Complex Numbers

(-5 + 4i) - (-12 + 3i) - (10 + 19i)

Distribute Minus symbols, and solve by combining like terms. -5 + 4i + 12 - 3i - 10 - 19i A: - 3 - 18i

Denominator

The bottom number in a fraction. an imaginary number (i) should never be left in the denominator. so multiply the numerator and denominator by i. this will cause the denominator to be i-squared, converting it to - 1. i-squared = - 1

5 over 3 + 2i

5 over 3 + 2i = Multiply the denominator with the top by its conjugate. So, 3 - 2i x 3 and 3-2i x 3 - 2i

(2 + 5i)(7 - 3i)

(2 + 5i)(7 - 3i) = this problem is the product of two binomials, so FOIL it. 2 x 7 = 14 ; 2 x - 3i = - 6i 5i x 7 = 35i ; 5i x - 3i = - 15i squared. 14 - 6i + 35i - 15i-squared. - 15i-sqaured = - 15 x - 1, (- times - equals 15. 14 - 6i + 35i + 15 = A: 29 + 29i

Square Root of - 75

- 75 is not a perfect number, so it will need to be split into two roots. 25 x 3 = 75; So, the Square root of 25 is 5. and the 3 is not a perfect number, so it's left alone, still inside of a root. A: 5i; square root of 3

Multiplying Complex Numbers

1. 5 (2i) = 10i 2. 7 (3 - 4i) = 21 - 28i 3. (- 7i) (2i) = - 14 (- 1) i's cannot be combined. Two i's together equal -1. thus, the problem would be set up like - 14 times (- 1), which equals 14 another example: (5i)(6i) = 30i-squared; which is 30 multiplied by - 1 A = - 30

- Square Root of - 54

54 is not a perfect number, so it must be broken into two Roots. 9 x 6 = 54; So, SR of 9 equals 3, and 6 is not perfect. now, remember the negative symbols on the inside and outside. the negative (-) from 54, now becomes an i and plugs in with 3; 3i then, the negative from the outside is applied to 3i, while 6 is left a SR. A: - 3i Square root of 6.

8i (3 + 7i)

8i (3 + 7i) = 8i times 3 = 24i 8i times 7i = 56i-squared. 56i-sqaured is 56 x - 1 = -56 now, standard form says real numbers first. A: - 56 + 24i

24 - 16i over 6i

Dividing CN: 24 - 16i over 6i = Separate by adding the denominator (6i) to each. 24 over 6i ; - 16i over 6i multiply the i in 6i by the top and bottom, this will covert the 6i to -6; resulting in 24i over - 6. As for 16i over 6i, the i's cancel out, resulting in: 16 over 6. this can be simplified (by 2). standard form: real; imaginary - 16(2) over 6(2) = 8 over 3 A: - 8 over 3 - 24i over -6

(3 + 5i) + (7 - 2i)

To add complex numbers, add like terms: a (real part) + bi (imaginary parts) (3 + 5i) + (7 - 2i) = First, combine the real parts, 3 + 7 = 10 Second, combine imaginary parts, 5i - 2i = 3i 10 (real) and 3i (fake) cannot be combined, so the answer is 10 + 3i

complex conjugates

Two complex numbers of the form: (a + bi) and (a - bi)

Square root

a number that when multiplied by itself equals a given number

cube root

a number that when multiplied three times equals a given number. 2 - 2 - 2: 2 x 2 =4, 4 x 2 = 8

(4 + 7i) - ( 10 - 4i)

add like terms. However, since there is a minus sign in the middle, that minus sign should be distributed to 10 and 4i. (- 10 + 4i) now, combine like terms and solve. 4 + 7i - 10 + 4i = A: - 6 + 11i

Binomials

an algebraic expression with two different variables

Cube root of - 64

because this problem is cubed, it does not matter if the inside number is negative or not. A: - 4 4 x 4 = 16 x 4 = 64

15 + 25i over 5

dividing complex numbers (fractions): 15 + 25i over 5 = its best practice to split and separate the denominator if its only one number. So, the problem should be; 15 over 5 + 25i over 5 Once, those two seperated fractions willl cancel out the denominator. A: 3 + 5i Standard form: Real number first; imaginary second.

14i over 2 =

dividing complex numbers: 14i over 2 = 14 divided by 2 equals 7. because i is in the numerator, it stays and is applied to 7. A: 7i

20i over 5i

dividing complex numbers: 20i over 5i = the i's cancel out. 20 divided 5 = 4

35 over 7i

dividing complex numbers: 35 over 7i = since i should not be in the denominator, multiply the top and bottom by i. this will square the i in the denominator, converting it to a negative number. So, 7i x i = 7i-squared. which converts to -7. now, 35 times i = 35i. 35i over - 7 = A: - 5i

i squared

i times i or i-squared will always equal; Negative 1 (-1)

Dividing Complex Numbers

multiply the numerator and denominator by the conjugate of the denominator

Square root of -25

the square root of -25 = 5i First square root 25, which is 5. Then simply add the i. this is done every time a negative number is inside the square root.

Imaginary number

the square root of a negative number. A negative number inside of a square root is called "an imaginary number."

Numerator

the top number in a fraction. If the i (imaginary) is in the numerator, it stays.

Imaginary number cancellation

when ever dividing complex numbers, if an i is in the numerator and denominator, they cancel out.

- Square Root of - 16

when ever there is a Negative outside the square root, first solve the number inside the square root first; Square Root of -16 is 4i. Then simply apply the negative. A: - 4i