Find and Calculate Complex Numbers in a + bi Form

Identify what the problem asks you to find

A complex number has the form z=a+bi, where a and b are real numbers and i²=−1. Its real part is a and its imaginary part is b, not bi. In z=3−4i, the components are therefore 3 and −4. A purely real number such as 7 is also complex, with imaginary part zero. If a problem supplies an equation rather than the components themselves, rearrange or solve that equation first; a two-number arithmetic form does not determine unknown coefficients automatically.

Separate real and imaginary components in an equation

Suppose z+(2−3i)=5+i. Subtract 2−3i from both sides to get z=(5+i)−(2−3i)=3+4i. In the local form, enter the first number as real 5 and imaginary 1, select subtraction, then enter real 2 and imaginary −3 for the second number. The arithmetic result verifies the rearrangement. Substituting back gives (3+4i)+(2−3i)=5+i. This final substitution checks the original equation and catches a sign error that merely inspecting the result might miss.

Add or subtract like components

For addition, combine the real parts and combine the imaginary parts separately. For example, (3−4i)+(−1+6i)=2+2i. For subtraction, distribute the minus sign to both components of the second number: (3−4i)−(−1+6i)=4−10i. Enter a negative imaginary component as a signed number in its own field; do not type the whole expression 3−4i into the real field. Recording the four components before calculation makes the chosen operation and each sign unambiguous.

Multiply using i squared equal to negative one

Expand (2+3i)(4−i) just as a product of binomials: 8−2i+12i−3i². Replacing i² with −1 gives 11+10i. The general real component is ac−bd and the imaginary component is ad+bc for (a+bi)(c+di). The minus sign in ac−bd is essential; using ac+bd treats i² as positive one and changes the result. Verify the worked example with real/imaginary pairs 2,3 and 4,−1 and multiplication selected.

Divide by multiplying with a conjugate

The conjugate of c+di is c−di, and their product is the real number c²+d². Thus (3+2i)/(1−i) becomes ((3+2i)(1+i))/2=(1+5i)/2=0.5+2.5i. The denominator must not be zero in both components. A zero real component alone is acceptable when the imaginary component is nonzero; for example, division by 2i is defined. Check a quotient by multiplying it by the original divisor and recovering the dividend within displayed rounding.

Interpret the modulus as distance

The modulus of z=a+bi is |z|=sqrt(a²+b²), its distance from the origin in the complex plane. For 3+4i, the modulus is five. The point's horizontal coordinate is the real part and its vertical coordinate is the imaginary part, so this calculation is a right-triangle distance rather than ordinary addition of components. The modulus cannot be negative. The tool reports it for the arithmetic result, so distinguish that result from either of the two input numbers when comparing values.

Choose the argument using the correct quadrant

The argument is an angle locating a nonzero complex number from the positive real axis. An atan2-style calculation accounts for the signs of both components; simply using arctan(b/a) can select the wrong quadrant and is unsuitable when a is zero. The number −1+i has principal argument 135 degrees, or 3π/4 radians, rather than −45 degrees. Other arguments differ by whole turns. The argument of zero is undefined because the origin has no direction; its modulus is still zero.

Separate arithmetic from root finding and precision

Solving z²=−9 gives two solutions, 3i and −3i, whereas calculating a product of two supplied complex numbers is a different operation. Do not assume the local arithmetic form is a symbolic equation solver or a source of every branch of a complex function. Decimal output is rounded, and cancellation can leave small numerical residuals. Keep sufficient precision when using a result in another operation, and check an exact identity algebraically when the distinction between zero and a tiny nonzero value matters.

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