Until now, we have been dealing exclusively with real numbers. This chapter introduces a new topic--imaginary and complex numbers. Complex numbers are numbers of the form a + bi, where i = and a and b are real numbers. They are used in a variety of computations and situations. Complex numbers are useful for our purposes because they allow us to take the square root of a negative number and to calculate imaginary roots. The first section discusses i and imaginary numbers of the form ki. Here, the reader will learn how to simplify the square root of a negative number.
The focus of the next two sections is computation with complex numbers. Section two explains how to add and subtract complex numbers, how to multiply a complex number by a scalar, and how to multiply a complex number by another complex number. Section three introduces the concept of a complex conjugate and explains its use in dividing a complex number by another complex number. Complex numbers are useful in a variety of situations. They appear frequently in almost every branch of mathematics. We will use them in the next chapter when we find the roots of certain polynomials--many polynomials have zeros that are complex numbers.
Terms
Complex Conjugate - The complex conjugate of a + bi is a - bi. When two complex conjugates are multiplied, the result is a2 + b2.
Complex Number - A number of the form a + bi, where i =
and a and b are real numbers.
Imaginary Number - A number of the form
, where k < 0, or a number of the form ki, where i =
.
Imaginary Part - The term bi of a complex number a + bi.
Real Part - The term a of a complex number a + bi.
Imaginary Numbers
Imaginary Numbers
Until now, we have been dealing with real numbers. We have not been able to take the square root of a negative number because the square root of a negative number is not a real number. Instead, the square root of a negative number is an imaginary number--a number of the form , where k < 0. Imaginary numbers are represented as ki, where i = . For example, = 5i and = i . We can simplify square roots of negative numbers by factoring out = i and simplifying the resulting root.
Examples:
- Simplify
.
- Simplify
.
- Simplify
.
Observe the following:
| i1 | = | i | |
| i2 | = | ( )2 = - 1 | |
| i3 | = | i2i = - 1(i) = - i | |
| i4 | = | i3i = - i(i) = - i2 = - (- 1) = 1 | |
| i5 | = | i4i = 1(i) = i | |
| i6 | = | i5i = - 1 | |
| i7 | = | i6i = - i | |
| i8 | = | i7i = 1 | |
| i9 | = | i | |
| | ... | | |
Thus, we can find
in using the following:
- If n÷4 leaves a remainder of 1, in = i.
- If n÷4 leaves a remainder of 2, in = - 1.
- If n÷4 leaves a remainder of 3, in = - i.
- If n÷4 leaves no remainder, in = 1.
Examples:
- What is i54?
54÷4 = 13R2.
Thus, i54 = - 1. - What is i103?
103÷4 = 25R3.
Thus, i103 = - i.
Complex Numbers
Complex Numbers
A complex number is a number of the form a + bi, where i = and a and b are real numbers. For example, 5 + 3i, - + 4i, 4.2 - 12i, and - - i are all complex numbers. a is called the real part of the complex number and bi is called the imaginary part of the complex number. In the complex number 6 - 4i, for example, the real part is 6 and the imaginary part is -4i. Adding and Subtracting Complex Numbers
To add two complex numbers, add their real parts and add their imaginary parts: (a1 + b1i) + (a2 + b2i) = (a1 + a2) + (b1 + b2)i.
Examples:
(12 + 6i) + (11 + 5i) = (12 + 11) + (6 + 5)i = 23 + 11i
(5 - 7i) + (4 + i) = (5 + 4) + (- 7 + 1)i = 9 - 6i.
(2 - 4i) + (- 6 - 5i) = (2 - 6) + (- 4 - 5)i = - 4 - 9i.
To subtract two complex numbers, subtract their real parts and subtract their imaginary parts: (a1 + b1i) - (a2 + b2i) = (a1 - a2) + (b1 - b2)i.
Examples:
(4 + 5i) - (2 + 3i) = (4 - 2) + (5 - 3)i = 2 + 2i.
(3 - 7i) - (4 + 6i) = (3 - 4) + (- 7 - 6)i = - 1 - 13i
(- 4 + 2i) - (3 - 11i) = (- 4 - 3) + (2 - (- 11))i = - 7 + 13i
(6 - 9i) - (- 3 - 4i) = (6 - (- 3)) + (- 9 - (- 4))i = 9 - 5i
Multiplying a Complex Number by a Scalar
To multiply a complex number by a scalar, multiply the real part by the scalar and multiply the imaginary part by the scalar: c(a + bi) = ca + cbi.
Examples:
4(2 + 5i) = 4(2) + 4(5i) = 8 + 20i
(6 - 9i) = (6) + (- 9i) = 2 - 3i.
-2(11 - 2i) = - 2(11) + (- 2)(- 2i) = - 22 + 4i.
2i(5 + 7i) = 2i(5) + 2i(7i) = 10i + 14i2 = 10i + 14(- 1) = - 14 + 10i.
Multiplying Complex Numbers
To multiply two complex numbers, use the FOIL method and treat each complex number as an ordinary binomial. Then simplify the i2 term (i2 = - 1) and combine like terms.
| (a1 + b1i)(a2 + b2i) | = | a1a2 + a1b2i + a2b1i + b1b2i2 | |
| | = | a1a2 + (a1b2 + a2b1)i + b1b2(- 1) | |
| | = | (a1a2 - b1b2) + (a1b2 + a2b1)i. | |
Examples:
(2 + 3i)(5 + 2i) = ?
| | = | 10 + 4i + 15i + 6i2 | |
| | = | 10 + 19i - 6 | |
| | = | 4 + 19i. | |
(3 - 4i)(6 + i) = ?
| | = | 18 + 3i - 24i - 4i2 | |
| | = | 18 - 21i + 4 | |
| | = | 22 - 21i. | |
(7 - 2i)(6 - 2i) = ?
| | = | 42 - 14i - 12i + 4i2 | |
| | = | 42 - 26i - 4 | |
| | = | 38 - 26i. | |
(2 + 3i)2 =?
| | = | (2 + 3i)(2 + 3i) | |
| | = | 4 + 6i + 6i + 9i2 | |
| | = | 4 + 12i - 9 | |
| | = | -5 + 12i. | |
(5 + 4i)(5 - 4i) = ?
| | = | 25 - 16i2 | |
| | = | 25 + 16 | |
| | = | 41. |
Complex Conjugates and Dividing Complex Numbers
Complex Conjugates
Every complex number has a complex conjugate. The complex conjugate of a + bi is a - bi. For example, the conjugate of 3 + 15i is 3 - 15i, and the conjugate of 5 - 6i is 5 + 6i. When two complex conjugates a + bi and a - bi are added, the result is 2a. When two complex conjugates are subtracted, the result if 2bi. When two complex conjugates are multiplied, the result, as seen in Complex Numbers, is a2 + b2. Dividing Complex Numbers
To find the quotient of two complex numbers, write the quotient as a fraction. Then multiply the numerator and the denominator by the conjugate of the denominator. Finally, simplify the expression:
Examples:
(3 + 2i)÷(4 + 6i) =?
(6 + 3i)÷(7 - 2i) =?
(3 - i)÷(- 5 + i) =?
Its a very good article. I liked it becuase in this article the theory is explained with the help of examples. So it will work as an ebook in itself. Very useful for beginners and for those who are revising their concepts.
I thought it will be useful to you so posted it here. If you think it helped you in any way to rate kar dena and please leave ur valuable comments.
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