Division by a complex number is a very similar process to ‘rationalising’ surds – we call it ‘realising’

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Argand Diagrams

We can represent complex numbers on an Argand diagram. This similar to a normal set of x and y axes except that the x axis represents the real part of the number and the y axis represents the imaginary part of the number.

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The argand diagrams allow complex numbers to be expressed in terms of an angle (the argument) and the length of the line joining the point z to the origin (the modulus of z). Hence the complex number can be expressed in a polar form. The argument is measured from the real axis and ranges...

...
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....

...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...

...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...

...Natural Convection Flow of Fluids of Diﬀerent Prandtl Numbers
in the Stokes Problem for a Vertical Porous Plate
Basant K. JHA and Clement A. APERE
J. Phys. Soc. Jpn. 81 (2012) 064401
# 2012 The Physical Society of Japan
Person-to-person distribution by the author only. Not permitted for publication for institutional repositories or on personal Web sites.
FULL P APERS
Journal of the Physical Society of Japan 81 (2012) 064401
DOI: 10.1143/JPSJ.81.064401
MHD Natural Convection Flow of Fluids of Diﬀerent Prandtl Numbers
in the Stokes Problem for a Vertical Porous Plate
Basant K. JHAÃ and Clement A. APEREy
Department of Mathematics, Ahmadu Bello University, Zaria, Nigeri
(Received October 19, 2010; accepted April 6, 2012; published online May 23, 2012)
This paper investigates the eﬀects of diﬀerent Prandtl numbers on the MHD natural convection ﬂow of ﬂuids in the
Stokes problem for the impulsive motion of a porous vertical plate. Both cases of the applied magnetic ﬁeld being ﬁxed
either to the ﬂuid or to the moving plate are considered. Uniﬁed closed form expressions are obtained for the
temperature and the velocity, which are used to compute the Nusselt number and the skin friction respectively. These
equations are solved semi-analytically using the Laplace transform technique along with the Riemann-sum
approximation method. The inﬂuence of diﬀerent ﬂow parameters such as Prandtl...

...The story "My Oedipus Complex" by Frank O'Connor deals exclusively with a little boy named Larry and his feelings towards his father. When his father returns home from World War II, Larry is resentful and jealous of losing his mother's undivided attention, and finds himself in a constant struggle to win back her affections.
<br>
<br>I really enjoyed "My Oedipus Complex," because it reminded me a great deal of my elementary school days. My brother Brian was born when I was five, and from that day on there was never a moment of peace in the house. He was constantly underfoot, and after he was old enough, spent all his time trying to sweet talk my mother into whatever it was he wanted at the moment. Kissing her hand and lavishing praise on her mothering skills was one of his favorites, and it was usually pretty effective, too. My mother was oblivious to the terror he had inflicted on the rest of the household - me, two dogs, and a school of goldfish - and saw only her sweet, perfect baby boy. For this reason, I saw Larry as a tamer but equally spoiled version of my brother, and his mother as remarkably similar to my own.
<br>
<br>I have many memories of my brother's most unforgettable acts, many of them occurring around the time my mother's new boyfriend Rodney started showing up at the house. After his first introduction to my brother, it was was remarkable that he ever came back. "Just a moment ... Do be quiet ... Don't interrupt again!" (97)...

...(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,...

...Frank O'Connor
Everyone shows traces of the little green monster, known as jealousy, but some more than others. This specific trait has had a huge effect on the world over time, sometimes destroying, sometimes rebuilding trust, friendships, and even business opportunities. Jealousy is an extremely prominent element in most of Frank O'Connor's writings and is often shown through different literary concepts such as conflict, characterization, and obsessive love. In writing “My OedipusComplex”, O'Connor investigates the issue of jealousy through the various actions of his characters and the conflicts they get themselves tangled in, more importantly the ones involving their childlike obsessive love.
To fully understand O'Connor's stories, you first have to delve deep into his background. Frank O'Connor was born in Cork, Ireland, on September 17, 1903 to Michael and Minnie O'Donovan. He was born under the name of Michael O’Donovan, but later created the pseudonym “Frank O’Connor” that he would use for all of his writings (Gale). It was there in Cork that he experienced the horrors and distress of living in poverty. Even when the family had a small amount of money, O'Connor's father would regularly go out on drinking sprees and return home violent and cruel (Gale). O'Connor, being the only son, learned to help provide for his mother when his father's priorities fell short. O'Connor's education was minimal as he only attended formal school for a short...