7th. integers

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1. Natural numbers

Natural numbers are counting numbers but these set of numbers do not include zero. This is because you cannot count zero. So numbers,
1,2,3,4,5,6……etc are all natural numbers

ABCD

2. Whole numbers

All natural numbers along with zero are called whole number. For example
0, 1, 2, 3, 4, 5, 6………etc are all whole numbers.
NOTE:- Thes types of numbers do not include fractions.
From the definition of natural numbers we can conclude that every natural or counting number is a whole number.

3. Integers

Integers include all natural numbers, zero and negative numbers for example,
-4, -3, -2, -1, 0, 1, 2, 3, ………. etc are all integers.
So now we have,

1. Positive integers:- 1, 2, 3, ……
2. Negative integers:- -1, -2, -3, …….
3. 0 (zero):- which is an integer that is neither negative nor positive.

Important Points
On the number line when we

1. Add a positive integer, we move to the right.
2. Add a negative integer we move to the left.
3. Subtract a positive integer we move to the left.
4. Subtract a negative integer, we move to the right.

Properties of integers

• 1. Closure under Addition

  For the closure property the sum of two integers must be an integer then it will be closed under addition.

  Example

  2 + 3 = 5

  2+ (-3) = -1

  (-2) + 3 = 1

  (-2) + (-3) = -5

  As you can see that the addition of two integers will always be an integer, hence integers are closed under addition.

  If we have two integers p and q, p + q is an integer.

  2. Closure under Subtraction

  If the difference between two integers is also an integer then it is said to be closed under subtraction.

  Example

  7 – 2 = 5

  7 – (- 2) = 9

  - 7 – 2 = – 9

  - 7 – (- 2) = – 5

  As you can see that the subtraction of two integers will always be an integer, hence integers are closed under subtraction.

  For any two integers p and q, p - q is an integer.

  3. Commutative Property

  a. If we change the order of the integers while adding then also the result is the same then it is said that addition is commutative for integers.

  For any two integers p and q

  p + q = q + p

  Example

  23 + (-30) = – 7

  (-30) + 23 = – 7

  There is no difference in answer after changing the order of the numbers.

  b. If we change the order of the integers while subtracting then the result is not the same so subtraction is not commutative for integers.

  For any two integers p and q

  p – q ≠ q – p will not always equal. 

  Example

   23 - (-30) = 53

  (-30) - 23 = -53

  The answer is different after changing the order of the numbers.

  4. Associative Property

  If we change the grouping of the integers while adding in case of more than two integers and the result is same then we will call it that addition is associative for integers.

  For any three integers, p, q and r

  p + (q + r) = (p + q) + r

  Example

  If there are three integers 3, 4 and 1 and we change the grouping of numbers, then

  

  The result remains the same. Hence, addition is associative for integers.

  5. Additive Identity

  If we add zero to an integer, we get the same integer as the answer. So zero is an additive identity for integers.

  For any integer p,

  p + 0 = 0 + p =p

  Example

  2 + 0 = 2

  (-7) + 0 = (-7)

  Multiplication of Integers

  Multiplication of two integers is the repeated addition.

  Example

  • 3 × (-2) = three times (-2) = (-2) + (-2) + (-2) = – 6

  • 3 × 2 =  three times 2 = 2 + 2 + 2 = 6

  

  Now let’s see how to do the multiplication of integers without the number line.

  1. Multiplication of a Positive Integer and a Negative Integer

  To multiply a positive integer with a negative integer, we can multiply them as a whole number and then put the negative sign before their product.

  So the product of a negative and a positive integer will always be a negative integer.

  For two integers p and q, 

  p × (-q) = (-p) × q = - (p × q) = - pq

  Example

  4 × (-10) = (- 4) × 10 = - (4 × 10) = - 40

  

  2. Multiplication of Two Negative Integers

  To multiply two negative integers, we can multiply them as a whole number and then put the positive sign before their product.

  Hence, if we multiply two negative integers then the result will always be a positive integer.

  For two integers p and q,

  (-p) × (-q) = (-p) × (-q) = p × q

  Example

  (-10) × (-3) = 30

  3. The Product of Three or More Negative Integers

  It depends upon the number of negative integers.

  a. If we multiply two negative integers then their product will be positive integer

  (-3) × (-7) = 21

  b. If we multiply three negative integers then their product will be negative integer

  (-3) × (-7) × (-10) = -210

  If we multiply four negative integers then their product will be positive integer

  (-3) × (-7) × (-10) × (-2) = 420

  Hence, if the number of negative integers is even then the result will be a positive integer and if the number of negative integers is odd then the result will be a negative integer.

  Properties of Multiplication of Integers

  1. Closure under Multiplication

  In case of multiplication, the product of two integers is always integer so integers are closed under multiplication.

  For all the integers p and q

  p×q = r, where r is an integer

  Example

  (-10) × (-3) = 30

  (12) × (-4) = -48

  2. Commutativity of Multiplication

  If we change the order of the integers while multiplying then also the result will remain the same then it is said that multiplication is commutative for integers.

  For any two integers p and q

  p × q = q × p

  Example

  20 × (-30) = – 600

  (-30) × 20 = – 600

  There is no difference in answer after changing the order of the numbers.

  3. Multiplication by Zero

  If we multiply an integer with zero then the result will always be zero.

  For any integer p,

  p × 0 = 0 × p = 0

  Example

  9 × 0 = 0 × 9 = 0

  0 × (-15) = 0

  4. Multiplicative Identity

  If we multiply an integer with 1 then the result will always the same as the integer.

  For any integer q

  q × 1 = 1 × q = q

  Example

  21 × 1 = 1 × 21 = 21

  1 × (-15) = (-15)

  5. Associative Property

  If we change the grouping of the integers while multiplying in case of more than two integers and the result remains the same then it is said the associative property for multiplication of integers.

  For any three integers, p, q and r

  p × (q × r) = (p × q) × r

  Example

  If there are three integers 2, 3 and 4 and we change the grouping of numbers, then

  

  The result remains the same. Hence, multiplication is associative for integers.

  6. Distributive Property

  a. Distributivity of Multiplication over Addition.

  For any integers a, b and c

  a × (b + c) = (a × b) + (a × c)

  

  Example

  Solve the following by distributive property.

  I. 35 × (10 + 2) = 35 × 10 + 35 × 2

  = 350 + 70

  = 420

  II. (– 4) × [(–2) + 7] = (– 4) × 5 = – 20 And

  = [(– 4) × (–2)] + [(– 4) × 7]

  = 8 + (–28)

  = –20

  So, (– 4) × [(–2) + 7] = [(– 4) × (–2)] + [(– 4) × 7]

  b. Distributivity of multiplication over subtraction

  For any integers a, b and c

  a × (b – c) = (a × b) – (a × c)                      

  Example

  5 × (3 – 8) = 5 × (- 5) = – 25

  5 × 3 – 5 × 8 = 15 – 40 = – 25

  So, 4 × (3 – 8) = 4 × 3 – 4 × 8.

  Division of integers

  1. Division of a Negative Integer by a Positive Integer

  The division is the inverse of multiplication. So, like multiplication, we can divide them as a whole number and then place a negative sign prior to the result. Hence the answer will be in the form of a negative integer.

  For any integers p and q,

  ( – p) ÷ q = p ÷ (- q) = - (p ÷ q) where, q ≠ 0

  Example

  64 ÷ (- 8) = – 8

  2. Division of Two Negative Integers

  To divide two negative integers, we can divide them as a whole number and then put the positive sign before the result.

  The division of two negative integers will always be a positive integer.

  For two integers p and q,

  (- p) ÷ (- q) = (-p) ÷ (- q) = p ÷ q where q ≠ 0

  Example

  (-10) ÷ (- 2) = 5

  Properties of Division of Integers

  For any integers p, q and r

  Property	General form	Example	Conclusion
  Closure Property	p ÷ q is not always an integer	10 ÷ 5 = 2 5 ÷ 10 = 1/2 (not an integer)	The division is not closed under division.
  Commutative Property	p ÷ q ≠ q ÷ p	10 ÷ 5 = 2 5 ÷ 10 = 1/2	The division is not commutative for integer.
  Division by Zero	p ÷ 0 = not defined 0 ÷ p = 0	0 ÷ 10 = 0	No
  Division Identity	p ÷ 1 = p	10 ÷ 1 = 10	Yes
  Associative Property	(p ÷ q) ÷ r ≠ p ÷ (q ÷ r)	[(–16) ÷ 4] ÷ (–2) ≠ (–16) ÷ [4 ÷ (–2)] (-8) ÷ (-2) ≠ (-16) ÷ (-2) 4 ≠ 8	Division is not Associative for integers.

   Exercise 1

  Question 1.
  Some integers are marked on the number line. Which is the biggest and which is the
  smallest?
  
  Solution:
  The numbers marked = -3 and 2
  biggest number = 2
  smallest number = -3

  Question 2.
  Write the integers between the pairs of integers given below. Also, choose the biggest and
  smallest intergers from them.
  (i) – 5, – 10
  (ii) 3, – 2
  (iii) – 8, 5
  Solution:
  i) – 5, – 10
  Integers between – 5, – 10 are – 6, – 7, – 8, – 9
  smallest Integer – 9
  biggest integer = – 6

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