Imaginary numbers and real numbers make up the set of complex numbers. Complex numbers are written in the form: a + bi. Real part ~ -- Imaginary part
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1 C Complex Numbers Imaginary numbers and real numbers make up the set of Complex numbers are written in the form: a + bi Real part ~ -- Imaginary part Rules for Simplifying Complex Numbers 1. Identify the real part of the number. Identify what will become the imaginary part. 2. Simplify the imaginary part of the complex number. 3. Rewrite the complex number in the form a + bi. Rewrite the complex number. v in the form a + bi Step 1 Identify the real part of the number. Identify what will become the imaginary part. Step 2 Simplify the imaginary part of the complex number. Step 3 Rewrite the complex number in the form a + bi. Practice A Rewrite each complex number 1. V-18-2 Identify the real part of the number. Identify what will become the imaginary part. Simplify the imaginary part of the complex number. Rewrite the complex number in the form a + bi. 2. ~+4 3. v v v-81 = imaginary part 3 = real part v-81 = vcf x V8T V-81 = 9i 3 + 9i in the form a + bi. = imaginary part = real part V-18 = vcf x v-18 = v v v Algebra ~ Saddleback Educational Publishing 2006 (888)
2 C Adding Complex Numbers Imaginary numbers and real numbers make up the set of Complex numbers are written in the form: Real part a + bi ~ Imaginary part You can apply what you know about operations with real numbers to any questions with Rules for Adding Complex Numbers 1. For each complex number (in the form a + bi) identify the real part and the imaginary part. 2. Group the real parts together; group the imaginary parts together. 3. Simplify. Express the sum in terms of a + bi. Add. (3 + 4;) + (-2 + 6;) Step 1 For each complex number (in the form a + bi) identify the real part and the imaginary part. Step 2 Group the real parts together; group the imaginary parts together. Step 3 Simplify. Express the sum in terms of a + bi. Real parts = +3 and -2 Imaginary parts = +4i and +6i (3-2)+(4i+6i) 1 + loi Practice :B Add. 1. (-5-4i) + (3 + 6i) For each complex number (in the form a + bi) identify the real part and the imaginary part. Group the real parts together; group the imaginary parts together. Simplify. Express the sum in terms of a + bi. 2. (4 + 4i) + (3 - i) 3. (-7+2i)+(6-6i) 4. (12-3i)+(-12-6i) Real parts = -5 and Imaginary parts = -4i and (-5 + ) + (-4i+ ) (8 + 6i) + (8-6i) 6. (12-3i) + (-9 + i) 7. (4 + v-16) + (2 + v-25) Saddleback Educational Publishing 2006 (888)
3 C Subtracting Complex Numbers You can apply what you know about operations with real numbers to any operation with Rules for Subtracting Complex Numbers 1. For the complex number to the right of the minus sign, change the sign in front of the real part of the number complex and in front of the imaginary part. Change the minus sign (between the two complex numbers) to a plus. 2. For each complex number, identify the real part and the imaginary part. 3. Group the real parts together; group the imaginary parts together. Separate the real part from the imaginary part with a plus sign. 4. Simplify. Express the difference in terms of a + bi. Subtract. (3 + 4;) - (-4 + 2;) Step 1 For the complex number to the right of the minus sign, change the sign in front of the real part of the number complex and in front of the imaginary part. Change the minus sign (between the two complex numbers) to a plus. Step 2 For each complex number, identify the real part and the imaginary part. Step 3 Group the real parts together; group the imaginary parts together. Separate the real part from the imaginary part with a plus sign. Step 4 Simplify. Express the difference in terms of a + bi. (3 + 4i) - (-4 + 2i) = (3 + 4i) + (4-2i) Real parts: 3 and 4 Imaginary parts: 4i and -2i (3+4)+(4i-2i) (3 + 4) + (4i - 2i) = 7 + 2i Practice C!. Subtract. 1. (9 + 4i) - (2 + 5i) Step 1 (9 + 4i) - (2 + 5i) 9+4i+ Step 2 Real parts: 9 and ; imaginary parts: 4i and Step 3 (9 ) + (4i Step 4 2. (10 + 2i) - (4 + i) (2 + 6i) 3. (5-3i) - (-3-2i) 5. (6 + 3i) - 3i 64 Algebra ; Saddleback Educational Publishing 2006 (888)
4 C Multiplying Complex Numbers You can apply what you know about operations with real numbers to any operation with Rules for Multiplying Complex Numbers For two imaginary numbers: 1. Multiply the whole numbers; multiply i by i, if applicable. 2. Remember, i x i = P = -1. For two complex numbers: 1. Use the FOIL method. 2. Remember, i x i = P = Simplify by combining like terms. Multiply. (5 + 7;)(-2 + 6;) Step 1 Use the FOIL method. Step 2 Remember, i xi = P = -1. Step 3 Simplify by combining like terms. Practice J) Multiply. 1. (3+6i)(4-8i) Use the FOIL method. Remember, i x i = i2 = -1. (5 + 7i)(-2 + 6i) = i + (-l4i) + 42P (30i+ (-14i)) + 42(-1) (30i + (-14i)) - 42 = i - 42 = i (3 + 6i)(4-8i) = 12-24i i+ + = 12-24i+ + Simplify by combining like terms = 2. (5 + 6i)(3-4i) 3. (3 + 2i)(5 + 3i) 4. (7 + 3i)(4-2i) 5. (9 + 4i)(3 + 4i) Saddleback Educational Publishing 2006 (888) vwvw.sdlback.com 65
5 C Dividing Complex Numbers You can apply what you know about operations with real numbers to any operation with Rules for Dividing Complex Numbers 1. Multiply numerator and denominator by the complex conjugate of the denominator. Use the same complex number, but with the opposite operation sign between the real and imaginary parts. 2. Follow the rules for multiplying two 3. Simplify. Express the quotient in a + bi form. Solve I2/ Step 1 Multiply numerator and denominator by the complex conjugate of the denominator. Use the same complex number, but with the opposite operation sign between the real and imaginary parts. Step 2 Follow the rules for multiplying two Step 3 Simplify. Express the quotient in a + biform. Practice E:. Solve. 1 5-i i Multiply numerator and denominator by the complex conjugate of the denominator. Use the same complex number, but with the opposite operation sign between the real and imaginary parts. Follow the rules for multiplying two The complex conjugate of 3 - i is 3 + i i X 3 + i 3-i 3+i 2 + 2i X 3 + i = 6 + 2i + 6i + 2i2 3-i 3+i 9-3i+3i-l" i + 6i + 2i2 = 4 + 8i = -±- +!i = 1+ii 9-3i+3i-i The complex conjugate of i is 5-i 2+4ix--- --x 5-i -2 +4i Simplify. Express the quotient a + biform. in =-----= (2 + 4i) -7- (6 + i) 3. (3 + 3i) -7- (2 - i) 4. (-2 + i) -7- (3-2i) 5. (2 + 5i) -7- (8 + i) 66 Saddle back Educational Publishing 2006 (888)
6 C Absolute Value and Complex Numbers You can apply what you know about real numbers, including absolute value, to complex numbers. The absolute value of a complex number is its distance from the origin on the complex number plane. Rules for Finding the Absolute Value of a Complex Number 1. Write the complex number in the form I a + bi I. 2. Put the real number and the coefficient of the imaginary number into the formula Va2 + b2. 3. Simplify. The absolute value is always a positive number. Find 14-3;1. Step 1 Write the complex number in the form 1 a + bi I. 14-3il Step 2 Put the real number and the coefficient of the imaginary number into the formula "';~a2-+-b-2. Step 3 Simplify. The absolute value is always a positive number. Practice F Find the absolute value il Write the complex number in the form I a + bi I. Put the real number and the coefficient of the imaginary number into the formula "';-a-2-+-b ij 14-6il =V + 2 Simplify. The absolute value is always a positive number il il il il 6. j5i i + 21 Saddleback Educational Publishing 2006 (888)
7 C Finding a Complex Solution to a Simple Quadratic Equation You can use the concept of imaginary numbers to help find the solution to certain quadratic equations. The solution of some quadratic equations is a complex number. Rules for Finding Complex Solutions to Simple Quadratic Equations 1. Isolate the term with the i2 variable on one side of the equation. 2. Simplify the equation so you have i2 on one side. 3. Find the square root of each side. The result will be x = ± Solve. 5x = 0 Step 1 Isolate the term with the i2 variable on one side of the equation. Step 2 Simplify the equation so you have i2 on one side. Step 3 Find the square root of each side. The result will be x = ± complex number. Practice Go Solve. 1. 4i = 0 Isolate the term with the i2 variable on one side of the equation. Simplify the equation so you have i2 on one side. Find the square root of each side. The result will be x = ± complex number. 5i = i2 = i = i2 = -25 R =v-25 =v-1 x 25 x= ±i x 5 = ±5i 4i2+32 =0 4i2= 4i2-o- = -o- i2= R=v =v x=± 2. i = i2+48=0 4. i = i2 + 5 = Saddleback Educational Publishing 200G (888)
8 C Finding a Complex Solution to a Quadratic Equation You can apply what you know about operations with real numbers to any operation with You can use the concept of imaginary numbers to help find the solution to certain quadratic equations. The solution of some quadratic equations is a complex number. Rules for Finding Complex Solutions to Quadratic Equations 1. Write the quadratic equation in standard form (ax2+ bx + c = 0). 2. Determine the values for a, b, and c. 3. Plug a, b, and c into the quadratic formula. 4. If the number under the square root sign is negative, apply the concept of imaginary numbers to simplify. Solve. x2 = -4x - 29 Step 1 Write the quadratic equation in standard form (axl + bx + c = 0). Step 2 Determine the values for a, b, and c. Step 3 Plug a, b, and c into the quadratic formula. xl + 4x+ 29 = 0 a = 1, b = 4, c = 29 -b ± ~ -4 ±,/42-4(1)(29) x = 2a 2(1) Step 4 If the number under the square root sign is negative, apply the concept of imaginary numbers to simplify. Practice H Solve. 1. xl = 2x - 26 Write the quadratic equation in standard form (axl + bx + c = 0). Determine the values for a, b, and c. Plug a, b, and c into the quadratic formula. If the number under the square root sign is negative, apply the concept of imaginary numbers to simplify. 2. 6xl = -4x xl-4x=-10 xl - 2x+ 26 = 0 a=,b=,c= -b±~ x= 2a = x=---- ±V- ± 4. 4xl=-16x xl = -10x+ 14 Saddle back Educational Publishing 2006 (888)
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