Study guides › Level 8 · Algebra 2

The number i

Level 8 · Algebra 2 · for questions like “i is the number with i² = −1. What is i²³?”

💡 The idea

i² = −1. So the powers of i go round in a loop of four: i, −1, −i, 1, then repeat.

🪜 The steps

1. Divide the power by 4 and look at the remainder.
2. Remainder 1 → i, 2 → −1, 3 → −i, 0 → 1.
3. For (a + bi)(a − bi): the middle terms cancel, leaving a² + b².

✏️ Worked examples

1. i²³: 23 ÷ 4 leaves 3, so the answer is −i.

2. i⁴⁰: 40 ÷ 4 leaves 0, so the answer is 1.

3. (3 + 2i)(3 − 2i): 9 − 4i² = 9 + 4 = 13.

⚠️ Watch out

• i² is −1, not 1. That's what flips the minus into a plus in example 3.

🎯 Try these

1. i¹⁰
2. i⁵³
3. (5 + i)(5 − i)

Answers: 1) −1 2) i 3) 26

This guide is also in the 10MathQ app: tap “📖 Study guide” after a question.