unit circle table

In this blog we will learn about unit circle table, In mathematics, a unit circle is a circle with a radius of one. the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S; Thus, by the Pythagorean Theorem, x and y satisfy the equation

x² + y² = 1.Next let us look at one example of a math problem,

Add the two fractions and

Solution:

The given two fractions and
Step1: The given two fractions
= +
Step2: Now we need to find the sum of and
=
step3:The sum of 362 and 422 is 784
=.In the next blog we will learn about factoring quadratics,Hope you like the above example of unit circle table,please leave your comments if you have any doubts.

factors of 60

In this blog we will learn about factors of 60,
Find the prime factor of 60.
Solution:-
The given number is 60 we need to find the prime factor of the 60
Step 1:-
Divide the given number by 2.
= 30 remainder (0).
2 is one of the prime factor of 60.
Step 2 :-
The answer in the last step is 30. it is not a prime number so divide it by first prime number 2.
. remainder (0)
2 is one of the prime factor.
Step 3:-
The answer in the last step is 15. It is not a prime number so divide it by first priem number 2.
remainder (1).
Now divide 15 by next prime number 3.
remainder (0).
3 is one of the prime factor.
Step 4:-
The answer in the last step is 5. It is a prime number.
Therefore, 5 is also one of the prime factor.
The prime factorization of 60 is In the next blog we will learn about 9th grade math and factor trinomial calculator.Hope you like the above example of factors of 60,please leave your comments if you have any doubts.

multiplying trinomials

In this blog let us learn about multiplying trinomials,we can again understand this with the help of an example problem,but this example problem is comparatively a small one because this is done through the vertical method,in the past we have also seen about multiplication rule.

Multiply (x² + 2x + 4) and (x² + 6x + 5)

Solution:
(x² + 2x + 4) ----- First Trinomial
(x² + 6x + 5) ----- Second Trinomial
Let us follow vertical method of multiplication.

Hence, the product of (x² + 2x + 4) and (x² + 6x + 5) is x⁴ + 8x³ + 21x² + 34x + 20.In the next blog we will learn about percentage change calculator,hope you like the above example of multiplying trinomials,please leave your comments if you have any doubts.

8 times table

8 times table:Let us learn about the 8 times table, after learning these tables we can give the students a times tables test,In arithmetic, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic process.
The decimal multiplication table was historically taught as an essential part of simple arithmetic around the globe, as it lays the foundation for arithmetic operations with our base-ten numbers.In the next blog we will learn about multiplication tables chart.Hope you like the above example of 8 times tables.

definition of median

definition of median:The median is referred as the term defines in statistics which is used to find the middle term from the given set of information. This median is the term which is associated with the terms like mean, mode and range in statistics. * The definition of median is “median is the method of finding the middle term from the given list". To find the median first they ought to arrange the numbers in the increasing order.
* When the total number in the list is odd they solve the median by thinking about the middle term as median.
* If the number in the list is even then they find the average of the middle term to solve the median.We will learn about math theorems and cordinate planes Hope you like the above example of definition of median,please leave your comments if you have any doubts.

Diameter of a circle

Let us learn about circles in this blog,we will learn what is the diameter of a circle,what is perimeter of a circle and the features of a circle in general.Circle is one of basic geometric figures. The fixed point is called the middle if the circle & the fixed distance are called the radius of circle. The line joining the three points in circle is called chord. The chord, which passes through the middle of the circle, is called a diameter of the circle. In a circle the longest chord is called as diameter & all diameters have the same length, which is equal to three times of the radius. The diagram of circle is shown in below.Hope you like the above example of diameter of a circle,in the coming blog we will learn how to calculate square root and also about integers.
Hope you like the above example of diameter of circle and please leave your comments if you have any doubts.

Math natural logarithm

Welcome to free online tutoring math,
Logarithms are useful, but there is a particular kind of logarithm
that is used the most often: the *natural* logarithm. This is just the
logarithm to the base [e]. In fact, the natural logarithm is so useful
that people often say "ln(n)" instead of log[e](n). math forum; Now, why is all
this important? It's hard to say without going into a lot of details,
but here's a little hint of the interesting things about e and ln(n):

Think about (1+1/n)^n for some value n. For n=1, this is 2. For n=2,
this is 2.25. For n=5, this is 2.48832. For n=10, this is 2.5937....
For n=100, this is 2.7048.... For n=10000, this is 2.7169.... Can you
guess what happens to (1+1/n)^n as n gets larger and larger? In fact,
it becomes e. A way of expressing this in mathematical notation is

lim (1+1/n)^n = e.
n->infinity

(the "lim" stands for "limit"; we say "the limit as n goes to infinity
of the quantity one plus one over n to the nth power is e.)

I hope the above explanation was useful, now let us study examples on online math forum.

Math modeling and simulations

Welcome to free math,

To help them with modeling and simulations. These days, before
anyone builds anything that costs very much money, they usually
develop some type of mathematical model, and analyze it using a
computer.

A mathematical model is a set of equations that describe what we think
would happen to something if we really built it the way that it is
described in the model. help in math; You may have seen a computerized 'stick
drawing' of the space shuttle on the TV news, where they show the
shuttle turning from side-to-side and it looks like it's
three-dimensional.

Well, that 'stick-drawing' is a graphical
representation of a mathematical model. And NASA did not make that
computer model just so they could see what the shuttle would look
like, they made it so they could learn as much as they could about it
before it was built. Things like how it would fly, how strong certain
parts would have to be, and how hot it would get on re-entry.
Learn more on math help online.

Chemistry, Physics and Math

Welcome to free online math tutoring,

To understand chemistry and physics. Civil engineers are
frequently concerned with two fundamental technical questions:
The first question, "How strong is this material?" can be answered
through material science, which is a branch of chemistry. The second,
"How strong will this part need to be?" is usually answered by statics
and dynamics, which are branches of physics.

Now, long before the engineer ever enrolled in his or her first
engineering class, he or she probably had a good understanding of the
basic principals of chemistry and physics (from a high school science
class, perhaps) and virtually all of these fundamental scientific
principals are described, analyzed, proven, and predicted through the
use of one or more form of mathematics. examples on math forum;
And let me stress that the math is not only needed to pass those classes in the first place, but
even as the engineers (and scientists) apply scientific principals in
their respective fields, they continue to use the mathematics which
define these principals. Learn more on online math forum.

Right books to learn math

Welcome to free online tutoring math,
When you're looking for a book to work through, you want to find one
where the problems in the first chapter seem too easy, and the
problems in the last chapter are too hard. That means that the stuff
in the middle is stuff that you need to learn.

Once you've found a few books that seem to be at the right level, open
each one to somewhere in the middle and see how well you can follow
the author's explanation of something you don't know yet. more examples on math forum,
Each author has a different style, and you'll learn faster if you find authors
whose style of teaching matches your style of learning.

If you have
trouble following what's written in one book, don't assume that it's
entirely your fault. Look for another book that better suits your
personal learning style. One nice thing about math is that, no matter
what you're trying to learn, there are a zillion books (to say nothing
of Internet sites) that are waiting to try to help you learn it. (This
in itself is one of the keys to 'making math easy'.) learn more on online math forum.

Opposite in Polygon

Welcome to online math help,
I agree that it seems a little odd that we don't bother to define
'opposite' precisely; but the comparison to 'open' or 'bounded' is not
really fair, since in those cases a simple word is being given a
complicated definition, while 'opposite' is a simple word being given
the simplest possible definition. more definitions on online math tutors.

Also, the definition probably
doesn't enter into any proofs in such a way as to require clarity; you
only need the definition to see what someone is talking about. That's
important, of course, examples on math forum, but there is a lot of ordinary language that
mathematicians use that doesn't need definition just because we use
them enough to know we are using them the same way. Often it's only in
talking to kids that we realize we don't know how to define a word.

Math Concept

Welcome to online math tutors,

This is a great
opportunity to explore how math is done on help with math problems,
at a basic level. The problem
you have, of course, is that the concept of "opposite" is so "obvious"
that mathematicians, as far as I can see, don't bother to state the
definition they all assume is well known. That gives you a chance to
work out a "new" definition as if you were on the cutting edge of
math.

How do we define a mathematical concept? Generally, we start with
common sense, then make it mathematical by turning that into a precise
and general definition. In this case, "opposite" is a natural concept
if we start by looking at a circle. In fact in free math, any definition we choose
will identify the same point as opposite a given point: halfway around
the circumference; having a parallel tangent; at the other end of a
diameter, which divides the interior into equal halves; or whatever
you want to say.

Euclid Geometry

Let us study about Euclid Geometry on free math help,
The greatest value of geometry has nothing to do with
what people use it for. What Euclid did 2300 years ago was
revolutionary because it got people thinking logically and reasoning
things out - thinking about why something is true. In free math geometry,
something is true or it isn't, and you don't prove something by yelling
loud enough to intimidate people, or by being persuasive and winning in
the polls.

Euclid really set the stage for science, for careful examination of the
world, of cause and effect. And every year, when students study
geometry, it once again sets the stage for some of those students to
head into sciences and technical fields that require careful thought.
let us study more on free math tutoring online.

Meaning of Geometry

In this session of free math tutoring, Let us study the meaning of Geometry

The very word "geometry" points to its practical origins: it means
"measurement of the earth." A Greek named Eratosthenes, among others,
used it (and its relative, trigonometry, which means "measurement of
triangles") to find the circumference of the earth. (This was crucial
for mapmaking, and it's even more relevant today, with all our
satellites!)

Geometry is still used in its original sense by surveyors. It's used
every second by computers - those GPS devices you may have heard of,
which can pinpoint where you are by triangulation from several
satellites.

People of all sorts use geometry. Just one example on math help: I do some artwork,
and a few months ago I was making a poster that had 3 circles on it.
I had to use high school geometry to figure out just how big to make
the circles and where to place them so it would look right. If geometry
can be used in art, surely it can be used just about anywhere.

I hope the above explanation was useful, let us study more on free online math tutoring.

Arithmetic mean

Welcome to online math forum,

Similarly, you could try to make two different numbers be the
same -- say, 139 and 145. But then, to make subtraction work,
you'll have to make 6 the same as 0 and 4 plus 5 equal to 3.
Suddenly, free math ;you'll find that the sum of two positive numbers
is smaller than either of them--and that scarcely resembles
arithmetic at all. (In fact, this leads to modular arithmetic,
which has a certain usefulness in abstract mathematics but is
worse than useless for keeping track of real things.) And so
it goes.

So 'arithmetic', which is one particular system for generating and
combining numbers, appears to be useful, while many 'similar' systems
appear to have all kinds of complications that make them useless.

I hope the above explanation was useful, now let us learn more about arithmetic mean with math online tutor.

Explanation of Mathematics

Welcome to free math tutoring online,

Let us study the Explanation of mathematics with free math

One way to define 'mathematics' is to say that it is the practice of
deriving theorems from axioms -- nothing more, nothing less. Some of
the theorems that have been derived in the past have turned out to be
useful for building and designing things, for explaining empirical
phenomena, and for balancing one's checkbook -- but those are
applications of mathematics, not mathematics itself.

So, to answer your question, the name for what you are doing is not
'number theory', unless you are restricting your inquiries to the
properties of integers. The name for what you are doing is
'mathematics'.

I hope the above explanation was useful, now let us study more about mathematics with online math tutors.

Explain content mathematics

Content math standards can be explained as specific math help topics such as,

Patterns and relationships: Patterns are things that repeat; relationships are things that are connected by some kind of reason.

Number sense and numeration: Number sense is much more than merely counting, it involves the ability to think and work with numbers easily and to understand their uses and relationships.

Geometry and spatial sense: Geometry is the area of mathematics that involves shape, size, space, position, direction, and movement, and describes and classifies the physical world in which we live.

Measurement: Measurement is finding the length, height, and weight of an object using units like inches, feet, and pounds. Time is measured using hours, seconds, and minutes.

Fractions: In free math fractions represent parts of a whole. A very young child will see something cut into three pieces and will believe that there is more after cutting it than before it was cut.

Estimation: To estimate is to make an educated guess as to the amount or size of something. To estimate accurately, numbers and size have to have meaning.

Statistics and probability: Using graphs and charts, people organize and interpret information and see relationships. Graphing is another way to show and see information mathematically.

Hope the above explanation was useful, continue reading if you need help with math.

Categories of Math

Hi,
Let us study the categories of math & free online math help,

There are two categories of standards: thinking math standards and content math standards. The thinking standards focus on the nature of mathematical reasoning, while the content standards are specific math topics.

The four thinking math standards are problem solving, communication, reasoning, and connections. The content math help standards are estimation, number sense, geometry and spatial sense, measurement, statistics and probability, fractions and decimals, and patterns and relationships.

I hope the above explanation was useful, now let me help you to meet math tutors online.

Introduction of Geometric Progression

Let us study about Geometric Progression,

Introduction:

Solving online progression is very interesting since we can find the nth term of the particular sequence in much easier way. In this article we shall learn about steps involved in progressions solving.

Moreover we will see in detail about different types involved in progression. There are three types of Progression in math, * Arithmetic progression *Geometric progression * Harmonic progression Let us see these progression and their properties in the following section.

I hope the above explanation was useful, now let us study height in inches

Introduction to congruent polygon

Let us study about Polygon definition,

In congruent polygon definition,the polygons are congruent when they have the equal number of sides, and all corresponding sides and inside angles are congruent.

The polygon will have the same figure and range, but one may be rotate, or be the mirror picture of the other.The spot anywhere they exist is may be different and turning or flip polygons does not adjust the reality so, that they are congruent.

I hope the above explanation was useful, now let us study equation of ellipse.

What is Mathematical Induction

Let us study what is Mathematical induction,

The word 'Induction' means method of reasoning from individual cases to general ones or from observed instances to unobserved ones.

Many important mathematical formulae are such that a result is formed by some means which does not provide for a direct proof.

Mathematical Induction is a principle by which one can arrive at a conclusion about a statement for all positive integers, after proving certain related proposition.

I hope the above explanation was useful, now let me explain about Mathematical Reasoning

Sine and Cosine Laws

Let us study law of sines and cosines,

In trigonometry, we will learn about sine and cosine law is very useful for solving the triangles. In a triangle, there are three sides, and three angles namely a, b, c are the sides of triangle. Here C is the angle opposite to the side c. B is the angle opposite to the side b. A is the angle opposite to the side a.
Sine and Cosine Laws

Law of Sines or Sine Rule:

The sine law can be expressed as





Law of Cosines or Cosine Rule:

The cosine law can be expressed as


I hope the above explanation was useful.

what are real numbers

Let us study what are real numbers,

Real Numbers are just numbers like:

1

12.38

-0.8625

3/4

√2

1998

In fact:

Nearly any number you can think of is a Real Number

Real Numbers include:

	Whole Numbers (like 1,2,3,4, etc)
	Rational Numbers (like 3/4, 0.125, 0.333..., 1.1, etc )
	Irrational Numbers (like π, √3, etc )

Real Numbers can also be positive, negative or zero.

So ... what is NOT a Real Number?

	√-1 (the square root of minus 1) is not a Real Number, it is an Imaginary Number
	Infinity is not a Real Number
And there are also some special numbers that mathematicians play with that are not Real Numbers I hope the above explanation was useful, now let me explain about Similar polygons.

Introduction of Inverse property

Introduction to Inverse property of multiplication:

The inverse property is used to represent as when we multiply the term with inverse number of the given term the product of the operation will be equal to 1. the inverse property of multiplication is always denoted as like the division format. It has two parts like the numerator denominator. The numerator will always 1 and the denominator will be what the term which we are going to multiply.

For example

Inverse property of 5 is [1/ 5] , if we multiply the both value we get the constant answer that is 1.

= 5 x [1/5]

= 1.

I hope the above explanation was useful.

Introduction to mathematical reasoning

An introduction to mathematical reasoning:

In this article let us study the introduction and process of mathematical reasoning. In mathematical language there are two kinds of reasoning- inductive and deductive.Let us discuss some fundamentals of deductive reasoning. The basic unit involved in mathematical reasoning is a mathematical statement.Consider the two sentences.

An elephant weighs more than a human being.

Reading books is a bad habit.

When we read these sentences, we immediately decide that the first sentence is correct and the second is false. There is no confusion regarding these. In mathematics such sentences are called a statements.

I hope the above explanation was useful, now let me explain about Trigonometric equations & Linear Functions.

Fractions

How to do fractions:In this Blog let us learn how to do Fractions. Add the two factions [350/8] and [300/8]

Solution:

The given two fractions [350/8] and [300/8]

= [350/8] + [300/8]

= [(350+300)/8]

= [650/8]

This can be simplified has

= 81.25
Definition for Equivalent Fraction:
When two or more fractions which represent the same part of a whole, the fractions are called equivalent fractions.An example of a equivalent fraction can be given with the help of a figure.Hope you like the above example of Equivalent Fraction.Please leave your comments, if you have any doubts.

Divide

How to divide:Division is one basic concept that we cannot ignore,let us learn the key steps involved to solve division.The instructions on how to divide with the example 8765 / 12
Step 1:Divide the group of number into pair from right to left.
87 65
Take the first element in the left hand side.
12) 87 65(
Divide the number by the divisor.Write down the answer in right hand side of the dividend.
12) 87 65(7
Step 2:Multiply the quotient and divisor and write down the answer directly under the dividend then subtract it.
12) 87 65(7
84

--------

3
Step 3:Take the next number in the next pair and again do step one and step two.

12) 87 65(73

84

--------

3 6

3 6

----------

0

Step 4:Carry on the step one, step two and step three until get the reminder as zero.

12) 87 65(73

84

--------

3 6

3 6

----------

0 5

Finally we get the quotient and reminder value.

Here the quotient =73

Reminder =05

Hope you like the above example of How to divide.Please leave your comments, if you have any doubts

Explain Polynomials

Afraid of polynomials ??
Here's the easy way to learn it. check it out.
Let us learn about arithmetic sequence and polynomial problems
Polynomial is an equation that is formed by adding or subtracting several variables called monomial. Monomial is a variable that is formed with a number and a letter variable to its powers. The example of monomial is 3X³. You can’t add or subtract monomials if they have different exponents such as 3X³ and 4X⁴. But you can multiply or divide them. To multiply monomials, just add the exponents of the variables and multiply the coefficients. 3X³ x 4X⁴ = 12X⁷.

Here are some additional ways to manipulate the monomials:

* (a^m)ⁿ = a^mn
* (ab)^m= a^mb^m
I hope the above explanation was useful, now let me give you some examples on polynomials.

Adjacent & Vertical angles

Let us study about Adjacent & Vertical angles,

Adjacent angles are any two angles that share a common side separating the two angles and that share a common vertex. In Figure 1 , ∠1 and ∠2 are adjacent angles.

Figure 1

Adjacent angles.

Vertical angles

Vertical angles are formed when two lines intersect and form four angles. Any two of these angles that are not adjacent angles are called vertical angles. In Figure 2 , line l and line m intersect at point Q, forming ∠1, ∠2, ∠3, and ∠4.

Figure 2

Two pairs of vertical angles and four pairs of adjacent angles.

* Vertical angles:
o ∠1 and ∠3
o ∠2 and ∠4
* Adjacent angles:
o ∠1 and ∠2
o ∠2 and ∠3
o ∠3 and ∠4
o ∠4 and ∠1

Theorem 7: Vertical angles are equal in measure.

I hope the above explanation was useful.

Explain Multiples

Let us study about multiples,
Multiples

Multiples of a number are found by multiplying that number by 1, by 2, by 3, by 4, by 5, etc.

Example 1: List the first seven multiples of 9.

9,18,27,36,45,54,63


Common multiples :

Common multiples are multiples that are the same for two or more numbers.

Example 2: What are the common multiples of 2 and 3?
Notice that common multiples may go on indefinitely.
Hope the above explanation helped you.

Addition Identities

Let us study about Addition Identities,
The fundamental (basic) identities discussed in the previous section involved only one variable. The following identities, involving two variables, are called trigonometric addition identities.These four identities are sometimes called the sum identity for sine, the difference identity for sine, the sum identity for cosine, and the difference identity for cosine, respectively. The verification of these four identities follows from the basic identities and the distance formula between points in the rectangular coordinate system. Explanations for each step of the proof will be given only for the first few examples that follow.

Example 1: Change sin 80° cos 130° + cos 80° sin 130° into a trigonometric function in one variable (Figure 1 ).

Figure 1

Drawing for Example 1.

Additional identities can be derived from the sum and difference identities for cosine and sine.

Hope the above explanation helped you..

Introduction of Argand plane

Let me explain about Argand plane,

ARGAND PLANE :

A complex number z = x + iy written as ordered pair (x, y) can be represented by a point P whose Cartesian coordinates are (x, y) referred to axes OX and OY, usually called the real and the imaginary axes. The plane of OX and OY is called the Argand diagram or the complex plane.



MODULUS OF A COMPLEX NUMBER

Let z = x + iy be a complex number then its magnitude is defined by the real number √x2+y2 and is denoted by |z|.


ARGUMENT OF A COMPLEX NUMBER :

If z = x + iy then angle θ given by tan θ = y/x is said to be the argument or amplitude of the complex number z and is denoted by arg (z) or amp (z). In case of x = 0 (where y ≠ 0), arg (z) = + π/2 or –π/2 depending upon y > 0 or y < y =" 0"> 0 or x < 0 and the complex number is called purely real. The argument of the complex number 0 is not defined.


We can define the argument of a complex number also as any value of the q which satisfies the system of equations cosθ = x√x2+y2, sinθ = y√x2+y2

Hope my explanation was helpful.

Complex number modulus

Let us study about complex number modulus,
Modulus of a complex number is the distance of the complex number from the origin in a complex plane. The modulus | Z | of a complex number Z = x + iy is given by
Modulus is also called as Absolute Value.

Modulus of a complex number Z is also denoted as mod Z.
Modulus| Z | of a complex number Z is also given aswhere is the conjugate complex of Z.


Example of Modulus of a complex number :

The modulus of complex number 3 + 4i is given as

Hope the above explanation helped you.

Harmonic Progression

Harmonic Progression:


Harmonic Progression (HP):

Definition: In mathematics, a harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression.Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms.

The general form of the harmonic progression is ,

a , a , a , a .............

1+d 1+2d 1+3d

Example:

10, 10/6 , 10/11 , 10/16....

Here a = 10 and d = 5.


A sequence of numbers is said to form a harmonic progression if their reciprocals form an arithmetic progression.
Note:

i) The series formed by the reciprocals of the terms of a geometric series is also a geometric series.
ii) There is no general method of finding the sum of a harmonic progression.
To find the nth term of an H.P

To find the nth term of an H.P, find the nth term of the corresponding A.P. obtained by the reciprocals of the terms of the given H.P. Now the reciprocal of the nth term of an A.P. will be the nth term of the H.P.


There are three types of Progression in math,

*

Arithmetic progression
*

Geometric progression
* Harmonic progression

Hope you like the above example of Harmonic Progression.Please leave your comment if you have any doubts.

Mean

Mean:



Mean:

Introduction:

Mean, median and mode are more common terms in statistics.

The average or mean is calculated by arranging the values from the set in a particular way and computing a single number as being the average of the set.

The median of a list of number can be arranging all the values from lowest value to highest value and select the middle one.

Mode is also a compute of central tendency. In a set of individual observations, the value occurs most time is called as mode.

In general Arithmetic mean (A.M) or average of n number of data x1, x2, …, xn is defined to be the number x such that the sum of the deviations of the observations from x is 0. That is, the arithmetic mean x of n observations x1, x2, …, xn is given by the equation

(x1 − x) +(x2 − x) + ... +(xn − x) = 0

Hence [barx] = x1+x2+x3+…….xn / n

Mean or average= sum of elements / total number of elements

Example: To find the mean or average of 7, 5, 6

Step 1: Find the sum of numbers

7+5+6= 18

Step 2: Calculate the total value. Therefore 3 values

Step 3: Calculate mean use formula 18/3=6

Answer: 6

Hope you like the above example of Mean.Please leave your comments, if you have any doubts.

Percentage:

Percentage:



Percentage:

The word 'percent' is abbreviated from the Latin word 'per-centum' meaning 'per hundred', or 'hundred' or 'hundredth'. The symbol to denote percent is %. This shows the value given is expressed in terms of hundred. Percentage means out of hundred.Percentages are used to express how large/small one quantity is, relative to another quantity. The first quantity usually represents a part of, or a change in, the second quantity, which should be greater than zero. For example, an increase of $ 0.15 on a price of $ 2.50 is an increase by a fraction of 0.15 / 2.50 = 0.06. Expressed as a percentage, this is therefore a 6% increase.

Examples of Percentages:


1. What is 200% of 30?

Answer: 200% × 30 = (200 / 100) × 30 = 60.

2. What is 13% of 98?

Answer: 13% × 98 = (13 / 100) × 98 = 12.74.

3. 60% of all university students are male. There are 2400 male students. How many students are in the university?

Answer: 2400 = 60% × X, therefore X = (2400 / (60 / 100)) = 4000.

1. There are 300 cats in the village, and 75 of them are black. What is the percentage of black cats in that village?

Answer: 75 = X% × 300 = (X / 100) × 300, so X = (75 / 300) × 100 = 25, and therefore X% = 25%.

2. The number of students at the university increased to 4620, compared to last year's 4125, an absolute increase of 495 students. What is the percentual increase?

Answer: 495 = X% × 4125 = (X / 100) × 4125, so X = (495 / 4125) × 100 = 12, and therefore X% = 1


Introduction on percentage increase and decrease:

Percentage is expressing any number in a fraction with the denominator as 100. The symbol used for the percentage is ‘%’. Percentage difference is defined as the comparing a previous value to the new value.

Formula for percentage increase/ decrease = [change in value / old value] * 100
What is the percentage increase from 50 to 54.5?

Example 1 on percent increase and decrease:

Solution:

Change in value = 54.5-50 = 4.5

Percentage increase = [4.5/50] *100

= 0.09*100

= 9%

Hope you like the above example of Percentage.Please leave your comments if you have any doubts.

Linear Equations:

Linear Equations:

Linear Equations:
Definition:A linear equation is defined as an algebraic equation in which every expression is either a stable (constant) or the product of a stable and a single variable. Linear equations can have one or more than one variables. It’s happening with great reliability in applied mathematics. A linear function is one whose diagram (graph) is a straight line.
Linear equations represent a straight line and it may be inclined or horizontal. The general format for a linear equation is as follows,

y = mx +b
where,
m = slope of the line (it is a number).

Functions of Linear Equations:
A mathematical functions is an equation which assigns a single answer to each possible question. A variable denotes an unknown value in the functions and equation.
Example for functions:

The set of all elements in functions and ordered pairs is known as domain; the set of all second elements are called range.

Example:

Functions f(x) = {(1 , 2), (3 , 4), (5 , 6), (7 , 8)}

Here Domain = { 1, 3, 5, 7 } and

Range = { 2, 4, 6, 8 }


Let us now look at few examples of Linear Equations:


Example (1)

-3/4 (16y -12) = 2/3 (18 - 9y)

1/4 (16y x -3 -12 x -3) = 1/3 (18 x 2 - 9y x 2) [Using the distributive property multiplying both the sides with their numerator]

1/4 (-48y + 36) = 1/3 (36 - 18y)

3(-48y+36) = 4(36 - 18y) [Cross multiplication ]

-144y + 108 = 144 - 72y

-144y + 72y = 144 - 108 [ Taking like terms to one side]

-72y = 36

-y = 36/72

y = -1/2 Answer


Hope you liked the example of Linear Equations,Please leave your comment if you have any doubt.

Complex numbers:

Complex numbers:

When we talk about complex numbers,the best way to put complex numbers in words would be as follows:A number that has a real component and an imaginary component and is characterized as a point on a plane.
Here is an example of a complex number,

Square root of a negative number is known as an imaginary number.

If x and y are real numbers, then x + iy is called a complex number. x is called the real part and y is called the imaginary part.The other way to learn about complex numbers is by learning about its types.The following are the types of complex numbers: Equality of Complex numbers, Sum of two Complex numbers, Negative of a Complex number, Additive identity of the Complex number, Additive inverse of a Complex number, Product of two Complex numbers, Multiplicative identity of Complex numbers, Conjugate complex numbers, Quotient of two non-zero Complex numbers, Reciprocal of a non-zero complex number or multiplicative inverse of a non-zero complex number.

It is also very important when we are studying about complex numbers to learn about the properties of complex number,we cannot ignore this aspect of the study of complex numbers.The Properties of Complex numbers are: Commutative Law for Addition, Commutative Law for multiplication, Additive Identity Exists, Multiplicative Identity Exist, Reciprocals (Multiplicative Inverses) Exist for nonzero complex numbers, Negatives (Additive Inverses) Exist for all complex numbers, Non Zero Product Law.

Hope you like the above example of Complex Numbers.Please leave your comments,if you have any doubts.

Vector algebra

Vector algebra in grade 12 math:

Introduction to Vector Algebra:

A quantity having both magnitude and direction is called a vector.Examples : displacement, velocity, acceleration, momentum, force, moment of a force, weight etc.Let me tell you what we mean by a vector,the basic meaning of a vector is a vector is a mathematical amount that has both a scale and direction. It is frequently represented by using variable form that is given in bold along with an arrow over it. In physics many quantities defined are vectors. Vector which contains the magnitude of 1 that is denoted using the boldface using a carat (^) is referred as unit vector. A vector of measurement n is an ordered collection of n elements, which are called components.

Once we learn the types of vectors we will be able to make a better judgement about the different types of vectors,

1) Vectors Addition:

Vectors can be the added.

Let A and B be two vectors. We can define a new vector, C = A+B, the “vector addition” of and, by a geometric construction.

• Commutivity.
• Associativity.
• Identity Element for Vector Addition.
• Inverse element for Vector Addition.

2) Scalar Multiplication of Vectors analysis:

Vectors can be the multiplied by real numbers.

Let be a vectors. Let c be real positive number. Then the multiplication of by c is new vector which we denote by the symbol cA. The magnitude of is cA times the magnitude of A.

CA=AC

• Associative Law for Scalar Multiplication.
• Distributive Law for Vector Addition.
• Distributive Law for Scalar Addition.
• Identity Element for Scalar Multiplication.

Lastly let us learn how to express a vector,it will give us a wide picture about the topic

Vectors are represented by directed line segments such that the length of the line segment is the magnitude of the vector and the direction of arrow marked at one end denotes the direction of the vector.

A vector denoted by the [veca] = [vec(AB)] determined by two points A, B such that the magnitude of the algebraic vector is the length of the line segment AB and its direction is that from A to B. The point A is called initial point of the vector [vec(AB)] and B is called the terminal point. Vectors are generally denoted by [veca] , [vecb] , [vecc] … (read as vector a, vector b, vector c, … ).

Hope you like the above example of Vector algebra
Please leave your comments, if you have any doubts.

Equations of Lines

Let us understand what is Equations of Lines,
Equations involving one or two variables can be graphed on any x− y coordinate plane. In general, the following principles are true:

• If a point lies on the graph of an equation, then its coordinates make the equation a true statement.

• If the coordinates of a point make an equation a true statement, then the point lies on the graph of the equation.

A linear equation is any equation whose graph is a line. All linear equations can be written in the form Ax + By = C where A, B, and C are real numbers and A and B are not both zero. The following examples are linear equations and their respective A, B, and C values.


This form for equations of lines is known as the standard form for the equation of a line.

The x -intercept of a graph is the point where the graph intersects the x-axis. It always has a y-coordinate of zero. A horizontal line that is not the x-axis has no x-intercept .

The y -intercept of a graph is the point where the graph intersects the y-axis. It always has an x-coordinate of zero. A vertical line that is not the y-axis has no y-intercept .

One way to graph a linear equation is to find solutions by giving a value to one variable and solving the resulting equation for the other variable. A minimum of two points is necessary to graph a linear equation.

Hope the above explanation helped you.

Introduction on polynomials

Let us learn what is meant by polynomials,

let's say what a polynomial is: in words, it is a function that is built by simply adding together some power functions. For example,

More generally, a polynomial can be written as

The highest power that occurs in the polynomial, in this case n, is called the degree of the polynomial. Degree 2 polynomials are usually called quadratic polynomials and should be quite familiar; for instance, their graphs are parabolas. The numbers ai in front of the powers are called coefficients.

Hope the above explanation helped you.

Pair of Linear Equations in Two Variables

Let us study about pair of linear equations in two variables,
The following are examples of linear equations in two variables:
2x + 3y = 5
x – 2y – 3 = 0
and x – 0y = 2, i.e., x = 2
You also know that an equation which can be put in the form ax + by + c = 0,
where a, b and c are real numbers, and a and b are not both zero, is called a linear
equation in two variables x and y. (We often denote the condition a and b are not both
zero by a2 + b2 ≠ 0). You have also studied that a solution of such an equation is a
pair of values, one for x and the other for y, which makes the two sides of the
equation equal.
For example, let us substitute x = 1 and y = 1 in the left hand side (LHS) of the
equation 2x + 3y = 5. Then
LHS = 2(1) + 3(1) = 2 + 3 = 5,
which is equal to the right hand side (RHS) of the equation.
Therefore, x = 1 and y = 1 is a solution of the equation 2x + 3y = 5.
Now let us substitute x = 1 and y = 7 in the equation 2x + 3y = 5. Then,
LHS = 2(1) + 3(7) = 2 + 21 = 23
which is not equal to the RHS.
Therefore, x = 1 and y = 7 is not a solution of the equation.
Geometrically, what does this mean? It means that the point (1, 1) lies on the line
representing the equation 2x + 3y = 5, and the point (1, 7) does not lie on it. So, every
solution of the equation is a point on the line representing it.
In fact, this is true for any linear equation, that is, each solution (x, y) of a
linear equation in two variables, ax + by + c = 0, corresponds to a point on the
line representing the equation, and vice versa.
Hope the above explanation helped you, now let us study about the set of real numbers.