Natural Numbers | Whole Numbers:
- The numbers 1, 2, 3,... which we use for counting are known as natural numbers.
- If we add 1 to a natural number, we get its successor. If you subtract 1 from a natural number, you get its predecessor.
- Every natural number has a successor. Every natural number except 1 has a predecessor.
- The natural numbers along with zero form the collection of whole numbers.
- Every whole number has a successor. Every whole number except zero has a predecessor.
- All natural numbers are whole numbers, but all whole numbers are not natural numbers.
Properties of Whole Numbers:
Closure property :
Whole numbers are closed under addition and also under multiplication. i.e. We can say that sum of any two whole numbers is a whole number. Also, product of any two whole numbers is a whole number.
E.g. 2 + 3 = 5 , a whole number.
2 * 3 = 6, a whole number.
Notes:
1. Whole numbers are not closed under division. E.g. Division of whole number 5 by 7 is not a whole number.
2. Division of a whole number by 0 is not defined.
3. Whole numbers are not closed under subtraction. E.g. 7-8 i.e. -1 is not a whole number.
Commutativity of addition and multiplication:
We can add two whole numbers in any order. We say that addition is commutative for whole numbers. This property is known as commutativity for addition..
E.g. 4+ 6 = 10 = 6+4.
Similarly, You can multiply two whole numbers in any order. E.g. 4*3=12=3*4
Thus, addition and multiplication are commutative for whole numbers.
Notes:
1. Subtraction is not commutative for whole numbers.
2. Division is not commutative for whole numbers.
Associativity of addition and multiplication:
Addition and multiplication, both, are associative for whole numbers.
E.g. (2 + 3) + 4 = 5 + 4 = 9. Also, 2 + (3 + 4) = 2 + 7 = 9
Similarly, (5 × 6) × 2 is same as 5 × (6 × 2).
The associative property of multiplication is very useful in the calculations.
Distributivity of multiplication over addition:
Multiplication is distributive over addition for whole numbers.
E.g. 2 × (3 + 5) = (2 × 3) + (2 × 5)
Identity (for addition and multiplication):
Zero is called an identity for addition of whole numbers or additive identity for whole numbers. i.e. Addition of zero to any whole number results in same whole number again.
E.g. 0+3 =3, 5+0=5, etc.
Zero has a special role in multiplication too. Any number when multiplied by zero becomes zero!
E.g. 5 *0 =0; 4*0=0, etc.
1 is the identity for multiplication of whole numbers or multiplicative identity for whole numbers. i.e Product of any whole number and 1 is always the same whole number again.
E.g. 5*1=5, 8*1=8, etc.
Integers:
Integers form a bigger collection of numbers which contains whole numbers and negative numbers.
The collection of numbers..., – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, ... is called integers. So, – 1, – 2, – 3, – 4, ... called negative numbers are negative integers and 1, 2, 3, 4, ... called positive numbers are the positive integers.
Notes:
- Every positive integer is larger than every negative integer.
- Zero is less than every positive integer.
- Zero is larger than every negative integer.
- Zero is neither a negative integer nor a positive integer.
- Farther a number from zero on the right, larger is its value.
- Farther a number from zero on the left, smaller is its value.
Rules for Addition of Integers:
1. When we have the same sign, add and put the same sign.
(i) When two positive integers are added, we get a positive integer. [e.g. (+ 3) + ( + 2) = + 5].
(ii) When two negative integers are added, we get a negative integer. [e.g. (–2) + ( – 1) = – 3].
2. When one positive and one negative integers are added, we subtract them as whole numbers by considering the numbers without their sign and then put the sign of the bigger number with the subtraction obtained. The bigger integer is decided by ignoring the signs of the integers
e.g. (+4) + (–3) = + 1 and (–4) + ( + 3) = – 1.
Numbers such as 3 and – 3, 2 and – 2, when added to each other give the sum zero. They are called additive inverse of each other.
On a number line when we,
(i) add a positive integer, we move to the right.
(ii) add a negative integer, we move to the left.
(iii) subtract a positive integer, we move to the left.
(iv) subtract a negative integer, we move to the right.
Notes:
1. When two positive integers are added we get a positive integer.
2. When two negative integers are added we get a negative integer.
3. The additive inverse of any integer a is – a and additive inverse of (– a) is a.
4. Subtraction is opposite of addition.
5. For any two integers a and b,
a – b = a + additive inverse of b = a + (– b)
a – (– b) = a + additive inverse of (– b) = a + b
Rules for Subtraction of Integers:
The subtraction of an integer is the same as the addition of its additive inverse. Hence, to subtract an integer from another integer it is enough to add the additive inverse of the integer that is being subtracted, to the other integer.
E.g. -8 - (-10)= ?
Sol: – 8 – (– 10) is equal to – 8 + 10 as additive inverse of –10 is 10.
Properties of Addition and Subtraction of Integers:
Closure under Addition:
Since addition of integers gives integers, we say integers are closed under addition. In general, for any two integers a and b, a + b is an integer.
E.g. (i) 17 + 23 = 40, is an integer. (ii) 19 + (-25) = -6, is an integer.
Closure under Subtraction:
If a and b are two integers then a – b is also an integer. So, integers are closed under subtraction, unlike whole numbers.
E.g. 7 - (-9) = -2 , is an integer.
Commutative Property:
Addition is commutative for integers. In general, for any two integers a and b, we can say a + b = b + a. E.g. (– 8) + (– 9) and (– 9) + (– 8) are equal.
However, subtraction is not commutative for integers.
Associative Property:
Addition is associative for integers. In general for any integers a, b and c, we can say a + (b + c) = (a + b) + c.
Additive Identity:
When we add zero to any integer , we get the same integer. Thus, Zero is an additive identity for integer.
In general, for any integer a, a + 0 = a = 0 + a.
MULTIPLICATION OF INTEGERS:
Rules:
1. Product of a positive and a negative integer is a negative integer, whereas the product of two negative integers is a positive integer. For example, – 2 × 7 = – 14 and – 3 × – 8 = 24.
In general, for any two positive integers a and b, (– a) × (– b) = a × b
2. Product of even number of negative integers is positive, whereas the product of odd number of negative integers is negative.
Properties of Multiplication Of Integers:
1. Integers are closed under multiplication. That is, a × b is an integer for any two integers a and b.
2. Multiplication is commutative for integers. That is, a × b = b × a for any integers a and b.
3. The integer 1 is the identity under multiplication, i.e., 1 × a = a × 1 = a for any integer a.
4. Multiplication is associative for integers, i.e., (a × b) × c = a × (b × c) for any three integers a, b and c. i.e. the product of three integers does not depend upon the grouping of integers.
5. 1 is the multiplicative identity for integers. In general, for any integer a we have, a × 1 = 1 × a = a
6. Like whole numbers, distributivity of multiplication over addition is true for integers also. In general, for any integers a, b and c, a × (b + c) = a × b + a × c
DIVISION OF INTEGERS:
Division is the inverse operation of multiplication.
E.g. 3 × 5 = 15 That implies 15 ÷ 5 = 3 and 15 ÷ 3 = 5.
So, We can say for each multiplication statement of whole numbers there are two division statements.
Rules:
1. When we divide a negative integer by a positive integer, we divide them as whole numbers and then put a minus sign (–) before the quotient. We, thus, get a negative integer.
E.g. (–12) ÷ 2 = (– 6)
2. When we divide a positive integer by a negative integer, we first divide them as whole numbers and then put a minus sign (–) before the quotient. That is, we get a negative integer. In general, for any two positive integers a and b, a ÷ (– b) = (– a) ÷ b where b ≠ 0.
E.g. 72 ÷ (–8) = –9
3. When we divide a negative integer by a negative integer, we first divide them as whole numbers and then put a positive sign (+). That is, we get a positive integer. In general, for any two positive integers a and b , (– a) ÷ (– b) = a ÷ b , where b ≠ 0.
E.g. (–12) ÷ (– 6) = 2
PROPERTIES OF DIVISION OF INTEGERS:
1. Integers are not closed under division. E.g. 3 divided by 8 results in 3/8.
2. Division is not commutative for integers.
3. Like whole numbers, any integer divided by zero is meaningless and zero divided by an integer other than zero is equal to zero i.e., for any integer a, a ÷ 0 is not defined but 0÷ a = 0 for a ≠ 0.
4. When we divide a whole number by 1 it gives the same whole number. Also, negative integer divided by 1 gives the same negative integer. So, any integer divided by 1 gives the same integer.
In general, for any integer a, a ÷ 1 = a.
5. Division is not associative for integers.
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