Enter a Complex Expression
Use +, -, *, /, ^, parentheses, i or j. Implicit multiplication works in 2i and (1+i)(2-i).
Example Data Table
| Expression | Simplified form | Action |
|---|---|---|
| (3+4i)+(2-i) | 5 + 3i | |
| (2+3i)(4-i) | 11 + 10i | |
| (4+2i)/(1-i) | 1 + 3i | |
| i^12 | 1 | |
| 2(3-i)-i | 6 - 3i |
Formula Used
Every complex result is written as a + bi. The real component is a. The imaginary coefficient is b.
For division, the denominator is multiplied by its conjugate. This makes the denominator real before the final simplification.
How to Use This Calculator
- Type a complex expression using numbers, i or j, and supported operators.
- Use parentheses to keep grouped values together.
- Choose the decimal precision and a small zero threshold.
- Select your preferred output unit and result view.
- Choose working steps when you need an arithmetic trail.
- Select Simplify Expression to place the result above this form.
- Use CSV or PDF downloads when you need a saved report.
Understanding Standard Complex Form
Complex numbers combine a real part with an imaginary part. Standard form writes every value as a plus bi. Here, a is real. The letter b is also real. The symbol i represents the square root of negative one. This notation separates both components clearly. It makes addition, subtraction, multiplication, and division easier to check. A calculator can reduce long expressions without hiding the important structure. You still see the final real component and imaginary component. That result helps with algebra, engineering, circuits, signals, and geometry.
Reading Terms Carefully
Start by reading parentheses from the inside outward. Replace every occurrence of i squared with negative one. Then combine matching real terms and matching imaginary terms. For example, three plus four i added to two minus i becomes five plus three i. The real values combine together. The imaginary coefficients combine together. This rule also works when one part is zero. A result such as seven plus zero i is simply seven. A result such as zero minus five i is normally written as negative five i.
Products, Quotients, and Powers
Multiplication needs special attention because i squared changes the sign. Multiply each term as you would multiply two binomials. Then replace i squared by negative one. For example, multiply two plus three i by four minus i. The raw products include eight, negative two i, twelve i, and negative three i squared. Replacing the last term gives positive three. The simplified result is eleven plus ten i. Keeping the terms visible helps prevent sign errors.
Division uses a conjugate. The conjugate of c plus di is c minus di. Multiply the numerator and denominator by that conjugate. The denominator becomes c squared plus d squared. This value is real, which removes i from the denominator. Then divide both numerator components by the real denominator. This method works for every nonzero complex denominator. The calculator applies this same rule before presenting your simplified form.
Powers can be simplified through repeated multiplication. Integer powers are supported because their results remain predictable. Positive powers multiply the base repeatedly. Negative powers first find the reciprocal. Powers of i follow a cycle of four. The sequence is i, negative one, negative i, and one. Recognizing that cycle can simplify an expression quickly. Parentheses remain important. They show which complete complex value receives the exponent.
Precision and Verification
Use a sensible rounding setting when decimal input is involved. Exact integers are usually best for learning. Decimal results can appear after division. Check the displayed ordered pair as a second view. Its first value is a. Its second value is b. The modulus and angle add useful context, but they do not replace standard form. Enter a valid expression, review the steps, and verify signs. Use results to check each learning attempt. Small checks expose misplaced parentheses. They quickly catch incorrect signs and invalid denominators before final answers. Clear notation produces dependable complex-number results in every calculation.
Frequently Asked Questions
1. What does a + bi mean?
It is the standard representation of a complex number. The value a is the real part. The value b is the coefficient of the imaginary unit i.
2. Does the calculator accept j?
Yes. You may enter i or j. The calculator treats both symbols as the imaginary unit and displays the selected output symbol.
3. Can I type implicit multiplication?
Yes. Write 2i, 2(1+i), or (1+i)(2-i). The calculator inserts multiplication between adjacent values automatically.
4. Which powers are supported?
Use real whole-number powers from -100 to 100. This includes common forms such as i^4, (2+i)^3, and (1-i)^-2.
5. Why does i squared become negative one?
The imaginary unit is defined so that i² equals -1. This identity turns products of imaginary terms into real values.
6. How is complex division simplified?
The calculator uses the conjugate of the denominator. This removes the imaginary unit from the denominator and leaves a real divisor.
7. What happens with a zero denominator?
The calculator stops and shows an error. Dividing by zero is undefined for both real and complex numbers.
8. Can I use decimal values?
Yes. Decimal inputs are accepted. Use the precision setting to control displayed rounding after the expression is simplified.
9. Does the result include modulus and argument?
Yes. The result panel includes modulus and the principal argument in degrees. These are useful checks for polar relationships.
10. Why display the ordered pair?
The pair (a, b) shows the real component first and the imaginary coefficient second. It offers a quick second view of standard form.
11. Should I round during manual work?
Clear notation keeps complex calculations accurate, readable, and useful.