
  • Free lesson

Aligned to N-CN.A.1 — Common Core State Standards for Mathematics.

What this lesson teaches

There is a number i defined by i^2 = −1, so i = √(−1). Every complex number can be written as a + bi, where a is the real part and b is the imaginary part, both real numbers.

Worked examples

Example 1

Write √(−25) in the form a + bi. Split it as √(25) × √(−1) = 5 × i = 5i. So √(−25) = 0 + 5i, with real part 0 and imaginary part 5.

Example 2

Find i^27. The powers of i repeat every four: i¹ = i, i² = −1, i³ = −i, i⁴ = 1. Divide the exponent by 4 and keep the remainder: 27 = 4(6) + 3, so i^27 = i³ = −i. Only the remainder matters, because i⁴ raised to any power is just 1.

Example 3

Common mistake: multiplying two negative radicands as if the rule still held. √(−4) × √(−9) is NOT √36 = 6. The rule √a × √b = √(ab) requires non-negative a and b. Convert first: √(−4) = 2i and √(−9) = 3i, so the product is 6i² = −6. Writing every negative root in the form bi BEFORE multiplying prevents the sign error.

Practice questions

  1. Write √(−49) in the form a + bi.
  2. Identify the real and imaginary parts of 3 − 7i.
  3. Express √(−16) + 2 as a complex number a + bi.

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En español

La Unidad Imaginaria i

Existe un número i definido por i^2 = −1, así que i = √(−1). Todo número complejo se puede escribir como a + bi, donde a es la parte real y b es la parte imaginaria, ambas números reales.

Ejemplo 1: Escribe √(−25) en la forma a + bi. Sepáralo como √(25) × √(−1) = 5 × i = 5i. Así que √(−25) = 0 + 5i, con parte real 0 y parte imaginaria 5.

Ejemplo 2: Halla i^27. Las potencias de i se repiten cada cuatro: i¹ = i, i² = −1, i³ = −i, i⁴ = 1. Divide el exponente entre 4 y quédate con el residuo: 27 = 4(6) + 3, así que i^27 = i³ = −i. Solo importa el residuo, porque i⁴ elevado a cualquier potencia es simplemente 1.

Ejemplo 3: Error común: multiplicar dos radicandos negativos como si la regla siguiera valiendo. √(−4) × √(−9) NO es √36 = 6. La regla √a × √b = √(ab) exige a y b no negativos. Convierte primero: √(−4) = 2i y √(−9) = 3i, así que el producto es 6i² = −6. Escribir toda raíz negativa en la forma bi ANTES de multiplicar evita el error de signo.

More Number & Quantity lessons

Meaning of Rational ExponentsN-RN.A.1Radicals and Rational ExponentsN-RN.A.2Operations with Complex NumbersN-CN.A.2Quadratics with Complex SolutionsN-CN.C.7

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