Addition, subtraction and multiplying complex numbers and simplifying powers of i

Addition, subtraction and multiplying complex numbers and simplifying powers of i

Complex Numbers Basics

  • A complex number is a number in the form a + bi where ‘a’ is the real part and ‘bi’ is the imaginary part.
  • ‘a’ and ‘b’ are real numbers and ‘i’ is the imaginary unit, defined by i^2 = -1.
  • Two complex numbers are equal only if their real and imaginary parts are equal.

Addition and Subtraction of Complex Numbers

  • To add or subtract complex numbers, combine like terms by adding or subtracting the real parts and the imaginary parts separately.
  • For example, to add (3 + 2i) + (1 + 4i), you would add the real parts 3 and 1 to get 4, and the imaginary parts 2i and 4i to get 6i, resulting in 4 + 6i.

Multiplication of Complex Numbers

  • To multiply complex numbers, apply the distributive property (also known as the FOIL method) and simplify by using the fact that i^2 = -1.
  • For instance, multiplying (3 + 2i)(1 + 4i) involves multiplying each term in the first complex number by each term in the second complex number, which yields (31 - 24) + (34 + 21)i = -5 + 14i.

Powers of i

  • i raised to a power cycles every four powers: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, and then it repeats.
  • To simplify an expression with i raised to a larger power, divide the power by 4 and use the remainder to determine the equivalent power of i. For example, i^73 is equivalent to i because 73 divided by 4 has a remainder of 1.

Complex Conjugates

  • The conjugate of a complex number is obtained by changing the sign of its imaginary part. The conjugate of a + bi is a - bi.
  • Multiplying a complex number by its conjugate results in a real number. This concept is useful for division of complex numbers.

Division of Complex Numbers

  • To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator, simplify, and separate into real and imaginary parts. For instance, (3 + 2i) / (1 + 4i) turns into (3 + 2i)(1 - 4i) / (1 + 4i)(1 - 4i), which simplifies to (14 - 11i) / 17, or 14/17 - 11/17i.