ALGEBRAArithmeticMathematics Calculator
i

Powers of i: Imaginary Unit Cycle

i = √(-1). i²=-1. Powers cycle: i⁰=1, i¹=i, i²=-1, i³=-i, i⁴=1. i^n = i^(n mod 4). Euler: e^(iπ/2)=i.

Concept Fundamentals
-1
i²
1,i,-1,-i
Cycle
i^(n mod 4)
i^n
i
e^(iπ/2)
Compute Powers of iEnter exponent n

Why This Mathematical Concept Matters

Why: Imaginary unit i appears in complex numbers, electrical engineering (phasors), and quantum mechanics. The 4-cycle simplifies i^n for any n.

How: i²=-1. So i³=i²·i=-i, i⁴=i²·i²=1. Cycle repeats. i^n = i^(n mod 4). For negative n: i^(-n)=1/i^n.

  • ●i⁰=1, i¹=i, i²=-1, i³=-i, i⁴=1. Cycle of 4.
  • ●i^n = i^(n mod 4). e^(iπ/2)=i.
  • ●i^(-1)=1/i = -i (since i·(-i)=1).
Sources:Khan AcademyNCTM Standards

📐 Examples — Click to Load

Enter Power

powers_of_i.sh
CALCULATED
$ i_pow 7
i^7
-i
n mod 4
3
Complex form
0-1i
Cycle
1, i, -1, -i
Powers of i Calculator
i^7 = -i
numbervibe.com
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Cycle (1, i, -1, -i)

Active Value

📐 Step-by-Step Breakdown

INPUT
i^n
i^7
n mod 4
7 mod 4 = 3
ext{Cycle} ext{length} ext{is} 4
RESULT
Result
-i
Complex form
0-1i
a + ext{bi} ext{representation}

⚠️For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

i

i² = -1. i^n cycles 1,i,-1,-i

— Imaginary unit

🔄

i^n = i^(n mod 4)

— Four-cycle

📋 Key Takeaways

  • • i = √(-1), so i² = -1
  • • Cycle: i^0=1, i^1=i, i^2=-1, i^3=-i, i^4=1
  • • i^n = i^(n mod 4) — use remainder when dividing by 4
  • • Euler: e^(iπ/2) = i; e^(iπ) = -1

💡 Did You Know?

🔢i^7 = i^3 = -i (7 mod 4 = 3)Source: Cycle
📐i represents 90° rotation on the complex planeSource: Geometry
🔄i^(-1) = -i (multiplicative inverse)Source: Algebra
📊AC circuits: engineers use j instead of i to avoid confusion with currentSource: Engineering
⚡e^(iπ) + 1 = 0 connects five fundamental constantsSource: Euler's Identity
📏|i^n| = 1 for all integer nSource: Magnitude

📖 How Powers of i Work

Since i⁴ = 1, the powers of i repeat every 4: i^0=1, i^1=i, i^2=-1, i^3=-i. For any integer n, i^n = i^(n mod 4). For negative n, use ((n mod 4) + 4) mod 4 to get a positive remainder.

📝 Worked Example: i^7

Step 1: 7 mod 4 = 3

Step 2: i^7 = i^3 = -i

Complex form: 0 - 1i

⚠️ Common Mistakes to Avoid

  • i² = 1: Wrong. i² = -1.
  • Negative powers: i^(-1) = -i, not 1/i. Use mod: (-1+4) mod 4 = 3, so i^(-1) = i^3 = -i.
  • Forgetting cycle: i^100 = i^0 = 1. Don't compute 100 multiplications.

🎯 Expert Tips

💡 Mod 4 Shortcut

For large n, only n mod 4 matters. i^999 = i^3 = -i.

💡 Negative Exponents

Use ((n mod 4) + 4) mod 4 for negative n to get 0–3.

❓ FAQ

What is i?

i = √(-1), the imaginary unit. i² = -1.

Why does i^n cycle?

i^4 = 1, so i^(n+4) = i^n. Only 4 possible values: 1, i, -1, -i.

What is i^0?

i^0 = 1. Any non-zero number to the power 0 equals 1.

What is i^(-1)?

i^(-1) = -i. The multiplicative inverse of i is -i since i(-i) = -i² = 1.

⚠️ Disclaimer: Uses standard i² = -1. For electrical engineering, j is often used instead of i.

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