Complex Numbers and Powers of i

Key Concepts

Imaginary Numbers

• An imaginary number is any number that can be written in the form ( bi ), where ( i ) is the imaginary unit ( i^2 = -1 ) and ( b ) is a real number.

Complex Numbers

• A complex number is any number that can be written in the form ( a + bi ), where ( a ) and ( b ) are real numbers.
• The set of complex numbers includes all real numbers and all imaginary numbers.

Addition and Subtraction

• Combine like terms: real parts with real parts and imaginary parts with imaginary parts.
• Example: ((2 - 3i) + (4 + 6i) = (2 + 4) + (-3 + 6)i = 6 + 3i)

Multiplication

• Express square roots of negative real numbers in terms of ( i ).
• FOIL method is used for multiplying two complex numbers.
• Example: ((2 + 3i)(4 + 8i) = 8 + 16i + 12i + 24i^2 = 8 + 28i - 24 = -16 + 28i)

Division

• Multiply the numerator and the denominator by the complex conjugate of the denominator.
• Simplify and express in standard form ( a + bi ).

Powers of i

• Powers of ( i ) cycle through a pattern every four terms: ( i, -1, -i, 1 ).
• Use this cycle to simplify powers of ( i ).
• Example: ( i^{25} ) is simplified by dividing 25 by 4 to get a remainder of 1, hence ( i^{25} = i^1 = i ).

Practice Problems

1. Simplify ((4 + 2i) + (-3 - 5i))
2. Multiply ((-3 + 4i)(5 + 2i))
3. Multiply ((-8 - 7i)(5 - 4i))
4. Multiply ((3 - 2i)(5 + 4i))
5. Multiply ((3 - 4i) \times 2)
6. Evaluate ((3 - 2i)(5 + 4i) - (3 - 4i) \times 2)
7. Write ( i^{35} + i^{73} ) in standard form.
8. Simplify ( i^{925} )
9. Simplify ( i^{460} )
10. Write ( i^{25} + i^{41} ) in standard form.

Answers

1. (1 - 3i)
2. (-8 + 2i)
3. (-13 - 29i)
4. (23 + 2i)
5. (-7 - 24i)
6. (30 + 26i)
7. (\frac{3}{17} + \frac{2}{17}i)
8. (i)
9. (1)
10. (\frac{2}{29} + \frac{2}{29}i)
11. (4i)
12. (2\sqrt{2}i)
13. (-6)
14. (4 + 5i)
15. (-3 + 2i)