Natural Numbers

natural numbers

Numbers which come naturally when we count are called Natural numbers. So, counting numbers are called Natural numbers. e.g., 1, 2, 3…

According to the Set theory, Natural number is defined as the set of positive integers, or simply, the set of numbers starting from 1 N = {1, 2, 3, …}

Predecessor of a Natural Number

↪ Predecessor of any natural number is the number that comes before it.

↪ To get the predecessor of a given natural number, we subtract one from it,

e.g., Predecessor of 2 = 2−1 = 1

Successor of a Natural Number

↪ Successor of any natural number is the number that comes after it.

↪ To get the successor of a given natural number, we add one to it,

e.g., Successor of 2 = 2+1 = 3

⇪ Is there any natural number which has no predecessor?

⇪ Does 1 have both a successor & predecessor?

Successor of 1 =1+1 = 2

Predecessor of 1 = 1-1 = 0   → 0 is not a Natural number

So, 1 has no predecessor in natural numbers.

Is there any natural number which has no successor? OR

Is there a last natural number?

Every natural number has successor as there are infinite natural numbers & hence there is no largest natural number.

Comparing Natural Numbers

↪ The greater the number of digits, the greater is the number.

9567 > 456 > 73 >  5

↪ If two numbers have the same number of digits, the number with the bigger digit on the left hand side is greater.

9342 >  5632, 345 < 672

↪ If the leftmost digits are the same we compare the next digit to the right and keep doing this until the digits are different.

3468 < 3495,

875 > 871,

31 > 30

Forming a Greatest or a Least number from the given digits without repetition

↪ To form greatest number from the given digits without repetition, we arrange the digits in descending order from left to right.

↪ To form least / smallest number from the given digits without repetition, we arrange the digits in ascending order from left to right.

e.g., Let’s form the largest and the smallest number from digits 4, 7, 2, and 9 without repetition.

Arranging digits in ascending order, we get

2479 → smallest possible number formed by using given digits without repetition.

Arranging digits in descending order, we get

9742 → largest possible number formed by using given digits without repetition.

Greatest / Smallest number according to the number of digits

Smallest one digit number = 1

(∵ 1 has no predecessor in natural number)

Largest one digit number = 9

(∵ Successor of 9 is 10 which is a two digit number, ∴ 9 is the largest one digit number)

Smallest two digit number = 10

(∵ Predecessor of 10 is 9 which is a one digit number, ∴ 10 is the smallest two digit number)

Largest two digit number = 99

(∵ Successor of 99 is 100 which is a three digit number, ∴ 99 is the largest two digit number)

Similarly,

Smallest three digit number = 100

Largest three digit number = 999

To obtain smallest n digit number put (n−1) zeroes after 1.

To obtain greatest n digit number write 9s.

In exponential form

Smallest n digit number = 10n-1

Largest n digit number = 9n

Properties of natural numbers

Closure property

When a mathematical operation (+ or − or × or ÷) is performed on any two natural numbers & if the result so obtained is also a natural number then the numbers are said to possess the closure property for that operation. If the result is not a natural number then natural numbers are not closed for that operation.

↬ Addition of two natural numbers always result in a natural number, therefore natural numbers are closed under addition.     e.g.,

1+3 = 5, (Operands(1 & 3) & result (5), all are natural numbers).

10+100 = 110, (same as above).

↬ Multiplication of two natural numbers always result in a natural number, therefore natural numbers are closed under multiplication. e.g.,

1×3 = 3, (Operands(1 & 3) & result (5), all are natural numbers).

10×100 = 1000, (same as above).

↬ Subtraction of two natural numbers not always result in a natural number, therefore natural numbers are not closed under Subtraction e.g.,

3−1 = 2

2−2 = 0,     (0 is not a natural number)

1−3 = ?,     (we can’t subtract larger no. naturally from the smaller no.( −2 is not a natural number)).

↬ Division of two natural numbers not always result in a natural number, therefore natural numbers are not closed under division. e.g.,

3÷1 = 3

2÷2 = 1

5÷3 = ?,  (not fully divisible (5⁄3 is not a natural number)).

1÷0 = ?,  (any number divided by 0 is not defined).

Commutative property :

If the result of a mathematical operation (+ or − or × or ÷)  on any two natural numbers does not change when the order is reversed then the numbers are said to possess the commutative property for that operation. If the result get changed when the order is changed then natural numbers are not commutative for that operation.

↬ Addition of two natural numbers will give same result when the order of numbers is reversed, therefore natural numbers are commutative under addition.     e.g.,

1+3 = 5 ⇔ 3+1 = 5

10+100 = 110 ⇔ 100+10 = 110.

↬ Multiplication of two natural numbers will give same result when the order of numbers is changed, therefore natural numbers are commutative under multiplication. e.g.,

1×3 =3 ⇔ 3×1 = 3

10×100 =1000 ⇔ 100×10 = 1000.

↬ Subtraction of two natural numbers will not give same result if the order is reversed , therefore natural numbers are not commutative under subtraction. e.g.,

3−1 = 2 ⇎ 1−3 = −2, we can’t subtract larger no. naturally from the smaller no.

7−4 = 3 ⇎ 4−7 = −3, we can’t subtract larger no. naturally from the smaller no.

↬Division of two natural numbers will not give same result if the order is reversed , therefore natural numbers are not commutative under division. e.g.,

3÷1 = 3 ⇎ 1÷3 = 1/3

4÷2 = 1 ⇎ 2÷4 = 1/2

Associative property :

If the result of mathematical operation (+ or – or × or ÷)  on any two or more natural numbers does not change when the grouping (how we use parenthesis)  is changed then the numbers are said to possess the associative property for that operation. If the result get changed when the grouping is changed then natural numbers are not associative for that operation.

↬Addition of natural numbers will give same result when the grouping of numbers is changed, therefore natural numbers are associative under addition.     e.g.,

(1+3)+5 =4+5 =9 ⇔ 1+(3+5) =1+8 =9

(10+11)+20 =21+20 =41 ⇔ 10+(11+20) =10+31 =41

↬Multiplication of natural numbers will give same result when the grouping of numbers is changed, therefore natural numbers are associative under multiplication. e.g.,

(1×3)×5 =3×5 =15 ⇔ 1×(3×5) =1×15 =15

(10×11)×5 =110×5 =550 ⇔ 10×(11×5) =10×55 =550

↬Subtraction of natural numbers will not give same result if the grouping is changed , therefore natural numbers are not associative under subtraction. e.g.,

(4−2)−1= 2−1 =1 ⇎ 4−(2−1) =4−1 =3

↬Division of natural numbers will not give same result if the grouping is changed, so natural numbers are not associative under division. e.g.,

(16÷4)÷2 =4÷2 =2 ⇎ 16÷(4÷2) =16÷2 =8

Distributive Property/ Distribution of multiplication over addition :

Multiplying a number with sum of two numbers will give same result if the number is first multiplied with each addend & products then added after.

2×(4+5) = 2×9 = 18

2×(4+5) =(2×4) + (2×5 )=8+10 =18 (using Distributive property)

↪ Can be applied for solving lengthy or harder products (Simplifying)

12×35 =12×(30+5) =360+60 =420

126×55+126×45 =126×(55+45) =126×100 =12600.

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