Chapter Notes and Summary
• Integers : collection of all natural numbers, 0 and negatives of natural numbers are called integers, i.e., we can represent integers on number line.
Thus integers range from {………… – 5, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5,…………}
• Absolute value of an integer : absolute value of an integer is numerical value of integer regardless of its sign.
• Successor and predecessor of an integer : Let a be an integer, then :
(a + 1) is called successor of a.
(a – 1) is called predecessor of a.
• Properties of addition on integers :
(i) Closure property of addition : sum of two integers is always an integer.
(ii) Commutative law of addition : a + b = b + a.
(iii) Associative law of addition : (a + b) + c = a + (b + c).
• Properties of subtraction of integers :
(i) Closure property : If a and b are two integers, then (a – b) is also an integer.
(ii) If a is any integer, then a – 0 = a.
(iii) If a, b, c are integers and a > b, then (a – c) > (b – c).
• Properties of multiplication of integers :
(i) Closure property : product of two integers is always an integer.
(ii) Commutative law : a × b = b × a.
(iii) Associative law : a × (b × c) = (a × b) × c.
(iv) Distributive law : a × (b + c) = a × b + a × c.
• Properties of division on integers :
(i) If a and b are two integers, then a ÷ b is not necessarily an integer.
(ii) If a ≠ 0, then a ÷ a = 1.
(iii) a ÷ 1 = a.
(iv) If a is non-zero integer, then 0 ÷a = 0, but a ÷ 0 is not meaningful.
(v) (a ÷ b) ÷ c ≠ a ÷ (b ÷ c) unless c = 1.
• Representation of Integer on Number line To represent integers on a number line, draw a line and mark some points at equal distances on it as shown in figure given below. Mark a point as ‘zero‘ on it. Points to right of zero are positive (+ve) integers and are marked as +1, +, 2, +3 etc. or simply 1, 2, 3 etc. Points to left of zero are negative (–ve) integers and are marked as –1, – 2, –3 etc.
• Ordering of Integers On number line, number increases as we move to right and decreases as we move to left.
Therefore, ……….. < –5 < – 4 < – 3 < – 2 < – 1 < 0 < 1 < 2 < 3 < 4 < 5 and so on.
• Addition of Integers To add two negative integers, we add corresponding positive integers and retain negative sign with sum.
e.g., find sum of –5 and –3.
–5 + (–3) = – (5 + 3) = – 8.
To add a positive integer and a negative integer, we ignore signs and subtract integer with smaller numerical value from integer with larger numerical value and take sign of larger one.
e.g., (a) Consider –6 and + 4 As 6 – 4 = 2, therefore –6 + (+4) = – 2
(b) Consider + 5 and – 2 5 + (–2) = 5 – 2 = 3 Two integers whose sum is zero are called additive inverse of each other.
To subtract an integer from a given integer, we add additive inverse of integer to given integer e.g., Subtract :
(a) 3 from – 4 (b) –3 from – 4
(a) additive inverse of 3 is –3 So, – 4 – 3 = – 4 + (–3) = – (4 + 3) = – 7
(b) additive inverse of –3 is 3 So, –4 – (–3) = – 4 + 3 = – 1
• Addition/Subtraction of Integers on Number Line Firstly, draw number line and represent first number on it. then, to add/subtract second number in first, we move left/right to first number according to second integer (either –ve or + ve) e.g., (i) To add –2 and –3, we move two steps to left of reaching –2. Then, we move three steps to left of –2 and reach –5.
Thus, –2 + (–3) = – 5
(ii) To subtract 2 from –3, we first move three steps to left of 0, reaching –3. Then we move 2 steps to left of 3 and reach –5.
So, –3 + (–2) = – 3 – 2 = – 5.