Complex Numbers 2-4

Imaginary Numbers Designed so negative numbers can have square roots. Imaginary numbers consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i 2 = -1. Example

Practice

Powers of i i 1 = i i 2 = -1 i 3 = -i i 4 = 1

Higher Powers of i 2  i 2 = -1 3  i 3 = -i 0  i 0 = i 4 = 1 Divide the exponent by 4, then determine the remainder. If the remainder is 1  i 1 = i 2  i 2 = -1 3  i 3 = -i 0  i 0 = i 4 = 1

Practice i 20 = i 27 = i 71 - i 49 = i 4444484044844444441 = i -2 =

Negative Exponents

Multiplying Imaginary Numbers  

Practice 24i -18     6 -10

Complex Numbers Complex numbers are the sum of a real number and an imaginary number. They are written in the form a + bi where a is a real number and bi is an imaginary number. Complex numbers include real and imaginary numbers since 3 = 3 + 0i or 4i = 0 + 4i Imaginary numbers follow the properties we have learned (commutative, associative, distributive…)

Imaginary Numbers b≠0 Complex Numbers a + bi Real Numbers b=0 PURE

The Big Picture where does everything fit 복소수 허수 순허수 Pure

Addition and Subtraction Combine the real parts and combine the imaginary parts. (6 + 3i) + (8-2i) = (6+8) + (3 – 2)i = 14 + i (6 + 3i) - (8-2i) = (6-8) + (3 – -2)i = -2 + 5i (6 - 3i) + (8-2i) = (6+8) + (-3 – 2)i = 14 - 5i

Practice 3i – (5 – 2i)= (-2 + 8i) – (7+3i)= 4 – 10i + 3i – 2 = 2 – 7i

Graphing Complex Numbers

Finding Absolute Values The absolute value of a complex number | a + bi | is its distance from the origin. So we use the distance formula or simplified as we ignore the i for the formula

Finding Absolute Values Practice - Find |6 – 4i | |-2 + 5i | |4i|     3) 4

Multiplying Complex Numbers Multiply like real numbers and treat i like a variable but i2 = -1 (3 + 2i)(4-7i) = 12 – 21i +8i -14i2 = 12 – 13i -14(-1) = 12-13i +14 = 26 – 13i

Practice 6 – 3i +4i – 2i2 = 6+i+2= 8+ i 4 + 2i -2i – i2 = 4 – (-1) = 5 4 – (-1) = 5 18 – 12i -15i -10= 8 - 27i

Division  

Complex Conjugates  

Practice    

Equations with complex numbers Two complex numbers are equal if their real part is equal and their imaginary part is equal. If a+bi = c+di then a=c and b=d 5x+1 + (3+2y)i = 2x-2 + (y-6)i real part 5x+1 = 2x-2, 3x = -3, x=-1 imaginary part 3+2y = y-6, y=-9

Conjugates 켤레복소수 In algebra, a conjugate is a binomial formed by negating the second term of a binomial. The conjugate of x + y is x − y, where x and y are real numbers. If y is imaginary, the process is termed complex conjugation: the complex conjugate of a + bi is a − bi, where a and b are real.

Complex Conjugates  

Practice    

Equations with complex numbers Two complex numbers are equal if their real part is equal and their imaginary part is equal. If a+bi = c+di then a=c and b=d 5x+1 + (3+2y)i = 2x-2 + (y-6)i real part 5x+1 = 2x-2, 3x = -3, x=-1 imaginary part 3+2y = y-6, y=-9

Graphing Points You have the real axis (x-axis) and the imaginary axis (y-axis). Plot point (a, b) You can plot inequalities by shading areas of the graph. How would we graph {a + bi | a≤3 and b ≤2} ?

Graph {a + bi | a≤3 and b ≤2}

What happens when we multiply a complex number, a + bi by i ?

More about Complex Numbers To solve an equation involving complex numbers, equate the real parts and equate the imaginary parts.

Product of a Complex Number and its Conjugate What happens when you multiply a complex number by it’s conjugate? (3+4i )(3-4i ) = So (a + bi )(a – bi ) = 9-(-16) = 25 a2 – (b2)(i2) = a2 – (-1)(b2) = a2 + (b2)

Dividing and Reciprocals When you divide by a complex number (a fraction with a complex number as a denominator) you multiply both the numerator and denominator by 1 by multiplying both by the conjugate of the denominator,

Example

Finding the Reciprocal To find the reciprocal of a complex number, you divide 1 by that complex number. The reciprocal of 3 + 2i is But now you need to rationalize the denominator by multiplying the numerator and denominator by the conjugate. So find the reciprocal of 3 + 2i