Numbers

Number

A number is an arithmetic value which we use in our daily life for counting and measuring things. Like- as you know there are 7 days in a week or there are 1,000 milli-litres in 1 litre. In these two examples, the arithmetic value 7 denotes counting of days and 1 denotes counting of the week. Similarly, 1,000 and 1 denote the measurement of any liquid.

Number system

Any of numerous sets of symbols and the rules for using them to denote numbers, which are used to state how many objects are there in a given set. Thus, the idea of “oneness” can be denoted by the Roman numeral I, by the Greek letter alpha α which was the first letter used as a numeral, by the Hebrew letter aleph which is the first letter used as a numeral, or by the modern number 1, which is nothing but a Hindu-Arabic in origin.

A concise treatment of these numeral systems follows. For further discussion, let us understand number systems: Number systems.

Type Of Numbers

Natural Numbers (N)
Whole Numbers (W)
Integers (Z)
Fraction
Rational numbers (Q)
Irrational Numbers
Real numbers (R)
Imaginary

Natural Numbers (N)

The counting numbers are called Natural numbers. Example: 1,2,3,4… and so on. The Natural numbers contain all positive numbers starting from 1 to ∞ (infinity).

The set of natural numbers are denoted by the capital letter “N” and are represented as N= {1, 2, 3, …..}.

Sometimes we are asked, if 0 (zero) a natural number or not?

The answer to this question is NO, because Natural numbers start from 1 and go up to infinity. They are all positive numbers. In fact, 0(zero) is a Whole Number.

Number line representation of Natural numbers.

Thus, Natural numbers are set of all the whole numbers except 0.

2. Whole Numbers ( W )

Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, ... (and so on)

No Fractions!

3. Integers

A number with no fractional part (no decimals).

Includes:

• the counting numbers {1, 2, 3, ...},

• zero {0},

• and the negative of the counting numbers {-1, -2, -3, ...}

We can write them all down like this: {..., -3, -2, -1, 0, 1, 2, 3, ...}

The integers are the natural numbers consisting of the set of real numbers, their additive inverses and zero.

{..., -9, -8, -7, -6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...}

4. Fraction

Fractions represent equal parts of a whole or a collection.

Fraction of a whole: When we divide a whole into equal parts, each part is a fraction of the whole.

For example,

5. Rational Number

A number that can be made by dividing two integers (an integer is a number with no fractional part).

The word comes from "ratio".

Examples:

• 1/2 is a rational number (1 divided by 2, or the ratio of 1 to 2)

• 0.75 is a rational number (3/4)

• 1 is a rational number (1/1)

• 2 is a rational number (2/1)

• 2.12 is a rational number (212/100)

• −6.6 is a rational number (−66/10)

• etc

But π (pi) is not a rational number, it is an "Irrational Number"

6. Irrational Numbers

An irrational number is those number that which cannot be written as a proportion (or fraction). The irrational number does not end or neither repeats in a decimal form. The ancient Greek mathematician discovered that not all integers are the rational number; there are equations that cannot be solved using ratios of integers.

The first step was to study the equation 2=x2. What number times itself equals to 2?

The square root of 2 is about 1.414, because 1.4142=1.999396, which is almost equal to 2.you will never get the exact number by squaring the fraction (or terminating decimal numbers). The square root of 2 is an irrational number, meaning it is decimal equivalent goes on forever, with no repetitious pattern:F

The golden ratio is another famous irrational number, a number which as great importance in biology:

1+5√2=1.61803398874989...

π(pi), the proportion of the perimeter of a circle to its diameter:

π=3.14159265358979...

and e, the most significant number in calculus:

e=2.71828182845904...

7. Real numbers

The real numbers are the set of numbers containing all the irrational numbers and all the rational numbers. The real numbers consist of “all the numbers” on the number scale. The real numbers are infinite numbers just as there are infinitely many numbers in each of the other sets of numbers. But, it can be explained that the infinity of the real numbers is a bigger infinity.

The "smaller numbers ", or countable infinity numbers of the integers and rational numbers are sometimes called ℵ0(alef-naught), and the uncountable infinity numbers of the reals are called ℵ1(alef-one).

There are even "bigger" infinities, but one should know to take a set theory class for that specific set.

8. Imaginary

The complex numbers are the set {a+bi | the real numbers are a and b}, where the imaginary unit is

i, −1−√-.

The complex numbers include the set of real numbers, that is, which includes the set of both rational and irrational numbers. The real numbers, in the complex system, are denoted in the form a+0i=a. a real number.

This set is always denoted by C in the short form. The set of complex numbers is significant because for any polynomial p(x) with real number coefficients, all the solutions of p(x)=0 will be in C.

Properties of Natural numbers

While performing various operations on numbers, we notice several properties of Natural numbers. These properties help us to understand the numbers better and they make calculations under certain operations very simple.

Properties of Natural numbers are classified into four categories which are as below:-

Closure property
Commutative property
Associative property
Distributive property

1. Closure property

The closure property of natural numbers says that Natural numbers are always closure under the addition and multiplication, meaning that, addition and multiplication of natural numbers will always result in a natural number.

While, they are not closure under subtraction and division, meaning that, subtraction and division of natural numbers may or may not always result in a natural number.

Lets verify the above statements by taking some example:

Addition: 5 + 2=7 ; 14 + 6=20. Here, the addition of two natural numbers result in a natural number that is 7 and 20 respectively.

Subtraction: 6 - 3 = 3; 5 - 5 = 0, Here, the subtraction in the first operation results in 3, which is a natural number, while, the second operation results in 0 which is not a natural number.

Multiplication: 2 x 3=6; 10 x 3=30. Here, the results of multiplication are 6 and 30 respectively which is a natural number.

Division: 9 ÷ 3=3; 3 ÷ 2=1.5; Clearly here also you can see natural numbers may or may not follow closure property for the division.

2. Associative property

The natural numbers hold associative property in case of addition and multiplication. i.e. a + ( b + c ) = ( a + b ) + c and a × ( b × c ) = ( a × b ) × c. While they do not have associative property for subtraction and division of natural numbers.

Examples for associative property of natural numbers:

Addition: a + ( b + c ) = ( a + b ) + c ; (5 + 7) + 3 = 12 + 3 = 15 and 5 + (7 + 3) = 5 + 10 = 15 So, (5 + 7) + 3 is equal to 5 + (7 + 3).

Subtraction: a – ( b – c ) ≠ ( a – b ) – c ; 2 - ( 5 – 4 ) = 1 and (2 – 5) – 4 = -7. So, 2 - ( 5 – 4 ) is not equal to (2 – 5) – 4.

Multiplication : a × ( b × c ) = ( a × b ) × c ; 2 × (3 × 4) = 24 and (2 × 3 ) × 4 = 24. So (2 × 3) × 4 is equal to 2 × (3 × 4).

Division : a ÷ ( b ÷ c ) ≠ ( a ÷ b ) ÷ c ; 16 ÷ (4 ÷2) = 8 and (16 ÷4) ÷2 = 2. So 16 ÷ (4 ÷2) is not equal to (16 ÷4) ÷2 = 2.

3. Commutative Property

The natural numbers show the commutative property for addition and multiplication. i.e. a+ b = b + a and a × b = b × a. for example: (a) 5+2=7 and 2+5=7 (b) 12 x 3 =36 and 3 x 12 =36.

While subtraction and division of natural numbers does not show the commutative property. i.e. a – b ≠ b – a and a ÷ b ≠ b ÷ a.

4 Distributive property

Multiplication of natural numbers is always distributive over addition.

i.e. a × (b + c) = ab + ac

for example 5 x (3 +2 )= 5 x5 =25 and (5 x 3) + (5 x 2) = 15+10 = 25.

Multiplication of natural numbers is also distributive over subtraction.

i.e. a × (b – c) = ab – ac

for example 5 x (3-2) = 5 x1 = 5 and (5 x 3) –(5 x 2) = 5.

Solved Examples

Q1. What is the smallest natural number?

Ans: 1 is the smallest natural number.

Q2. Is 0 a natural number?

Ans: No, because natural numbers start from 1.

Q3. Give examples of the first 10 natural numbers.

Ans: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

Q4. Identify natural numbers from given numbers: 13, 01, 10, 0, 5, 7.9, -4, 22.

Ans: Natural numbers are: 13, 01, 10, 5, 22.

Q5. Are all natural numbers a whole number?

Ans: Yes, because the whole number starts at 0 and goes up to infinity. That's why whole numbers are a combination of zero and natural numbers. So, all the natural numbers are the whole number.

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