Hotmath
Math Homework. Do It Faster, Learn It Better.

Operations with Complex Numbers

Toaddtwocomplex numbers,添加the real part to the real part and the imaginary part to the imaginary part.

( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i

Example 1:

( 2 + 7 i ) + ( 3 − 4 i ) = ( 2 + 3 ) + ( 7 + ( − 4 ) ) i = 5 + 3 i

Tosubtracttwo complex numbers, subtract the real part from the real part and the imaginary part from the imaginary part.

( a + b i ) − ( c + d i ) = ( a − c ) + ( b − d ) i

Example 2:

( 9 + 5 i ) − ( 4 + 7 i ) = ( 9 − 4 ) + ( 5 − 7 ) i = 5 − 2 i

Tomultiplytwo complex numbers, use theFOILmethod andcombine like terms.

( a + b i ) ( c + d i ) = a c + a d i + b c i + b d i 2 = a c + ( a d + b c ) i − b d ( Remember i 2 = − 1 ) = ( a c − b d ) + ( a d + b c ) i

Example 3:

( 3 + 2 i ) ( 5 + 6 i ) = 15 + 18 i + 10 i + 12 i 2 = 15 + 28 i − 12 = 3 + 28 i

Todividetwo complex numbers, multiply the numerator and denominator by the complexconjugate, expand and simplify. Then, write the final answer in standard form.

a + b i c + d i ⋅ c − d i c − d i = ( a c + b d ) + ( b c − a d ) i c 2 + d 2

Example 4:

3 + 2 i 4 − 5 i = 3 + 2 i 4 − 5 i ⋅ 4 + 5 i 4 + 5 i = 12 + 15 i + 8 i + 10 i 2 16 + 25 = 2 + 23 i 41 = 2 41 + 23 41 i