1. Chapter 1 Equations, Inequalities, and Mathematical Models 1.4 Complex Numbers

2. Objectives At the end of this session, you will be able to:  Define complex numbers.  Add and subtract complex numbers.  Multiply complex numbers.  Divide complex numbers.  Perform operations with square roots of negative numbers.

3. Contents 1. Imaginary Unit 2. Complex Numbers 3. Operations with Complex Numbers 3.1 Addition of Complex Numbers 3.2 Subtraction of Complex Numbers 3.3 Multiplication of Complex Numbers 3.4 Division of Complex Numbers 4. Roots of Negative Numbers 5. Summary

4. 1. Imaginary Unit In this section, we will study imaginary and complex numbers. Imaginary numbers allow us to find the square root of negative numbers. We will learn how to add, subtract, multiply, and divide complex numbers, as well as find the principle square root of negative numbers. We have learned that if x 2 = 4, then x could be 2 or it could be -2. But we also know that we can not get a negative result by squaring. That is, if we square a positive number, the result is positive, and if we square a negative number, the result is still positive. For example, (4) 2 = 16 and (-4) 2 = 16. Thus, as the square of a real number is never negative, there does not exist any real number x such that x 2 = -1. To provide a setting in which such equations have solutions, mathematicians invented an expanded system of numbers, called complex numbers. Euler was the first mathematician to introduce the symbol i (read as ‘iota’) for the square root of –1 with the property i 2 = -1. He also called this symbol as the imaginary unit. This imaginary number i is the basis of complex numbers. The Imaginary Unit i The imaginary unit i is defined as Powers of i We have, therefore, i 3 = i 2. i = (-1). i = - i and i 4 = ( i 2 ) 2 = (-1) 2 = 1. Note: i 0 = 1.

5. 1. Imaginary Unit (Cont…) Imaginary Numbers: A number whose square is negative is known as an imaginary number. For example:  -1,  -2,  -4 are imaginary numbers because (  -1) 2 = -1, (  -2) 2 = -2, and (  -4) 2 = -4. Using the imaginary unit i, we can express the square root of any negative number as a real multiple of i. For example, We can check this result by squaring 6 i. NOTE: For any two real numbers a and b, is true only when at least one of a and b is either 0 or positive. In fact, ; where a and b are positive real numbers. Therefore, is wrong. The correct result is For any positive real number a, we have

6. 2. Complex Numbers Complex Numbers: The set of all numbers of the form a +b i are called complex numbers, where a, and b are real numbers and i is the imaginary unit. For example, 7 + 2 i, -1+ i, 6 i, 2 + 0 i are complex numbers. Standard form of a complex number: A complex number is said to be in a standard or simplified form if it is expressed as a + b i. For example, 6 i can be written in the standard form as 0 + 6 i. Real and Imaginary parts of a complex numbers: Complex numbers are made up of a real number part and an imaginary number part.  If a + b i is a complex number, then the real number a is called the real part and the real number b is called the imaginary part of the complex number.  For example, consider the complex number -4 + 6 i. Comparing it with the general form a + b i, we have a = -4 and b = 6. So the real part is –4 and the imaginary part is 6. As the imaginary part b = 6  0, the complex number is an imaginary number. Note: Any one of the real or imaginary parts, that is, a or b, can be 0.

7. 2. Complex Numbers (Cont…) Purely Real and Purely Imaginary Complex Numbers:  A complex number is purely real if its imaginary part is equal to zero, that is, b = 0. For instance, the number 3 can be written in the standard form a +b i as 3 + 0 i. The real part is 3 and the imaginary part is 0. Because the imaginary part is zero for this complex number, so we can say that the number is purely real. It is not an imaginary number.  A complex number is purely imaginary if its real part is zero, that is, a = 0. For example, number 2 i can be written as 0 + 2 i. The real part is 0 and the imaginary part is 2. As the real part is zero, the number 2 i is purely imaginary. NOTE: If b is a radical, then we usually write i before b. For example, we write rather than because it could be easily be confused with. Equality of Complex Numbers:Two complex numbers expressed in a standard form a + b i are said to be equal only if their real parts are equal and their imaginary parts are equal. That is, a + b i = c + d i only if a = c and b = d.

8. We can compare the standard form of the complex number a + b i to a binomial a + bx. Therefore, we can add, subtract, and multiply complex numbers using the same methods as we used for binomials. But an important point to be kept in mind is that i 2 = -1. 3.1 Addition of complex numbers: To add two complex numbers together, add their real number parts together, then add their imaginary parts together, and write the result as a complex number in the standard form. That is, (a + b i ) + (c + d i ) = (a + c) + (b + d) i For example, add the following complex numbers and write the result in the standard form: (7 + 2 i ) + (9 – 5 i ) = 7 + 2 i + 9 – 5 i (Remove the parentheses) = 7 + 9 + 2 i – 5 i (Group the real and imaginary parts together) = (7 + 9) + (2 i – 5 i )(Add the real parts and add the imaginary parts ) = (16) + (– 3 i )(Complex number in the standard form) 3.2 Subtraction of complex numbers: To subtract one complex number from another complex number, subtract their real parts, then subtract their imaginary parts, and express the difference as a complex number in the standard form. That is, (a + b i ) - (c + d i ) = (a - c) + (b - d) i 3. Operations with Complex Numbers

9.  For example, subtract (-1 - i ) – (8 – 2 i ) (-1 - i ) – (8 – 2 i ) = -1 - i - 8 + 2 i (Remove the parentheses; note the change in sign for 2 i ) = -1 - 8 – i + 2 i (Group the real parts and imaginary parts together) = (-1 - 8) + (-1 + 2) i (Subtract the real parts and the imaginary parts) = -9 + i (Complex number in the standard form) 3.3 Multiplication of complex numbers: The steps to be followed for multiplying two complex numbers are as follows:  Step 1: Multiply the complex numbers in the same manner as polynomials. We multiply two complex numbers in the same way as we multiply two polynomials using distributive property and the FOIL method. Recall: Distributive property a(b + c) = ab + ac; a(b - c) = ab – ac The FOIL method (ax + b). (cx + d) = ax. cx + ax. d + b. cx + b. d  Step 2: Simplify the expression. Add the real parts together and the imaginary parts together. Replace i 2 with –1.  Step 3: Write the final answer in the standard form. 3. Operations with Complex Numbers (Cont…)

10. For Example: Multiply (3 + 2 i )(5 – 4 i ) (3 + 2 i )(5 – 4 i ) F O I L = 15 – 12 i + 10 i – 8 i 2 (Use the FOIL method) = 15 – 12 i + 10 i – 8(-1)( i 2 = -1) = 15 + 8 – 12 i + 10 i (Group the real parts and the imaginary parts together) = 23 – 2 i (Combine the real parts and the imaginary parts) Complex Conjugate: Complex conjugates are used for the division of complex numbers. In the division of two complex numbers, we multiply the numerator and the denominator by the complex conjugate of the denominator to obtain a real number in the denominator. (The process is similar to the one used for rationalizing the denominator of a polynomial.)  We find the conjugate of a complex number by changing the sign between the two terms, keeping the same order of the terms.  For the complex number a + b i, we define its complex conjugate to be a – b i.  a + b i and a - b i are conjugates of each other. For example, the complex conjugate of 3 – 5 i is 3 + 5 i.

11.  The multiplication of the complex conjugates results in a real number, as: (a + b i ) (a – b i ) = a 2 – ab i + ab i – b 2 i 2 (Using the FOIL method ) = a 2 – b 2 (-1)( i 2 = -1) = a 2 + b 2 (Minus. Minus = Plus) Thus, we state the following result: Conjugate of a complex number The complex conjugate of the number a + b i is a – b i, and the complex conjugate of a – b i is a + b i. The multiplication of the complex conjugates gives a real number. (a + b i ) (a – b i ) = a 2 + b 2 (a – b i ) (a + b i ) = a 2 + b 2 3. Operations with Complex Numbers (Cont…)

12. 3.4 Division of complex numbers: The steps to be followed for division of two complex numbers are as follows:  Step 1: Find the conjugate of the denominator.  Step 2: Multiply the numerator and the denominator of the fraction by the conjugate found in Step 1. Keep in mind that as long as we multiply the numerator and denominator by exactly the same non-zero number, the fractions will be equivalent. When we multiply complex conjugates together, we get (a + b i ) (a – b i ) = a 2 + b 2.  Step 3: Simplify the expression. Add the real numbers and the imaginary numbers separately. Replace i 2 with –1.  Step 4: Write the final answer in the standard form. Let us solve an example to illustrate the above steps: Example: Divide and express the result in the standard form:  Step 1: In general, the conjugate a + b i is a – b i and vice versa. So the conjugate of the denominator 5 – 2 i is 5 + 2 i.

13. 3. Operations with Complex Numbers (Cont…)  Step 2 and 3: Multiply the numerator and the denominator of the fraction by the conjugate found in Step 1.  Step 4: Write the final answer in the standard form.

14. Let us find the square of 4i and –4i (4 i ) 2 = 16 i 2 = 16(-1) = -16( i 2 = -1) (-4 i ) 2 = 16 i 2 = 16(-1) = -16 We observe that the square of 4 i and the square of –4 i both result in –16. Hence, in the complex number system, –16 has two square roots, namely, 4 i and –4 i. We call 4 i the principal square root of –16. Principal Square Root of a Negative Number: For any positive real number b, the principal square root of its negative, -b, is defined by Example: Simplify 4. Roots of Negative Numbers

15. 4. Roots of Negative Numbers (Cont…) Solving problems involving square roots of negative numbers: We follow these steps for solving problems involving square roots of negative numbers:  Step 1: Express the square root of any negative number in terms of i. In other words, use the the definition of principle square roots of negative numbers before performing any operations.  Step 2: Perform the indicated operation.  Step 3: Write the final answer in the standard from. Example: Perform the indicated operation and write the answer in the standard form.

16. 5. Summary Let us recall what we have learned so far: The imaginary unit i is defined as The set of all numbers of the form a +b i is called complex numbers, where a and b are real numbers and i is is the imaginary unit. The real number a is called the real part and the real number b is called the imaginary part of the complex number. A complex number is purely real if its imaginary part is equal to zero, that is, b = 0. A complex number is purely imaginary if its real part is zero, that is, a = 0. A complex number is said to be in a simplified form if it is expressed in a standard form a + b i. Equality of Complex Numbers: a + b i = c + d i only if a = c and b = d. Operations with complex numbers:  Addition of complex numbers: (a + b i ) + (c + d i ) = (a + c) + (b + d) i  Subtraction of complex numbers: (a + b i ) - (c + d i ) = (a - c) + (b - d) i  Multiplication of complex numbers: We use the following steps for multiplying two complex numbers: Step 1: Multiply the complex numbers in the same manner as for polynomials. Step 2: Simplify the expression. Step 3: Write the final answer in the standard form.

17. 5. Summary (Cont…)  Conjugate of a complex number The complex conjugate of the number a + b i is a – b i, and the complex conjugate of a – b i is a + b i. The multiplication of the complex conjugates results in a real number. (a + b i ) (a - b i ) = a 2 + b 2 (a - b i ) (a + b i ) = a 2 + b 2  Division of complex numbers: We follow these steps for dividing two complex numbers: Step1: Find the conjugate of the denominator. Step 2: Multiply the numerator and the denominator of the fraction by the conjugate found in Step 1. Step 3: Simplify the expression. Step 4: Write the final answer in the standard form. Principal square root of a negative number: For any positive real number b, the principal square root of the negative number, -b, is defined by