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7.5 part 1: Complex Numbers This is the graph of the equation y = x2 + 4x + 5. Using the quadratic formula, we find the roots (zeros, x-intercepts) of the graph are Complex Numbers Imaginary Unit Conjugate Pair Complex Conjugates Group 1: The Classification of Numbers. Natural numbers are the counting number. Integers are any natural number, 0, and the negative counting numbers. Rational numbers are any numbers that can be written as a faction, including integers. Every fraction can be written either as a: o Terminating decimal o Repeating decimal Irrational numbers are any numbers that cannot be written as a fraction. Their decimals are: o Non-terminating o Non-repeating Real numbers are any rational numbers or irrational numbers. An imaginary number is any number that is the square root of a negative or has “i.” Complex numbers are any real or imaginary number that can be written in the form a + bi. Example: State all the classifications of the number -16/2. The number is a negative, so it cannot be a natural. -16/2 = -8. It is a whole counting number, so the lowest classification is an integer. Answer: -16/2 is an integer, rational, real, and complex number. Example2: State all the classifications of the number 7.26. The number is not a counting number, so it cannot be an integer. The number is a terminating decimal, which can be converted into the fraction 7 26 , so the lowest classification 100 is a rational number. Answer: 7.26 is a rational, real, and complex number. State all the classifications of each number. a. 0 e. 3 7 b. 4.453423… c. 3+4i f. 4 g. -5.4 d. 4.453476 h. 5 3 i. 4 Group 2: Simplifying Complex Numbers Simplifying complex numbers is the same as simplifying radicals. Our goal is to write the numbers in the form a + bi, where a is the real number, and bi is the imaginary part. Example: Simplify 50 36 . The 36 is the real number part. The 50 is the imaginary part. So 36 simplifies to 6. 50 25 2 1 5 2i 5i 2 . 50 36 simplifies to 6 5i 2 . Example2: Simplify 80 . The negative is not under the radical, so this is a real number. 80 16 5 4 5 . Simplifying complex Numbers a. 25 b. 19 c. 45 d. 4 e. 7 f. 12 g. 9 6 h. 18 7 i. 24 75 Group 3: Graphing Complex Numbers The x-axis is now the real number, the y-axis represents the imaginary. Example: State the real number represented on the complex plane on the right. A = 5 – 4i B = -3i C = -3 – i D = -6 E = -2 + 4i F=1 G=1+i **Notice how these are numbers in the form a + bi, not ordered pairs. Graph each number on the complex plane below. a. 3+4i b. -2-i c. 3 d. -1+ 25 e. 8i f. 4 1 Group 4: Determining Powers of i. Powers of i are cyclic, meaning that there is a pattern. Note: i1 = i 1 i i2 = i2 = -1 i1 =i 1 i i3 = i2i = (-1)(i) i4 = i2i2 = (-1)(-1) i3 = -i i4 = 1 i2 =___ i3 =___ i4 =___ Now, notice the following. i5 = i4i1 = (1)(i) i6 =i4i2 = (1)(-1) i7=i4i3=(1)(-i) i8=(i4)2 = (1)2 i5 = i i6 = -1 i7=-i i8 = 1 The same pattern repeats with every group of 4. Because i4 = 1, then every power that is a multiple of 4 will also equal 1. So to determine the value of a power of i Write the power as a multiple of 4 and then the remainder. The multiple of 4 will equal 1, and then determine the value of the remainder. Example: Simplify i43. The closest multiple of 4 to 43 is 40. So, i43 = i40i3. i40 = (i4)10 = (1)10 = 1 i43 = (1)(i3) = i3 = -i Answer: i43 = -i Simplify each power of i. a. i24 = b. i35 = c. i50 = d. i13 = e. i102 = f. i37 = Sharing: Complex Numbers With your home groups, share the most important points to remember from your expert group. Classification of Numbers Simplifying Complex Numbers Graphing Complex Numbers Powers of i Home Group Practice With your group, complete the following practice problems together. 1. Classify each number in as many ways as possible. a. 2. b. -6 d. 4 – 19i c. 0 Simplify each complex number. a. 3. 9.489 36 e. 3 30 b. 5 98 Plot the following complex numbers on the complex plane below. a. -5 b. -3 + 2i c. 49 1 d. 9 e. 5i + 8 4. Simplify each power of i. a. i27 b. i96 c. i34 d. i62 e. i13 Back to Solving Quadratic Equations There is no more “no solution” answers anymore. EVERY PROBLEM will have a solution!! Solve the quadratic equations. Leave answers in simplest form. a. x2 = -36 b. x2 = -28 c. -(x – 3)2 = 25 d. (2x + 7)2 – 15 = -28 e. 4(x – 11)2 + 27 = 3 f. -5(5x – 1)2 = 18 Now use the quadratic formula to solve these equations. a. b. Name: _______________________ Date: ___________ Homework: 7.5 part 1 Complex Numbers Simplify. 1. 108 5. i83 2. 144 3. 6. i57 32 15 4. 54 28 7. i66 8. i21 List all the classifications of each number. 9. 28 13. 100 45 10. 72 11. 14. 8.57389… Plot each number on the complex plane to the left. 17. 5 – 3i 18. i 19. -8 20. 2 + 5i 21. 4 64 72 15. 8.235 12. 100 16. 58 47 Solve each quadratic equation. Leave answers in simplest form. c. a. d. b. c.