Students can go through AP Board 7th Class Maths Notes Chapter 1 Integers to understand and remember the concepts easily.
AP State Board Syllabus 7th Class Maths Notes Chapter 1 Integers
→ Number System:
Natural Numbers:
a) Counting numbers 1, 2, 3, 4, 5, 6, …… are called natural numbers.
b) The set of all natural numbers can be represented by N = {1, 2, 3, 4, 5, ……}
→ Whole Numbers:
a) If we include ‘O’ among the natural numbers, then the numbers 0, 1, 2, 3, 4, 5, …… are called whole numbers.
b) The set of whole numbers can be represented by W = {0, 1, 2, 3, ……}
c) Clearly, every natural number is a whole number but ‘O’ is a whole number which is not a natural number.
→ Integers:
a) All counting numbers and their negatives including zero are known as integers.
b) The set of integers can be represented by Z or I = {……, -4, -3, -2,-1, 0, 1, 2, 3, 4, ……}
- Positive Integers:
The set I+ = {1, 2, 3, 4, ……} is the set of all positive integers. Clearly positive integers and natural numbers are same. - Negative Integers:
The set I– = {-1, -2, -3, ……} is the set of all negative integers. ‘0’ is neither positive nor negative. - Non-Negative Integers:
The set {0, 1, 2, 3, ……} is the set of all non-negative integers.
→ Properties of integers:
For any three integers a, b, c
i) a + b is also an integer – closure property w.r.t addition.
ii) a – b is also an integer – closure property w.r.t subtraction.
iii) a . b is also an integer – closure property w.r.t multiplication.
iv) a + b = b + a – commutative law w.r.t addition. ‘
v) a . b = b . a – commutative law w.r.t multiplication.
vi) a + (b + c) = (a + b) + c – associative law w.r.t addition.
a . (b . c) = (a . b). c – associative law w.r.t multiplication.
vii) a + 0 = 0 + a = a – identity w.r.t addition.
viii) a . 1 = 1 . a = a – identity w.r.t multiplication.
ix) a.(b + c) = a.b + a.c – distributive property.
x) a ÷ 0 is not defined
a ÷ 1 = a
0 ÷ a = 0 (a ≠ 0)
→ On a number line when you add a positive integer you move right side on the number line; and if a negative integer is added you move to the left side on the number line.
→ On the number line if you subtract a positive integer you move to the left side and if you subtract a negative integer you move to the right side.
→ Product of any two positive integers or any two negative integers is always a positive integer.
→ Product of a positive integer and a negative integer is always a negative integer (i.e.,) two integers with opposite signs always give a negative product.
→ Product of even number of negative integers is always a positive integer.
→ Product of odd number of negative integers is always a negative integer.