![]() ![]() Multiply numerator and denominator by the conjugate of 5-i. The result should be written in standard form. The quotient is found by multiplying both the numerator and the denominator by the conjugate of the denominator. ![]() The conjugate of the divisor is used to find the quotient of two complex numbers. The following list shows several pairs of conjugates, together with their products. The product of a complex number and its conjugate is always a real number. The numbers 6 + 5i and 6 - 5i differ only in their middle signs for this reason these numbers are called conjugates of each other. We can use the method of Example 8 to construct the following table of powers of i.Įxample 7(c) showed that (6 + 5i)(6 - 5i) = 61. Since i^2=-1the value of a power of i is found by writing the given power as a product involving, i^2 or i^4. Powers of i can be simplified using the facts that i^2=-1 and 1^4=1. = 6^2-25i^2 Product of the sum and difference of two terms To find a given product, it is easier just to multiply as with binomials. The product of two complex numbers can be found by multiplying as if the numbers were binomials and using the fact that i^2=-1, as follows.īased on this result, the product of the complex numbers a + bi and c + di is defined in the following way. Using this definition of additive inverse, subtraction of complex numbers a + biand c + di is defined as The sum of a + bi and -a-bi is 0 + 0i, so the number -a-bi is called the negative or additive inverse of a + bi. Since (a + bi) + (0 + 0i) = a + bi for all complex numbers a + bi, the number 0 + 0i is called the additive identity for complex numbers. The sum of two complex numbers a + bi and c + di is defined as follows. OPERATIONS ON COMPLEX NUMBERS Complex numbers may be added, subtracted, multiplied, and divided using the properties of real numbers, as shown by the following definitions and examples. Root((-4)(-9)) is not equal to root(-4)*root(-9).ĬAUTION When working with negative radicands, be sure to use the definition root(-a)=i root(a) before using any of the other rules for radicals.įINDING PRODUCTS AND QUOTIENTS INVOLVING NEGATIVE RADICANDS The rule root(c)*root(d)=root(cd) is valid only when c and d are not both negative. Then the properties of real numbers can be applied, together with the fact that i^2=-1. Products or quotients with negative radicands are simplified by first rewriting root(-a) as i root(a) for positive numbers a. Write each expression as the product of i and a real number. Many of the solutions to quadratic equations in the next section will involve expressions such as root(-a), for a positive real number a, defined as follows. The list below shows several numbers, along with the standard form of each number. (b) 3i,-11i,i root(14), and 5+i are imaginary numbers and complex numbers. (a) -8,root(7), and PIare real numbers and complex numbers. The following statements identify different kinds of complex numbers (The form a + ib is used to simplify certain symbols such as i root(5), since root(5)i could be too easily mistaken for root(5i))įIGURE 2.3 Complex numbers (Real numbers are shaded.) which is an extension of Figure 1.5 in Section 1.1.) A complex number that is written in the form a + bi or a + ib is in standard form. Both the set of real numbers and the set of imaginary numbers are subsets of the set of complex numbers. A complex number of the form a + bi, where b is nonzero, is called an imaginary number. Each real number is a complex number, since a real number a may be thought of as the complex number a + 0i. Numbers of the form a + bi, where a and b are real numbers, are called complex numbers. To get such a set of numbers, the number i is defined as follows. For example, there are no real number solutions to the quadratic equationĪ set of numbers is needed that permits the solution of all quadratic equations. Solutions of quadratic equations may not be real numbers. In the next section we will solve quadratic equations, which have a term raised to the second power (for example, x^2- 4x + 3 = 0). ![]()
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