Integer

[Audio] Hello students! Welcome back to another video of Point of Indian Study YouTube channel. Today we are going to study about integers for class 7..

[Audio] Introduction of integers, Positive and negative numbers are called integers. For example -3, 2 et cetera.

[Audio] Whole numbers Whole numbers include zero and all natural numbers, in other words, 0, 1, 2, 3, 4, and so on..

[Audio] Negative Numbers The numbers with a negative sign and which lies to the left of zero on the number line are called negative numbers..

[Audio] The Number Zero The number zero means an absence of value..

[Audio] Integers Collection of all positive and negative numbers including zero are called integers. ⇒ Numbers …, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, … are integers..

[Audio] Absolute Value of an Integer Absolute value of an integer is the numerical value of the integer without considering its sign. Example: Absolute value of -7 is 7 and of plus 7 is 7..

[Audio] Ordering Integers On a number line, the number increases as we move towards right and decreases as we move towards left. Hence, the order of integers is written as…, –5, –4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5… Therefore, – 3 < – 2, – 2 < – 1, – 1 < 0, 0 < 1, 1 < 2 and 2 < 3..

[Audio] The Additive Inverse of an Integer The negative of any number is the additive inverse of that number. The additive inverse of 5 is ( 5) and additive inverse of ( 5) is 5..

[Audio] Properties of Addition and Subtraction 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 plus 3 = 5 2 plus (-3) = -1 (-2) plus 3 = 1 (-2) plus (-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 plus q is an integer.

[Audio] 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.

[Audio] 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 plus q = q plus p Example 23 plus (-30) = – 7 (-30) plus 23 = – 7 There is no difference in answer after changing the order of the numbers..

[Audio] 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..

[Audio] 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 plus (q plus r) = (p plus q) plus 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..

[Audio] 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 plus 0 = 0 plus p =p Example 2 plus 0 = 2 (-7) plus 0 = (-7).

[Audio] Multiplication of Integers Multiplication of two integers is the repeated addition. Example 3 × (-2) = three times (-2) = (-2) plus (-2) plus (-2) = – 6 3 × 2 = three times 2 = 2 plus 2 plus 2 = 6.

[Audio] 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.

[Audio] 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..

[Audio] 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.

[Audio] 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).

[Audio] 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..

[Audio] 6. Distributive Property Distributivity of Multiplication over Addition For any integers a, b and c a × (b plus c) = (a × b) plus (a × c) Example Solve the following by distributive property. I 35 × (10 plus 2) = 35 × 10 plus 35 × 2 = 350 plus 70 = 420 II. (– 4) × [(–2) plus 7] = (– 4) × 5 = – 20 And = [(– 4) × (–2)] plus [(– 4) × 7] = 8 plus (–28) = –20 So, (– 4) × [(–2) plus 7] = [(– 4) × (–2)] plus [(– 4) × 7].

[Audio] 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..

[Audio] 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.

[Audio] 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.

[Audio] Properties of Division of Integers. See the table.

[Audio] For worksheet you can check description box..

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