1. ## Whole Numbers

When we think about objects, we come across whole objects and we count it as whole. From here arises the notion of whole numbers. The first ten whole numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
Now we can count a group of objects and add them.
1. 1+2 = 3
2. 3+5 = 8
We can also subtract a group of objects from other group.
1. 2-1 = 1
2. 5-3 = 2
But when we do these subtractions we face difficulty.
1. 1-1 = ?
2. 3-3 = ?
To overcome this we add one more number to the group of whole numbers and it is 0.

2. ## Natural Numbers

When we deal with whole numbers and do the operation of subtraction we face difficulty when both the numbers of the operation are the same. To overcome this difficulty we added the number zero to the set of whole numbers. The group or the set is now called as natural numbers. The first ten natural numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Now the problem with subtraction of same two numbers is solved
1. 1-1 = 0
2. 3-3 = 0
We can add the numbers easily but face a different kind of difficulty when we subtract a larger number from a smaller number.
1. 1 - 2 = ?
2. 3 - 5 = ?
To overcome this we add more numbers to the group of natural numbers and call the whole set as integers.

3. ## Integers

When we deal with natural numbers and do the operation of subtraction we face difficulty when both the number to be subtracted is greater than the number from which it is to be subtracted. To overcome this difficulty we add a set of negative numbers to the set of natural numbers. The group or the set is now called as Integers. The first ten integers cannot be written and neither the last ten integers. we can write the numbers around zero as it extends to both sides of zero on the number line. Integers are ...,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,...
Now the problem is solved with subtraction
1. 1 - 2 = -1
2. 3 - 5 = -2
We can add and subtract the numbers easily but face a different kind of difficulty when we divide a number from another number.
1. 1/2 = ?
2. 3/5 = ?
To overcome this we add more numbers to the group of integers and call the whole set as the set of rational numbers.

4. ## Rational Numbers

A number which can be expressed in the form a/b is called a rational number. The numbers a and b must be integers(Z) and b is not equal to zero.
Now the problem is solved with division
1. 1/2 = .5
2. 3/5 = .6

### In terms of set theory

A set of rational numbers Q consists of
Q = {a/b: a∈Z, b∈Z and b≠0}

### Examples of rational Numbers

1/2, -2/3 , 7/14, 14/7,etc.
Every integer (Z) is a rational number.

### Properties of rational numbers

1. Q is closed under addition. By the term closed we mean 'When we add two rational numbers we get a rational number.'
• a/b + c/b = (a + c)/b
1/2 + 2/2 = (1+2)/2 = 3/2

• a/b + c/d = (ad + bc)/bd
1/2 + 3/4 = (1×4 + 3×2)/(2×4) = (4+6)/8 = 10/8

2. Q is closed under subtraction.
• a/b − c/b = (a − c)/b
1/2 − 2/2 = (1−2)/2 = −1/2

• a/b − c/d = (ad − bc)/bd
1/2 − 3/4 = (1×4 − 3×2)/(2×4) = (4−6)/8 = −2/8

3. Q is closed under multiplication.
• a/b × c/b = (a × c)/(b×b)
1/2 × 2/2 = (1×2)/(2×2) = 2/4 = 1/2

• a/b × c/d = ac/bd
1/2 × 3/4 = (1×3)/(2×4) = 3/8

4. Q is closed under division.
• a/b ÷ c/b = a/b × b/c = (a × b)/(b × c) = a/c
c/b ≠0
1/2 ÷ 2/2 = (1×2)/(2×2) = 2/4 = 1/2

• a/b ÷ c/d = a/b × d/c = (a×d)/(b×c)
c/d ≠ 0
1/2 ÷ 3/4 = (1×4)/(2×3) = 4/6 = 2/3

• In division, the second rational number must not be zero.

5. There exists a rational number between any two rational numbers. As we know integers are rational numbers and between two integers there is a rational number.

• Let a and b be rational numbers then (a+b)/2 is also a rational number.

• Let a/b and c/d be rational numbers then (a/b + c/d)/2 is also a rational number.

5. ## Irrational Numbers

A number which cannot be expressed in the form of a rational number is called an irrational number. An irrational number is a decimal that is neither repeating nor terminating.

### In terms of set theory

A set of rational numbers R−Q consists of
R−Q = {c ≠ a/b: a∈Z, b∈Z and b≠0}
R is the set of real numbers.

π,√2,√3,etc

### Rationalization

Rationalization is a process by which a fraction with an irrational denominator is converted into a fraction with rational denominator. An irrational denominator can be converted into a rational denominator with the help of a Rationalizing factor. The rationalizing factor has its product with the denominator a rational number.

1/√2 = (1/√2)×(√2/√2) = √2/2 here √2 is a rationalizing factor.

6. Real Number ⇒ A union set of rational and irrational numbers is called a set of real numbers.

• 0 (ZERO) ⇒ is called the additive identity. When this number is added to any number the number remains the same. Or in other words we obtain the original number.
1. 13+0 = 13
2. 23+0 = 23

⇒ gives the result zero when any number is multiplied to it.
1. 13×0 = 0
2. 23×0 = 0

• 1 (One) ⇒ is called the multiplicative identity. when this number is multiplied to any number the number remains the same. Or in other words we obtain the original number.
1. 13×1 = 13
2. 23×1 = 23

⇒increases the result when added to a number by unit.
1. 13+1 = 14
2. 23+1 = 24

• 2 (Two) ⇒ is the first even number.
⇒ The fraction of 2 is 1/2 and equals .5.
⇒ The square root of 2 is irrational.

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Aug 1, 2017, 7:14 PM