Whole numbers

Whole numbers:

The set of whole numbers is the set of natural numbers along with zero. so W = the set of whole numbers = 0,1,2,3,............

so Zero is the least number of the set of Whole numbers.

As the whole numbers is an infinite set we cant determine the highest number of this set.

The set of Whole numbers is a subset of Rational numbers.The best way to learn about whole numbers is to understand its properties,the properties of whole numbers are,it is advisable to use a number line inorder to understand the properties of whole numbers.

1. Number 3 <>

2. There is no whole number to the left of zero on the number line.So zero is the smaller number than each of the numbers to its right on the number line.That means 0 is the smallest or least of the whole numbers.

3. A whole number which is greater than a given whole number by 1 is said to be a successive whole number. 1 is the successive whole number to 0.Every whole number has one successor.

4. There is no whole number left to zero.hence 0 is not a successive whole number of any whole.

Hope you like the above example of Whole numbers.
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Pythagorean theorem:

Pythagorean theorem:

Introduction:

Let us learn about Pythagorean theorem,
Pythagorean theorem states that square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other sides.Consider the right angle triangle, a and b are the two smaller sides and c is the hypotenuse, so if we square a and b and add the both value we get the same answer as if we square c (c²)

This theorem can also be written as an equation relating the lengths of the sides a, b and c : where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.In the figure given above we can see the AC is called a Hypotenuse of the triangle and AB is called the opposite side of the triangle and BC is called the adjacent side of the triangle. This is the Pythagorean Theorem. It generally frames as

(Hypotenuse) ²= (Opposite side) ² + (Adjacent side) ²

Now that we have learned about the meaning of the pythagorean theorem let us also look at a few examples of thePythagorean Theorem,

Let a = 3 cm, b = 4 cm and c = 5cm.

By Pythagorean Theorem,

Thus 32+ 42= 9 +16 = 25 = 52

In a triangle ABC, AB = 5 cm, BC = 3 cm and AC = 4 cm

Examples 2:

Let a = 2.5 cm, b = 6 cm and c = 6.5cm

By Pythagorean Theorem,

Thus (2.5)2+ 62= 6.25 +36 = (6.5)2

In a triangle ABC, AB = 2.5 cm, BC = 6 cm and AC = 6.5 cm.

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

Trigonometric Ratios

Trigonometric Ratios

Before we learn about the Trigonometric Ratios let us learn about the meaning of Trigonometry in brief.

Trigonometry is a word consisting of three Greek words " Tri" means three, "Gon" means side, and "Metron" means measure. Thus, trigonometry is a study related to the measures of sides and angles of a triangle. Trigonometry is mainly used by captains of ships to find the direction and distance of islands and light houses from sea. Trigonometry is also used in astronomy, geography and engineering.
Trigonometric Functions and ratios can be studied with the help of examples:
The trigonometry radio co-ordinate plane, consider a point A on the +ve side of x-axis. The trigonometric ratios (circular functions) are defined as follows:

*

The sine of the angle [theta] is defined as the ratio r/r it is denoted by sin [theta]
*

sin [theta] =y/r ; cosecant values at [theta] =r/y = cosec [theta] ; y ≠ 0 and cos [theta] =x/r ; secant values at [theta] =r/x = sec [theta] ; x ≠ 0
*

tan [theta] =y/x ; cotangent values at [theta] =x/y = cot [theta] ; y ≠ 0

Let us learn some problems related to Trigonometric Ratios:

In any right-angled triangle ΔABC,That is given above,

let angle B = 90 o and angle C = Θ.

Line segment AC is the hypotenuse.

With reference to angle C, we can say that,

Line segment AB is the opposite side of

Line segment BC is the adjacent side of

Therefore, trigonometric ratios are given as,




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Arithmetic Progressions

Arithmetic Progressions

In mathematics, an Arithmetic Progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. For instance, the sequence 3, 5, 7, 9, 11, 13, … is an arithmetic progression with common difference 2.We usually come across many problems related to arithmetic progression,and we can solve these problems once we learn to identify the sequences.These are basically a set of numbers arranged in a definite order according to some definite rule is called a sequence.A sequence is a function whose domain is the set N of natural numbers.

Indicated sum of the terms in a sequence is called a series.The result of performing the additions is the sum of the series.It is easy to identify the sequence in arithmetic progression as soon as we look at the problem,let us now look at the examples of arithmetic progression.

Quantities are said to be in Arithmetic progression when they increase or decrease by a common difference.
Examples:

Each one of the following series form an Arithmetic progression
i) 1, 3, 5, 7, …

ii) 3, 7, 11, 15, …

iii) 15, 12, 9, …

iv) x, x - d, x - 2d, .....

The common difference is found by subtracting any term of the series from the immediate succeeding term.
In the above example, common difference in the first is 2, in the second it is 4, in the third it is -3, in the fourth it is -d and in the fifth it is d.

The general form of an A.P. is as follows:
a = first term, d = common difference, then A.P. is a, a+d, a+2d, a+3d,.....

Hope you like the above example of Arithmetic progression.
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