Complex Number System Arithmetic
A complex number is an expression in the form: a + bi where a and b are real numbers. The symbol i is defined as √ 1. a is the real part of the complex number, and b is the complex part of the complex number. If a complex number has real part as a = 0, then it is called a pure imaginary number. All real numbers can be expressed as complex numbers with complex part b = 0. -5 + 2i 3i 10 real part –5; imaginary part 2 real part 0; imaginary part 3 real part 10; imaginary part 0 complex number pure imaginary number real number

Addition and Subtraction To add/subtract two complex numbers, add/subtract the real part of the first number with the real part of the second number. Then add/subtract the imaginary part of first number with the imaginary part of the second number. 4 2 6 4 5 4 —2 3 4 6 2 4 5 2 4 3 2 – 6i 7 – 7i Multiplication To multiply two complex numbers, set up the complex numbers like two binomials and use the distributive property for binomials (FOIL method). Then use the fact that 1, and collect like terms. 3 2 · 4 12 3 8 2 12 3 8 2 1

− 14 + 5i

Complex Conjugate If is a + bi complex number, then it’s complex conjugate is a – bi. To form the conjugate of a complex number, simply negate the sign of the imaginary part of the complex number. One of the properties of the conjugate is that if you multiply a complex number by it’s conjugate, the result is a real number. Complex Complex Number Conjugate 2 – 4i 2 + 4i –3 + 2i –3 – 2i Division To divide two complex numbers, arrange the complex numbers into a fraction with the divisor as the numerator and the dividend as the denominator. Next, multiply the top and bottom of the fraction by the complex conjugate of the denominator, and collect like terms. 5 3 1

...Week 1 – Discussion
1. Counting Number :
Is number we can use for counting things: 1, 2, 3, 4, 5, ... (and so on).
Does not include zero; does not include negative numbers; does not include fraction (such as 6/7 or 9/7); does not include decimals (such as 0.87 or 1.9)
Whole numbers :
The numbers {0, 1, 2, 3, ...}
There is no fractional or decimal part; and no negatives: 5, 49 and 980.
Integers :
Include the negative numbers AND the whole numbers.
Example: {..., -3, -2, -1, 0, 1, 2, 3, ...}
Rational numbers:
It can be written as a fraction.
For example: If a is 3 and b is 2, then: a/b = 3/2 = 1.5 is a rational number
2. Give examples of correct and incorrect applications of the Order of Operations.
Problem: 3 + 4 x 2
Solution:
Correct
Incorrect
3 + (4 x 2)
= 3 + 8
= 11
( 3+4) x 2
= 7 x 2
= 14
3. Describe some real life applications of finance problems and geometric problems.
Finance Problem : Budgeting for daily expenses, such as groceries, paying my credit card bill, school supplies , etc
Geometric Problem : I think making a Birthday cake need Geometric in order to have a perfect shape and design.
Week 2 – Discussion
1. Explain the geometric sense of a linear system in two variables. Describe the possible cases.
2. Geometric sense of a linear system of inequalities in two variables.
3. How do the special...

...Department of Mathematics Center for Foundation Studies, IIUM Semester I, 2013/2014 SEF1124(MATH II) TUTORIAL 3 CHAPTER 3: POLAR COORDINATES & VECTORS Section 3.1 Polar Coordinates 3.2 The Complex Plane; De Moivre’s Theorem 3.3 Introduction to Vectors
3.4 The Dot Product 3.5 The Cross Product
Page Number 121-123 133-136 146-148
153-155 157-158
Questions 2, 15, 23, 33, 45, 53, 67, 75 15, 24, 33, 38, 46, 57, 71, 76, 81 9, 25, 33, 45, 61, 74, 75
11, 23, 31, 35, 40, 45 3, 16, 22, 24
*Required Textbook: - Salina Mohin et al , Mathematics & Statistics for Pre-University,
McGraw-Hill Education (Malaysia) Sdn Bhd.(2013)
EXTRA QUESTIONS 1. The rectangular coordinates of a point are given as coordinates for this point, for which (a) 2
Ans:
. Find the polar
(b)
2
Ans:
2. Evaluate
.
3
Ans: -5.832
3. Use DeMoivre’s Theorem to find the argument in degrees. 4. Given two complexnumbers roots of . 2 3
Ans:
, leaving your answer in polar form with
Ans:
2 2
2 and
. 2.
2 . Find the complex cube
,
2. . 3, 2 2. .
5. Find all solutions to the equation
Ans:
3 2 2 3 2 2 3 2 2 3 2 2
.
2 3 2 2 3 2 2 3 3 2 2 3 2 2
Page 1 of 2
6. Vector has magnitude 3, vector b has magnitude 4 and the angle between vectors a and b is 3 . What is the value of ?
Ans:
3
7. Given two vectors 2 and (a) Determine the vector . (b) Hence, find the possible values of if
,...

...1. Introduction
The purpose of this research paper is to introduce the topic of “Complex and Imaginary Numbers” and its applications. I chose the topic “Complex and Imaginary Numbers” because I am interested in mathematics that is hard to be pictured in your mind, unlike geometry or equations.
An imaginary number is the square root of a negative number. That is why they are called imaginary, what René Descartes called them, because he thought such a number could not exist. In this paper, I will discuss how complexnumbers and imaginary numbers were discovered, the interesting math of complexnumbers, and how they are used in other areas of mathematics and science. Complexnumbers are applied in engineering, control theory and improper integrals to take the place of certain imaginary values, as well as to simplify some explanations.
2. THe Concept
2.1 History of imaginary numbers
Long ago in ancient Greece, there was a society of mathematicians called Pythagoreans who believed the only numbers were natural numbers and positive rational numbers (Rusczyk 357). Later, Hippasus discovered irrational numbers such as√2, then 0 and negative numbers were introduced. This completes the real...

...
Complex Laws.
Let z1 = a + ib, and z2 = c + id, where a, b, c and d are real numbers.
Square root of a complexnumber.
Solve for x and y by inspection. If unable to do through inspection use the identity;
And then perform simultaneous equations.
Conjugate.
If then .
Adding vectors.
Complete Parallelogram Head to Tail
Subtracting vectors.
Complete Parallelogram
Modulus-Argument form.
If equation is not in correct , (eg, ) use the unit circle – just think of when the two conditions are met.
Rules of Modulus.
The modulus of a complexnumber is its length.
Rules of Argument.
The argument of a complexnumber is the angle made with respect to the positive x-axis.
Further vector properties.
If tail is at the origin, only one letter is used. .
However, if tail is not at origin, two letters are used. .
Note: goes from A to B. The arrow ‘starts’ at A and ‘ends’ at B.
Angel between two vectors.
Rules
Separate the argument like above.
Draw vectors z1 and z2.
Look at the heads of the vectors.
Determine the direction of angle.
Note: angle is between the two heads, and the head of the angle is on the ‘head’ line.
Rotations.
To rotate a complexnumber, z, by anticlockwise, multiply z by cis to get zcis....

...
Abstract
A complexnumber is a number that can be written in the form of a+bi where a and b are real numbers and i is the value of the square root of negative one. In the form a + bi, a is considered the real part and the bi is considered the imaginary part. The goal of this project is show how the use of complexnumbers originates in the history of mathematics.
Introduction
Complexnumbers are very important component of mathematics. They enable us to solve any polynomial equation of degree n. Simple equations like x3+1 would not have solutions if there were no complexnumbers. The complexnumber has enriched other branches of mathematics such as calculus, linear algebra (matrices), trigonometry, and you can find its applications in applied sciences, and physics. In this project we will present the history of complexnumbers and the long road to understanding the applications of this truly powerful number.
I) Ancient History
Russian Egyptologist V.S. Glenishchev traveled to Egypt in 1893 on routine business little did he know that what he would purchases would change how the world viewed Ancient Egyptian civilization, and shape mathematical landscape for years to come
Stolen from the valley of kings in 1878, at Deir el-Bahri, the Moscow...

...ComplexNumbers
All complexnumbers consist of a real and imaginary part.
The imaginary part is a multiple of i (where i =[pic] ).
We often use the letter ‘z’ to represent a complexnumber eg. z = 3 +5i
The conjugate of z is written as z* or [pic]
If z1 = a + bi then the conjugate of z (z* ) = a – bi
Similarly if z2 = x – yi then the conjugate z2* = x + yi
z z* will always be real (as i2 = -1)
For two expressions containing complexnumbers to be equal, both the real parts must be equal and the imaginary parts must also be equal.
If z1 = a + bi , z2 = x + yi and 2z1 = z2 + 3 then
2( a + bi) = x + yi + 3
hence 2a + 2bi = x + 3 + yi
so 2a = x + 3 (real parts are equal)
and 2b = y (imaginary parts are equal)
When adding/subtracting complexnumbers deal with the real parts and the imaginary parts separately
eg. z1 + z2 = a + bi + x + yi
= a + x + (b + y)i
When multiplying just treat as an algebraic expression in brackets
eg. z1 z2 = (a + bi)(x + yi)
= ax + ayi + bxi + byi2
= ax - by + (ay + bx)i (as i2 = -1)
Division by a...

...(by mohan arora)
Have you ever thought how this world of mathematics would be without irrational numbers? If the great Pythagorean hyppasus or any other mathematician would have not ever thought of such numbers?
Before ,understanding the development of irrational numbers ,we should understand what these numbers originally are and who discovered them? In mathematics, an irrational number is any realnumber that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals.
the best-known irrational numbers are: the ratio of a circle's circumference to its diameter π, Euler's number e, the golden ratio φ, and the square root of two √2.
The history of irrational numbers stated way back in 750-bc
It has been suggested that the concept of irrationality was implicitly accepted by Indian mathematicians since the 7th century BC, when Manava (c. 750 – 690 BC) believed that the square roots of numbers such as 2 and 61 could not be exactly determined. The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum),[7] who probably discovered them while identifying sides of the pentagram. However, Hippasus,...