MyMaths MyLife

We will have equations of one or two unknowns invariably in every problem. Some times we get three equations in three unknowns. In general, we need as many equations as the variables we will have to solve for. So, for solving for the values of two unknowns, we need two equations (or two conditions given in the problem) and for solving for the values of three unknowns, we need three equations.

One equation in one unknown:
An equation like 2x + 6 = 36 is an equation in one variable. We have only one variable x whose value we have to find out.

Two equations in two unknowns:
A set of equations like
2x + 3y = 10 ………… (1)
3x + 5y = 12 ………… (2)
Is called simultaneous equations in two unknowns. Here, we have two variables ( or unknowns) x and y whose values we have to find out.

And also we use three equations in three unknowns.

The basic number system we use in maths are:

Natural Numbers:
The numbers we use for counting are called natural numbers. The set of all natural numbers is denoted by “N”.
Hence N= {1,2,3,4……..}.

Whole Numbers:
When all the natural numbers and zero are put together, we get a new set of whole numbers. The set of whole numbers is denoted by “W”.
Hence W={0,1,2,3,4………}.

Integers:
The set containing positive numbers(1,2,3….), negative numbers(-1,-2,-3,…..) together with zero is called as set of integers. We denote the set of integers with “Z”.
Hence Z={……-3,-2,-1,0,1,2,3,……..}.

Rational Numbers:
A rational number is a number of the form a/b. Where a and b are integers, b≠0.
Example: {2/3, 4/7,7/9……..}

Prime Numbers:
Numbers which do not have any other factors except 1 and itself are called prime numbers.
Examples: {1,2,3,5,7,11,13,17,19,23,29,31,37,41……….}

Composite Numbers:
All numbers greater than 1 and except prime numbers are called composite numbers.
Examples: {4,6,8,9,10,12…….}.

Even Numbers:
Numbers which are exactly divisible by 2 are called even numbers.
Examples: {2,4,6,8,10…….}

Odd Numbers:
Numbers which are not exactly divisible by 2 are called even numbers.
(or)
Numbers which are not even numbers are called odd numbers.
Examples: {1,3,5,7,9……..}.

Important Facts and Formulae:

Results On Triangle

1.Sum of the angles of a triangle is 180 degrees.

2.The sum of any two sides of a triangle is greater
than third side.

3.Pythagoras Theorem:

In a right angled triangle (Hypotenuse)2 = (Base)2 +(Height)2

4.The line joining the mid point of a side of a triangle
to the opposite vertex is called the MEDIAN.

5.The point where the three medians of a triangle meet,
is called CENTROID. The centroid divides each of the
medians in the ratio 2:1

6.In an isosceles triangle, the altitude from the
vertex bisects the base

7.The median of a triangle divides it into two triangles
of the same area.

8.The area of the triangle formed by joining the mid points
of the sides of a given triangle is one-fourth of the area
of the given triangle.

Results On Quadrilaterals

1.The diagonals of a Parallelogram bisect each other.

2.Each diagonal of a Parallelogram divides it into two
triangles of the same area.

3.The diagonals of a Rectangle are equal and bisect
each other

4.The diagonals of a Square are equal and bisect each
other at right angles.

5.The diagonals of a Rhombus are unequal and bisect
each other at right angles.

6.A Parallelogram and a Rectangle on the same base
and between the same parallels are equal in area.

7.Of all he parallelogram of given sides the parallelogram
which is a rectangle has the greatest area.




Formulae:

1.Area of a RECTANGLE = length * breadth

Length = (Area/Breadth) and Breadth = (Area/Length)

2.Perimeter of a RECTANGLE = 2(Length + Breadth)

3.Area of a SQUARE = (side)2 = ½ ( Diagonal)2

4.Area of four walls of a room = 2(length + breadth) * height

5.Area of a TRIANGLE = ½ * base * height

6.Area of a TRIANGLE = √[s * (s-a) * (s-b) * (s-c)],
where a,b,c are the sides of the triangle and s = 1/2(a+b+c)

7.Area of EQUILATERAL TRIANGLE = √(3/4)* (side)2

8.Radius of in circle of an EQUILATERAL TRIANGLE of
side a = r / 2√3

9.Radius of circumcircle of an EQUILATERAL TRIANGLE
of side a = r / √3

10.Radius of incircle of a triangle of area ∆ and
semi perimeter S = ∆ / s

11.Area of a PARALLELOGRAM = (base * height)

12.Area of RHOMBUS = 1/2 (product of diagonals)

13.Area of TRAPEZIUM =
=1/2 * (sum of parallel sides)* (distance between them)

14.Area of a CIRCLE =  r2 where r is the radius

15.Circumference of a CIRCLE = 2r

16.Length of an arc = 2 rø / 360, where ø is central angle

17.Area of a SECTOR = ½ (arc * r) = r2ø / 360

18.Area of a SEMICIRCLE = r2 / 2

19.Circumference of a SEMICIRCLE = r

In any business environment the most important concern is about profit or loss of the transaction conducted.

The money paid by a shopkeeper to buy goods is called the cost price(C.P.) of the
shopkeeper. The price at which he sells the goods is called the selling price(S.P.) of the shopkeeper.

If the selling price(S.P.) is greater than the cost price(C.P.) then the shopkeeper gets profit. Profit = Selling price - Cost price.

If the selling price is less than the cost price then the shopkeeper gets loss.
Loss = Cost price - Selling price.

4. For comparison of profit or loss obtained in different transactions, they are they are expressed percentages.
Suppose
S.P = Selling price, C.P = Cost price,
P = Profit, L = Loss

Ratio of profit to cost price = P : C.P.
Percentage of profit = (P/c.p)*100.
Percentage of loss = (L/c.p)*100.

When two or more than two persons run a business jointly, they are called partners and the deal is known as partnership.


Ratio of Division of Gains:
1.When the investments of all the partners are do the same time, the gain or loss is distributed among the partners in the ratio of their investments.

Suppose A and B invest Rs x and Rs y respectively for a year in a business, then at the end of the year:(A's share of profit):(B's share of profit)=x:y

2.When investments are for different time periods, then equivalent capitals are calculated for a unit of time by taking (capital*number of units of time). Now gain or loss is divided in the ratio of these capitals.

Suppose A invests Rs x for p months and B invests Rs y for q months, then (A's share of profit):(B's share of profit)=xp:yq

3.Working and sleeping partners:A partner who manages the business is known as working partner and the one who simply invests the money is a sleeping partner.

Formula:
1.When investments of A and B are Rs x and Rs y for a year in a
business ,then at the end of the year
(A's share of profit):(B's share of profit)=x:y

2.When A invests Rs x for p months and B invests Rs y for q months,
then A's share profit:B's share of profit=xp:yq

Some Important definitions we use in sets.

Null Set:
A set is said to be a null set if it has no elements. It is also called an empty set or a void set and is denoted by ø.
Example:
{x | x is a perfect square and 4< x <9}

Finite and infinite sets:
A set ‘A’ is said to be finite if it is either an empty set or contains finite number of elements. Otherwise it is called an infinite set.
Example:
Set of natural numbers less than 100 is finite.

Cardinality of a finite set:
The number of distinct elements in a set is called the cardinality of the set. If a finite set A has n distinct elements, the cardinality of the set is n and is denoted by O(A) or n(A). The cardinality of an empty set is zero.
Example:
Cardinality of A = {a,e,I,o,u} is 5.

A set is a well defined collection objects. The objects of the set are called its elements. Sets are usually denoted by capital letters and the elements of the set are denoted by lower case. If an element x belongs to set A, it is denoted by x Є A. If x is not an element of A, it is denoted by striped(/)Є A.

A set, in general is represented in two forms:
1) In this form, a set is described by actually listing out all the elements. For example, the set of all odd natural numbers less than 10 is represented by {1,3,5,7,9}.

2) In this form, a set is described by a characterizing property. For example, the set of all odd natural numbers less than 10 is represented by {x Є N | x < 10 and x is odd}. The symbol | is read as “such that”.

My opinion is that mathematics is indeed a descriptive language. It is somewhat unique in this role though. The reason for this uniqueness is the fact that when people created this abstract system they wanted to use the most basic elements inherent in our reasoning, rather than just describe the observed. The result of this was a system that had very few and very basic axioms that seemed to match precisely the most basic patterns we observe in the Universe. It also meant that this system could evolve and create many patterns that appear to be greatly similar to those that we find in nature and can describe those.

What do I mean by basic elements ?
Well, one basic element of all reasoning systems we had so far is the existence of separate entities. Others are space, time, laws/relations that control
the entities and possibly more.