Complex Numbers n Understand complex numbers n Simplify

Complex Numbers n. Understand complex numbers n. Simplify complex number expressions

Imaginary Numbers n What is the square root of 9? n What is the square root of -9?

Imaginary Numbers n There is no real number that when multiplied by itself gives a negative number. n A new type of number was defined for this purpose. It is called an Imaginary Number. Imaginary numbers are NOT in the Real Set.

Imaginary Numbers n The constant, i, is defined as the square root of negative 1: n Multiples of i are called Imaginary Numbers

Imaginary Numbers n The square root of -9 is an imaginary number. . . n When we simplify a radical with a negative coefficient inside the radical, we write it as an imaginary number.

Imaginary Numbers n Simplify these radicals:

Multiples of i n Consider multiplying two imaginary numbers: n So. . .

Multiples of i n Powers of i:

Multiples of i n This pattern repeats:

Multiples of i n We can find higher powers of i using this repeating pattern: i, -1, -i, 1 What is the highest number less than or equal to 85 that is divisible by 4? 84 84 + ? = 85 1 So the answer is:

Powers of i - Practice 28 ni 41 75 ni 4 -i 113 ni 4 i 86 ni 4 -1 1089 ni 4 i

Solutions Involving i n. Solve:

Solutions Involving i n. Solve:

Solutions Involving i n. Solutions:

Complex Numbers n When we add a real number and an imaginary number we get a Complex Number. n Since the real and imaginary numbers are not like terms, we write complex numbers in the form a + bi n Examples: 3 - 7 i, -2 + 8 i, -4 i, 5 + 2 i

Complex Numbers: A/S n To add or subtract two complex numbers, combine like terms (the real & imaginary parts). n Example: (3 + 4 i) + (-5 - 2 i) = -2 + 2 i

Practice Add these Complex Numbers: n (4 + 7 i) - (2 - 3 i) n (3 - i) + (7 i) n (-3 + 2 i) - (-3 + i) = 2 +10 i = 3 + 6 i =i

Complex Numbers: M n To multiply two complex numbers, FOIL them: n Replace i 2 with -1:

Practice Multiply: n 5 i(3 - 4 i) = 20 + 15 i n (1 - 3 i)(2 - i) = -1 - 7 i n (7 - 4 i)(7 + 4 i) = 65

Complex Numbers: D n We leave complex quotients in fraction (rational) form: n But since i represents a square root, we cannot leave an i term in the denominator. . .

Complex Numbers: D n We must rationalize any fraction with i in the denominator. Monomial Denominator: Binomial Denominator:

Complex #: Rationalize n If the denominator is a monomial, multiply the top & bottom by i.

Complex #: Rationalize n If the denominator is a binomial multiply the numerator and denominator by the conjugate of the denominator. . .

Complex #: Rationalize n When you multiply conjugate complex numbers, the imaginary part drops out:

Practice n Simplify: