COMPLEX NUMBERS

Operations involving Complex Numbers

Designer

Name: Aqsa Syed                                                                                                                                    

 Overview

Complex numbers are a new system of numbers. Since in the world of real numbers it is impossible to take the square root of a negative number, in order to solve equation involving the square roots of real numbers the complex number system was developed.

A complex number is an expression of the form:

a + bi

where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property:

i ² = −1

The real number a is called the real part of the complex number, and the real number b is the imaginary part. When the imaginary part b is 0, the complex number is just the real number a.  Thus, the set of complex numbers includes all real numbers.

For example, 3 + 2i is a complex number, with real part 3 and imaginary part 2.

Complex numbers can be used to determine the roots of any quadratic equation including the roots of quadratic equations which were “undefined” or which had “no real roots”. The complex roots of a quadratic equation always occur in conjugate pairs. Two complex terms which differ only in the sign of the term containing i, are called Conjugates.

Complex numbers can be added, subtracted, multiplied, and divided.

Study Tips,  Methods  and or Advice

Advice/Tips:

v      Remember BEDMAS while solving equations.

v      Leave the answers in fractions if whole numbers are not obtained.

v      Remember to reduce fractions to the LOWEST TERMS.

v      Always place the REAL PART first and then the IMAGINARY PART.

v      Don’t forget to use FOIL [[First, Outer, Inner, Last]] when expanding binomials.

v      Don’t forget the Quadratic Formula :

v      Don’t forget the perfect squares of binomials.

Methods:

Addition, subtraction and multiplication of complex numbers is similar to the addition, subtraction and multiplication of numbers with variables. The ‘i’  in that case will be treated as a variable. But division of complex numbers is slightly different.

Addition:

As previously stated the addition of complex numbers is similar to the addition in algebra.  

v      For example: Simplify (7+6i) + (5-2i).

= 7+6i+5-2i

= 7+5+6i-2i

= 12+4i

Subtraction:

As previously stated the subtraction of complex numbers is similar to the addition in algebra.

v      For example: Simplify (4+7i)-(-6+2i)

= 4+7i+6-2i

= 4+6+7i-2i

= 10+5i

Multiplication:

Like addition and subtraction of complex numbers, multiplication of complex numbers also follows the rules of algebra.

v      For example: Simplify (5+3i).

Division:

The division process can be divided into two components: division by ‘di’ and division by ‘c + di’.

v   I. Division by di:

To simplify  multiply by .

For example: Simplify .

v   II. Division by c + di:

If you divide a complex number by c+di, then remember to multiply by the conjugate of the     denominator.

For example: Simplify.

[[In the last step, note how the fraction was split into two pieces. This is because a complex number has two parts; the real part and the i part. They aren't supposed to "share" the denominator.]]

The reason why we multiply byin order to simplify  and multiply by in order to simplify , is so that we can obtain a REAL number in the denominator.

Sample Questions and complete solutions

Extra Practice Questions and Answers

Self Reflection

Personal Reflection

I found Complex Numbers very interesting and easy to understand. I well-liked the concept of the existence of such a number system. I also found it very useful, because at times expression seemed impossible to solve because in the Real Number system the square root of a negative number was impossible to take, whereas with the help of Complex Numbers, answers could easily be obtained. On various occasions, although the answer of an expression was a real number yet the expression itself involved the square roots of negative numbers thus without the help of this number system i.e. Complex Numbers, one could not work the expression out. Not only this, another feature of  Complex Number system was that in terms of appearance, by using i in an expression made it the expression seem easy to handle as compared to the square root symbol with a negative number inside it.

As far I am concerned, the most challenging part of this topic was the T-Table Checks we had to do after solving the complex roots of a quadratic equation. Although the checks weren’t very difficult yet they were challenging because while doing the checks I ended up having so many terms that at times I mixed the plus-minus signs of different terms and thus my checks didn’t work. It was very frustrating at times because the whole expression seemed very confusing and when I put a lot of effort in it and yet was not able to get the correct answer, I just felt like giving up all at once! Not only this, when such errors occurred in the middle of tests and quizzes and even after going over my answer, I still could not find my mistake, I just felt like crying in the middle of it because I had studied so hard for the quiz or test and there I was sitting in the middle of it completely lost!  But hard work never goes waste, and due to my practices, eventually I was able to spot my error before the quiz time was over.

Through this web page, I have tried my best to include as many types of questions involving Complex Numbers as I could think of. Not only this, I have shown my working in the most efficient manner as I could manage for the convenience of those who would be using my page in the future. I have tried my best to explain this topic for students who have no idea of the topic and have also made the layout of this page as attractive as possible so as to encourage students to develop interest in this topic while studying from this page. I hope my work and effort does not go waste and is actually helpful for those who want to understand this topic. Hopefully this page will be helpful for many students involved with a course concerning this topic!