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So far in this course we have been dealing with real numbers. Real numbers contain rational numbers and irrational numbers. Irrational Numbers. Rational Numbers. Examples:. Examples:.
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So far in this course we have been dealing with real numbers. Real numbers contain rational numbers and irrational numbers. Irrational Numbers Rational Numbers Examples: Examples: We are now going to discuss Complex Numbers. Real Numbers contain rational and irrational numbers. Well, the Complex Numbers are a “bigger” set of numbers. The Complex Numbers contain the Real Numbers. Complex numbers which include the set of real numbers, as well as numbers that are even roots of negative numbers, like We can find a real number whose square is a positive number. memorize To provide a solution for the equation we must define the following: Definition No real number will work because the square of either a positive number or a negative number will result in a positive number. 52=25, and (-5)2=25
Definition Acomplex numberis any number that can be expressed in the form a + biwhere a and b are real numbers. The form a + bi is called the standard form of a complex number. The real number a is called the real part of the complex number, and b is called the imaginary part. (Note that b is a real number even though it is called the imaginary part.) Note: The i can be written before or after the radical. If it is written after the radical, just make sure the i does not appear to be under the radical sign. Examples: 7 + 5i Real part Imaginary part Real part Imaginary part Adding Complex Numbers Procedure: To add complex numbers: 1) add their real parts 2) add their imaginary parts Using symbols: (a + bi) + (c + di) = (a + c) + (b + d) i Note: If the procedure of working with real parts and imaginary parts may seem confusing, just remember, we can only add like terms.
Example 1. Add as indicated a) (4 + 3i) + (6 + 8i) b) (-5 + 3i) + (6 – 5i) Solution: Your Turn Problem #1 Simplify the following. a) (– 5 + 3i) + (7 – 6i) b) (3 + 8i) + (– 9 – i) To subtract complex numbers, c + di from a + bi, distribute the negative sign. Then combine “like terms”. (a + bi) – (c + di) = (a + bi) + (– c – di) = (a – c) + (b – d) i = –12 + 3i – 7 + 8i = –19+11i Example 2. Subtract as indicated a) (7 – 3i) – (18 – 5i) b) (– 12 + 3i) – (7 – 8i) Solution: Your Turn Problem #2 Simplify the following. a) (12 – 9i) – (14 – 6i) b) (2 – 3i) – (– 4 – 14i) =(-5 + 6) + (3 – 5) i = 1 – 2 i = (4 + 6) + (3 + 8) i = 10 + 11 i Answers a) 2 – 3i b) – 6 + 7i = 7 – 3i – 18 + 5i = –11 + 2 i Answers a.) – 2 – 3i b.) 6 + 11i
Simplifying Square Roots of Negative Numbers For any positive real number b, Example 3. Write in terms of i and simplify. Solution: Your Turn Problem #3 Write in terms of i and simplify. Answers
Solutions: Your Turn Problem #4 Simplify:
Your Turn Problem #5 Multiply Multiplication of Complex Numbers Procedure: Multiplying complex numbers • The procedure of multiplying complex numbers follows the same procedure as multiplying monomials, binomials and polynomials. • Simplify by replacing i2 with –1 and combining like terms. Express final result in a+bi form.
Your Turn Problem #5C illustrates an important situation. The two complex numbers (2+4i) and (2 – 4i) are conjugates of each other. In general, two complex numbers of the form (a+bi) and (a – bi) are called conjugates. In the previous chapter, we used conjugates to help us rationalize the denominator and simplify rational expressions such as: The product of a complex number and its conjugate is always a real number, which can be shown as follows. We will now simplify quotients such as where the denominator is a complex number. We will again have the same procedure: multiply both numerator and denominator by the conjugate of the denominator to write the quotient in standard form of a complex number. It is also necessary to write a complex number in standard form, a+bi. Division of Complex numbers To rationalize the denominator (get rid of the radical in the denominator), we multiplied the numerator and denominator by the conjugate of the denominator. Next Slide
Solution: Your Turn Problem #6 Step 1. Multiply top and bottom by the conjugate of the original denominator Step 2. Replace i2 with -1 and simplify. Step 3. Write in a+bi form (separate into fractions).
Solution: Your Turn Problem #7 Keep in mind that our goal is to not have an i in the denominator. Since the denominator is a 3i, if me multiply the denominator by i, we will have a 3i2 in the denominator. Well, since i2 = -1, the denominator will be -3. Don’t forget, whatever you multiply the denominator by, you must also multiply by the numerator. The End. B.R. 5-19-07
