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All courses Math II · N-CN.2 56 of 73
Add, subtract, and multiply complex numbers using i^2=-1 and algebraic properties.

Add complex numbers in standard form by combining real parts and imaginary parts

Problem
Add (3+4i)+(5+2i) by combining real parts 3+5 and imaginary coefficients 4+2. Write the sum in standard complex form.
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Hint

Group the real parts together and the imaginary parts together.

To add complex numbers in standard form, add real coefficients and imaginary coefficients separately.

Solution walkthrough

01

Group like component types

\[\text{real}~\text{parts}:~3~\text{and}~5;~\text{imaginary}~\text{terms}:~4i~\text{and}~2i\]

Complex addition combines real parts with real parts and imaginary coefficients with imaginary coefficients.

02

Add the real parts

\[3+5=8\]

The real component of the sum is 8.

03

Add imaginary coefficients

\[4i+2i=(4+2)i=6i\]

The common factor i remains while its real coefficients add.

04

Write and check standard form

\[(3+4i)+(5+2i)=8+6i\]

The result has real part 8 and imaginary coefficient 6 in a+bi order.

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Another way

  1. Treat complex numbers as ordered pairs: (3,4)+(5,2)=(8,6), then map back to 8+6i.

!

Common mistake

Do not add 4i and 2i as 6i squared. Addition preserves the factor i; exponents arise from multiplication.