Formula and Method for Imaginary Numbers
The imaginary unit i is defined by i = √-1, which means i² = -1. A number of the form bi (where b is any real number) is a pure imaginary number, and this identity is the key that unlocks every negative square root, imaginary-number operation, and power of i below.
How the calculation works
This calculator handles three related tasks built on i² = -1. First, it simplifies √(radicand): for a negative radicand -n, it applies √(-n) = √(-1) × √n = i√n, then pulls the largest perfect-square factor out of the remaining radical (for example √-48 = √(16 × 3) × i = 4√3 i); for a non-negative radicand it just returns the ordinary real square root. Second, it combines two pure imaginary numbers, b₁i and b₂i, using the operation you choose: addition and subtraction keep the result imaginary — (b₁ ± b₂)i — while multiplication and division convert i² into -1, so (b₁i)(b₂i) = -b₁b₂ and (b₁i) ÷ (b₂i) = b₁/b₂ are both real numbers. Third, it evaluates iⁿ for an integer exponent by reducing n modulo 4, since powers of i repeat in the fixed cycle i, -1, -i, 1.
Common mistakes
- Treating √-1 as undefined: in the real numbers it is undefined, but in the complex numbers it is defined as i — that is the entire point of the imaginary unit.
- Forgetting i² = -1 when multiplying: (3i)(4i) is not 12i; the two i's multiply to i² = -1, giving -12, a real number.
- Mis-simplifying radicals: √-48 is not 48i or 6.93i — factor out the largest perfect square first (16), giving 4√3 i.
- Losing track of the power-of-i cycle: i⁵ is not a new value; 5 mod 4 = 1, so i⁵ = i¹ = i.
Real-world applications
Imaginary and complex numbers describe quantities that a single real number line cannot, such as phase and impedance in AC electrical circuits, rotations and oscillations in signal processing, and the roots of polynomials that never cross the real axis (like x² + 1 = 0). Engineers and physicists write these as pure imaginary or complex values precisely so the algebra of i² = -1 can track rotation and phase shift automatically.
Frequently Asked Questions
What is an imaginary number?
How do you simplify the square root of a negative number?
What happens when you multiply or divide two imaginary numbers?
How do you evaluate powers of i?
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|---|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
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General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|---|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
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Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|---|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |