Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Field (mathematics) wikipedia , lookup
Cubic function wikipedia , lookup
Quadratic equation wikipedia , lookup
Eisenstein's criterion wikipedia , lookup
System of polynomial equations wikipedia , lookup
Factorization wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Real number wikipedia , lookup
Section 2.4 Complex Numbers What you should learn • How to use the imaginary unit i to write complex numbers • How to add, subtract, and multiply complex numbers • How to use complex conjugates to write the quotient of two complex numbers in standard form • How to find complex solutions to quadratic equations Real Number System Natural {1, 2, 3, 4,…} How many natural numbers are there? Real Number System Natural Whole {0, 1, 2, 3, 4,…} How many whole numbers are there? Real Number System Natural Whole Integers {...-3, -2, -1, 0, 1, 2, 3, …} How many integers numbers are there? Real Number System Natural Whole Integers Rational Fractions How many rational numbers are there? a a, b I , b 0 b Real Number System Natural Whole Integers Rational 2, , e How many irrational numbers are there? Irrational Real Number System Natural Whole Integers Each set is a subset of the Real Number System. The union of all these sets forms the real number system. The number line is our model for the real number system. Irrational Rational Real Numbers Definition of Square Root If a2 = n then a is a square root of n. 42 = (4)(4) = 16 4 is a square root of 16 (-4)2 = (-4)(-4) = 16 -4 is a square root of 16 What square root of -16? Whatever it is it is not on the real number line. Definition of i The number i is such that 1 i 2 2 1 i 1 i 2 i 1 Imaginary Unit b i b 16 i 16 4i Complex Numbers REAL a bi Imaginary Complex 3 2i 3 2 i 8 5 7 0i Definition of a Complex Number • If a and b are real numbers, the number a + bi is a complex number, and it is said to be written in standard form. • If b = 0 then the number a + bi = a is a real number. • If b ≠ 0, then the number a + bi is called an imaginary number. • A number of the form bi, where b ≠ 0 is called a pure imaginary number. Examples 16 4i 81 9i 7 i 7 If you square a radical you get the radicand 5 2 5 Whenever you have i2 the next turn you will have -1 and no i. i 1 2 i 1 2 2 Equality of Complex numbers If a + bi = c + di, then a = c and b = d. x 5i 7 yi x7 y 5 Is a negative times a negative always positive? Trick question. This is not a negative times a negative. 9 25 (3i )(5i ) 15i 15 2 Example 7 7 i 7 i 7 7i 2 7 Example 5 10 i 5 i 2 5 5i 2 2 5 2 Example 15 2 i 15 2 i 30 Example 32 i 32 2 i 2 16 4 Cancel the i factor Add Collect like terms. (3 5i) (4 7i) 7 2i First distribute the negative sign. Subtract Now collect like terms. (5 7i) (4 20i) 5 7i 4 20i 9 13i Multiplication (3 2i)(4 5i) 2 12 15i 8i 10i 12 7i 10 22 7i FO I L Simplify each expression. Express your answer in form. (5 4i)(3 7i) F-O-I-L 15 35i 12i 28i 2 Recall i2=-1 Combine like terms. 15 23i 28 43 23i Combine like terms. Write in the form a bi. 26 3 2i 26(3 2i ) 2 3 2i 3 2i 9 4i 2 26(3 2i ) 13 2(3 2i ) 6 4i Multiply by the conjugate factor. Powers of i 1 i raised to the 0 is 1. 1 Anything raised to the 1 is itself. i i 2 2 i 1 1 i 3 3 2 i i i i i (1)i i 0 Anything other than 0 Simplify as much as possible. i i i (1)(1) 1 4 2 2 i (i ) i (1)(1) 1 30 4 7 2 Use the Quadratic Formula 2 2 9 x 6 x 37 0 b b 4ac x a9 2a b 6 (6) (6) 2 4(9)(37) x c 37 2(9) 6 36i 6 36 1332 6 1296 18 18 18 6 36 1 i 2i 18 18 3 Homework Section 2.4 1-79, 83 odd