Algebra is a branch of mathematics that uses variables, numbers, and mathematical operations to represent problems or situations in the form of mathematical expressions.
Algebra involves manipulating abstract symbols, rather than numbers, with arithmetic.
For example, x+y=z and b-2=5 are algebraic equations, but 2+3=5 and 73*46=3,358 are not.
In algebra, variables like x, y, and z have unknown values and may change.
Mathematical operations like addition, subtraction, multiplication, and division are combined with variables to form a meaningful equation.
Algebra is a unifying thread of almost all of mathematics.
Elementary algebra deals with the manipulation of variables as if they were numbers and is therefore essential in all applications of mathematics.
Algebra Of Complex Numbers
Complex numbers are algebraic expressions that contain the factor i=√-1.
They are divided into two parts: the real part, denoted by Re(z), and the imaginary part, denoted by I(z).
For example, for a complex number z=2+3i,a=Re(z)=2 and b=Im(z)=3.
Here are some algebraic operations on complex numbers:
Multiplication is
defined as (π,π)×(π,π)=(ππ−ππ,ππ+ππ)
Addition is defined as (π,π)+(π,π)=(π+π,π+π)
Complex numbers
are useful in electrical circuits, where it is customary to use π for the
imaginary unit.
Graphical representation of complex numbers
Complex numbers can be represented graphically as a point in a coordinate plane. The plane where a complex number is assigned to each of its points is called a complex plane, also known as an argand plane or argand diagram.
The x-axis becomes the real axis and the y-axis becomes the imaginary axis. For example, the complex number x+iy is represented as a point in Figure 2.2.1.
If the number is in polar coordinates, r represents the distance from the origin to the point, and ΞΈ represents the angle that r makes with the positive x-axis (measuring counterclockwise).
The
modulus and argument of a complex number
The modulus and
argument of a complex number are:
Modulus and Argument - Newcastle University
The length of the line segment is called the
modulus of the complex number and is denoted |z| . The angle measured from the
positive real axis to the line segment is called the argument of the complex
number, denoted arg(z) a r g ( z ) and often labelled ΞΈ .
Numeracy, Maths and Statistics - Academic Skills Kit
How to Find the Modulus and Argument of a Complex Number
12-Jun-2021 — Modulus: The modulus of a
complex number z = a + b i is given by | z | = a 2 + b 2 . Argument: The
argument of a complex number z = a + b i is given by ΞΈ = tan − 1 where − Ο < ΞΈ ≤ Ο .
Study.com
Modulus and Argument of a Complex Number - Unacademy
The argument of a complex number is the
angle formed by the complex number with the positive axis of the argand plane,
while the modulus of a complex number is its distance from the origin.
Unacademy
The modulus and argument of a complex number - Mathcentre
The length of the line segment, that is OP,
is called the modulus of the complex number. The angle from the positive axis
to the line segment is called the argument of the complex number, z. The
modulus and argument are fairly simple to calculate using trigonometry.
Example.
Mathcentre
Find the modulus and argument of the complex number
The argument $\theta ={{\tan }^{-1}}\left(
\dfrac{b}{a} \right)$ is also called principal argument since tangent function
is periodic and all other arguments are given by $n\pi +\theta $ where $n$ is
any integer.
Vedantu
How to Find the Modulus and Argument of a Complex Number
14-Jan-2023 — number. we consider a general
complex number Z equals a plus b. i on the complex axes. it is here with real
component a and complex component B. now the modulus of Z means its length the
modulus is written like so and it is equal to the square root of a squared plus
b squared. this is the length of the line Z as calculated using Pythagoras's
Theorem now the argument of Z is the angle measured from the positive real
axis. for example the complex number shown has the argument measured from the
positive real axis as shown.
m.youtube.com
Modulus
The length of the line segment from the origin to the complex number.
Argument
The angle is formed by the complex number with the positive axis of the argand
plane.
The modulus and argument can be calculated using
trigonometry.
The cube root of unity
The cube root
of unity is a number that, when multiplied by itself three times, gives the
product as 1. It is written as 3√1 and has three roots: 1, \omega, \omega^2.
When multiplied
together, these three roots yield the answer unity. One of the roots is a real
root, while the other two are imaginary roots:
1 is the actual
root
π and π2 are imaginary roots
The product of the imaginary roots of the cube root
of unity is equal to 1, and the sum of the cube roots of unity is equal to
zero:
π⋅π2=π3=1
1+π+π2=0
The values of
the cube root of unity are:
1
π=−1+π√3/2
π2=−1−π√3/2
Arithmetic
Arithmetic is a
branch of mathematics that deals with operations on numbers. The four basic
operations in arithmetic are: Addition, Subtraction, Multiplication, Division.
The order of
these operations is given by the DMAS rule.
Arithmetic is
derived from the Greek word arithmos, which means "number". It
generally refers to the elementary aspects of the theory of numbers, arts of
mensuration (measurement), and numerical computation.
Arithmetic is
one of the important branches of mathematics that lays the foundation of the
subject "Maths" for students. The first known use of arithmetic was
in the 15th century.
An Indian mathematician and astronomer Brahmagupta
is known as the Father of Arithmetic.
What
are the 5 arithmetic operators?
Here are some
arithmetic operators:
Addition: (+)
Subtraction:
(-)
Multiplication:
(*)
Division: (/)
Modulo: (%)
Exponentiation:
(^)
Increment: (++)
Decrement: (--)
Unary plus: (+)
Many
programming languages support a combination of the assignment (=) and
arithmetic operators. For example, in C, the arithmetic operators include:
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