Complex Numbers Algebra 2A Lesson one notes

Complex number

a number in the form a+bi, where a is the real part, b is the imaginary part, and i is the imaginary unit of the number, where b≠0

Complex Conjugate

a pair of complex numbers that have real parts that are the same, and imaginary parts with opposite signs; number pairs of the form a+bi and a−bi are complex conjugates

Real number

any number in the set that includes both rational and irrational numbers

Real part of a complex number

for a complex number a+bi, the real number a

Imaginary part of a complex number

for a complex number a+bi, the real number b

Power of i (calculated)

i = sqrt -1 i^0 = 1 i^1 =i i^2 = -1 i^3= i^2*i = -1 * i= -i i^4 =i^3 *i = -i^2 = 1 i^5 = i^4 *i = i i^6 = i^5 * i = -1 i^7 = i^6 * i = -i i^8= i^7 * i = 1 ... i^10 = (i^2)^5 (-1)^5 = -1 .... i^15 = i^3 = -i i^20 = i^0 = 1 i^29= (i^2)^14 * i = (-1)^14 * i = 1*i=i According to the remainder, when dividing the exponent of i by 4, i^n is *1, if the remainder is 0. * i, if the remainder is 1. * −1, if the remainder is 2. * −i, if the remainder is 3.

Multiply i Powers

simplify: (9-i)(4+2i) 9(4+2i)-i(4+2i) Multiply by 9 36 + 18i - 4i - 2i^2 36 + 18i - 4i +2 38 + 14i

Divide i powers

simplify: 5-i/ 3+ 4i (5-1)(3-4i)/ (3+4i)(3-4i) Multiply out of the parenthesis; 5 * 3 15- 20i - 3i + 4i^2 / 9 - 12i + 12i - 16i^2 simplies: 15-23i - 4/ 9 +16 = 11- 23i/ 25 =11/25 - 23/25 i

Imaginary Number

the square root of any negative number, which has the form bi, where b is a real number

Imaginary unit

the square root of negative 1, I=√-1

Like terms

two or more terms that have the same variables with the same exponents in an algebraic expression