Problem 49 - College Algebra

Question

Explain how to divide complex numbers. Provide an example with your explanation.

Step-by-Step Solution

Answer

To divide complex numbers, write them in standard form, multiply top and bottom by the conjugate of the denominator and then simplify. For example, to divide (2 + 3i) by (1 - 2i), the answer is \( 4/3 - 7/3 * i \).

Step 1: Complex number definition

Identify the real (a and c) and the imaginary parts (b and d) of both complex numbers. For instance, let's say we want to divide (2 + 3i) by (1 - 2i).

Step 2: Find the Conjugate of the Denominator

In this case, the conjugate of the denominator (1 - 2i) is (1 + 2i) since we change the sign in the middle.

Step 3: Multiply Numerator and Denominator by the Conjugate

Carry out the multiplication, treating 'i' as a variable and remembering that \(i^2 = -1\). This should give you: \((2+3i)\times(1+2i)\) and \((1-2i)\times(1+2i)\) which translates to \((2+4i+3i+6i^2) \over (1+2i-2i-4i^2)\) after multiplication.

Step 4: Simplify the Equation

In the numerator, combine like terms and convert \(i^2\) to \(-1\) and do the same for the denominator. This simplifies to \( 2 + 7i - 6 \over 1 - 4 \), which simplifies further to \(-4 + 7i \over -3\).

Step 5: Final Answer

The final step is to divide every term by -3 (the real number in the denominator), the answer will be \( 4/3 - 7/3 * i \).