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Algebra 2 Complex numbers

Operations with complex numbers

20 practice questions 0 video lessons Theory + worked examples

Operations with Complex Numbers

California Algebra 2 • Standard N-CN.2 • Complex Numbers

Operations with Complex Numbers is a topic in Complex Numbers in the California Common Core State Standards. It is aligned to Standard N-CN.2, which requires students to use the relation i squared equals negative one and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Complex numbers add and subtract by parts, multiply with \(i^2=-1\), and divide by multiplying by the conjugate.

California Algebra 2 › Complex Numbers › Operations with Complex Numbers  —  Standard N-CN.2

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Theory

Complex numbers obey the usual algebra with \(i^2=-1\):

  • Add / subtract: combine real parts and imaginary parts.
  • Multiply: FOIL, then replace \(i^2\) with \(-1\).
  • Divide: multiply by the conjugate \(a-bi\) to clear \(i\) from the denominator.
The product of conjugates \((a+bi)(a-bi)=a^2+b^2\) is real.
Operations on a + bi Operations on a + bi Operations on a + bi add/subtract: combine parts multiply: FOIL, then i² = -1 divide: multiply by conjugate
The four operations on \(a+bi\).
Adding complex numbers Complex addition adds real and imaginary parts, like adding vectors tip-to-tail. Re Im 3+i 1+2i sum 4+3i
Addition adds the parts, like vectors.

Multiplication and conjugates:

\[(a+bi)(c+di)=(ac-bd)+(ad+bc)i,\qquad (a+bi)(a-bi)=a^2+b^2\]
multiply with FOIL and i squared equals negative one; conjugates give a real product
To divide, multiply by the conjugate of the denominator.

How to operate

  1. Add/subtract by combining like parts.
  2. Multiply with FOIL, then simplify \(i^2=-1\).
  3. Divide by multiplying by the denominator's conjugate.
  4. Write the result as \(a+bi\).
Example 1 — Add
Add \((3+2i)+(1-4i)\).
Solution

Combine real and imaginary parts.

\((3+1)+(2-4)i\)\(=\)\(4-2i\)
the sum is 4 minus 2 i
Example 2 — Subtract
Subtract \((5+i)-(2+3i)\).
Solution

Distribute the minus sign.

\((5-2)+(1-3)i\)\(=\)\(3-2i\)
the difference is 3 minus 2 i
Example 3 — Multiply
Multiply \((2+3i)(1-i)\).
Solution

FOIL, then replace \(i^2\) with \(-1\).

\(2-2i+3i-3i^2\)
\(=\)\(2+i+3\)
\(=\)\(5+i\)
the product is 5 plus i
Example 4 — Divide
Simplify \(\dfrac{4}{1+i}\).
Solution

Multiply top and bottom by the conjugate \(1-i\).

\(\dfrac{4(1-i)}{(1+i)(1-i)}\)\(=\)\(\dfrac{4-4i}{2}\)
\(=\)\(2-2i\)
the quotient is 2 minus 2 i

Common pitfalls

Replace \(i^2\) with \(-1\) after multiplying.
Use the conjugate to divide — never leave \(i\) in a denominator.
Combine only like parts: reals with reals, \(i\) with \(i\).

Frequently asked questions

How do you add complex numbers?

Add the real parts and the imaginary parts separately.

How do you multiply complex numbers?

FOIL, then replace \(i^2\) with \(-1\).

What is a complex conjugate?

\(a-bi\) is the conjugate of \(a+bi\); their product is real.

How do you divide complex numbers?

Multiply numerator and denominator by the conjugate of the denominator.

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