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Tag: KNOWING OUR NUMBERS
Integers | Study
KNOWING OUR NUMBERS | Study
KNOWING OUR NUMBERS | Study

Integers | Study
CBSE 6-10 | Math & Science | Study Premium
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English Version Integers | Speed Notes
Notes For Quick Recap
Whole Numbers The numbers 1,2, 3, ……which we use for counting are known as natural numbers. If you add 1 to a natural number, we get its successor. If you subtract 1 from a natural number, you get its predecessor. (Scroll down to continue …)
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Integers
Definition
Integers are the set of whole numbers that include positive numbers, negative numbers, and zero. The set of integers can be represented as: Integers={…,−3,−2,−1,0,1,2,3,…}Integers={…,−3,−2,−1,0,1,2,3,…}
Key Properties of Integers
- Closure Properties:
- Addition: The sum of any two integers is an integer.
- Examples:
- 2+3=52+3=5
- −1+4=3−1+4=3
- −2+(−3)=−5−2+(−3)=−5
- Examples:
- Subtraction: The difference between any two integers is an integer.
- Examples:
- 5−3=25−3=2
- −2−1=−3−2−1=−3
- 0−(−4)=40−(−4)=4
- Examples:
- Multiplication: The product of any two integers is an integer.
- Examples:
- 3×2=63×2=6
- −4×5=−20−4×5=−20
- −3×−2=6−3×−2=6
- Examples:
- Addition: The sum of any two integers is an integer.
- Identity Elements:
- Additive Identity: The integer 0 is the identity element for addition.
- Examples:
- 7+0=77+0=7
- −5+0=−5−5+0=−5
- 0+0=00+0=0
- Examples:
- Multiplicative Identity: The integer 1 is the identity element for multiplication.
- Examples:
- 4×1=44×1=4
- −3×1=−3−3×1=−3
- 0×1=00×1=0
- Examples:
- Additive Identity: The integer 0 is the identity element for addition.
- Inverse Elements:
- Additive Inverse: For every integer a, there exists an integer −a such that a+(−a)=a+−a=0.
- Examples:
- The additive inverse of 5 is -5: 5+(−5)=5+−5=0
- The additive inverse of -3 is 3: −3+3=0
- The additive inverse of 0 is 0: 0+0=0
- Examples:
- Multiplicative Inverse: Integers do not have multiplicative inverses within the set of integers (except for 1 and -1).
- Additive Inverse: For every integer a, there exists an integer −a such that a+(−a)=a+−a=0.
- Commutative and Associative Properties:
- Commutative Property:
- Addition: a+b = b+a
- Examples:
- 2+3=3+2
- −1+4 = 4+(−1) = 4-1 = 3
- 0+5 = 5+0 = 5
- Examples:
- Multiplication: a×b=b×a
- Examples:
- 3×4 = 4×3 = 12
- −2×1 = 1×−2 = -2
- 0×5 = 5×0 = 0
- Examples:
- Addition: a+b = b+a
- Associative Property:
- Addition: (a+b)+c = a+(b+c) = (a+c)+b
- Examples:
- (1+2)+3 = 1+(2+3) = (1+3)+2
- [0+(−4)]+2 = 0+[−4+2] = [(0+2)+(-4)]
- [-2+(−3)]+(-1) = -2+[−3+(-1)] = [-2+(−1)]+(-3)
- Examples:
- Multiplication: (a×b)×c=a×(b×c)(a×c)×b
- Examples:
- (2×3)×4 = 2×(3×4) = (2×4)×3
- (0×−1)×5 = 0×(−1×5) = (0×5)×−1
- (−2×3)×−1 = −2×(3×−1) = (−2×-1)×3
- Examples:
- Addition: (a+b)+c = a+(b+c) = (a+c)+b
- Commutative Property:
- Distributive Property:
- Multiplication distributes over addition:
- Example: a×(b+c)=(a×b)+(a×c) Or a×(b+c)=a×b+a×c
- Examples:
- 2×(3+4) = (2×3)+(2×4) = 6+12 = 14 Or (2×7) = 14
- −3×(1+2) = (−3×1)+(−3×2) = -3-6 = -9 Or −3×3 = −9
- 0×(5+7) = (0×5)+(0×7) = 0×(5+7) = 0×5+0×7 = 0+0 =0
- Examples:
- Example: a×(b+c)=(a×b)+(a×c) Or a×(b+c)=a×b+a×c
- Multiplication distributes over addition:
Ordering of Integers
- Integers can be ordered on a number line, where:
- Negative integers are to the left of 0.
- Positive integers are to the right of 0.
- Examples of ordering:
- …−3<−2<−1<0<1<2<3−3<−2<−1<0<1<2<3…
- −5,−2,0,4,3−5,−2,0,4,3 arranged in order: −5<−2<0<3<4−5<−2<0<3<4
Absolute Value
- The absolute value of an integer is its distance from zero on the number line, regardless of direction.
- Notation: ∣a∣∣a∣
- Examples:
- ∣3∣=3∣3∣=3
- ∣−3∣=3∣−3∣=3
- ∣0∣=0∣0∣=0
Conclusion
Understanding integers and their properties is fundamental in mathematics. They play a critical role in various areas, including algebra, number theory, and real-world applications. Mastery of integer operations is essential for higher-level mathematics.
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- Closure Properties:

KNOWING OUR NUMBERS | Study
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English Version Whole Numbers | Speed Notes
Notes For Quick Recap
Whole Numbers The numbers 1,2, 3, ……which we use for counting are known as natural numbers. If you add 1 to a natural number, we get its successor. If you subtract 1 from a natural number, you get its predecessor. (Scroll down to continue …)
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Every natural number has a successor. Every natural number except 1 has a predecessor.
Whole Numbers
Whole numbers are formed by adding zero to the collection of natural numbers. Hence, the set of whole numbers includes 0, 1, 2, 3, and so on.
Key Properties of Whole Numbers:
- Successors and Predecessors:
- Every whole number has a successor. For example:
- The successor of 0 is 1.
- The successor of 1 is 2.
- The successor of 2 is 3.
- Every whole number except zero has a predecessor. For example:
- The predecessor of 1 is 0.
- The predecessor of 2 is 1.
- The predecessor of 3 is 2.
- Every whole number has a successor. For example:
- Relationship with Natural Numbers:
- All natural numbers (1, 2, 3, …) are whole numbers, but not all whole numbers are natural numbers since whole numbers include 0.
- Number Line Representation:

Whole Number Line - To visualize whole numbers, we can draw a number line starting from 0:
- Mark points at equal intervals to the right: 0, 1, 2, 3, …
- This number line allows us to carry out operations:
- Addition: Moving to the right (e.g., 1 + 2 = 3).
- Subtraction: Moving to the left (e.g., 3 – 1 = 2).
- Multiplication: Making equal jumps (e.g., 2 × 3 means jumping twice the distance of 2, reaching 6).
- Division: Although division can be tricky, it involves partitioning. For example, 6 ÷ 2 means splitting 6 into 2 equal parts, resulting in 3.
Closure Properties:
- Adding two whole numbers always results in a whole number:
- Examples:
- 2 + 3 = 5
- 0 + 4 = 4
- 1 + 1 = 2
- Examples:
- Multiplying two whole numbers also results in a whole number:
- Examples:
- 2 × 3 = 6
- 0 × 5 = 0
- 1 × 4 = 4
- Examples:
- Whole numbers are closed under subtraction only if the result is non-negative:
- Examples:
- 2 – 1 = 1
- 5 – 3 = 2
- 3 – 3 = 0
- Yet, if the result is negative, they are not closed under subtraction:
- Example: 2 – 3 = -1 (not a whole number).
- So, the whole numbers are not not closed under subtraction.
- Examples:
- Division by whole numbers is defined only when the divisor is not zero, and the result is a whole number:
- Examples:
- 6 ÷ 2 = 3
- 8 ÷ 4 = 2
- 0 ÷ 5 = 0
- Division by zero is undefined (e.g., 5 ÷ 0).
- So, the whole numbers are not not closed under division.
- Examples:
Identity Elements:
- Zero acts as the identity for addition:
- Example: 5 + 0 = 5.
- The whole number 1 acts as the identity for multiplication:
- Example: 3 × 1 = 3.
Commutative and Associative Properties:
- Addition is commutative:
- Examples:
- 2 + 3 = 3 + 2
- 1 + 4 = 4 + 1
- 0 + 5 = 5 + 0
- Examples:
- Multiplication is also commutative:
- Examples:
- 2 × 3 = 3 × 2
- 1 × 4 = 4 × 1
- 0 × 5 = 5 × 0
- Examples:
- Both addition and multiplication are associative:
- Examples for addition:
- (1 + 2) + 3 = 1 + (2 + 3)
- (0 + 4) + 1 = 0 + (4 + 1)
- (2 + 2) + 2 = 2 + (2 + 2)
- Examples for multiplication:
- (1 × 2) × 3 = 1 × (2 × 3)
- (0 × 4) × 1 = 0 × (4 × 1)
- (2 × 2) × 2 = 2 × (2 × 2)
- Examples for addition:
Distributive Property:
- Multiplication distributes over addition:
- Example: 2 × (3 + 4) = 2 × 3 + 2 × 4.
Understanding these properties helps simplify calculations. It enhances our grasp of numerical patterns. These patterns are not only interesting but also practical for mental math.
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- Successors and Predecessors:

KNOWING OUR NUMBERS | Study
Pre-Requisires
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English Version Whole Numbers | Speed Notes
Notes For Quick Recap
Whole Numbers The numbers 1,2, 3, ……which we usefor counting areknown as natural numbers. If you add1 to a natural number, we get itssuccessor. If yousubtract 1 froma natural number, you get itspredecessor. (Scroll down to continue …)
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Every natural number has a successor. Every natural number except 1 has apredecessor. If we addthe number zeroto the collection of natural numbers, we get thecollection of whole numbers. Thus, the numbers 0, 1, 2, 3, … form the collection of whole numbers. Every wholenumber has asuccessor. Every whole number except zerohas a predecessor. All natural numbers are wholenumbers, but allwhole numbers arenot natural numbers. We take a line, mark a point on it and label it 0. We then mark out points to the right of 0, at equal intervals. Label them as1, 2, 3,…………………………………. Thus, we havea number linewith the whole numbers represented on it.We can easily perform the number operations of addition, subtraction and multiplication on the number line. Addition corresponds to moving tothe right onthe number line, whereas subtraction corresponds to moving to the left. Multiplication corresponds to making jumps of equal distance starting from zero. Addingtwo whole numbers always gives a whole number. Similarly, multiplying twowhole numbers always gives a whole number. We say that whole numbers are closed under addition and also under multiplication. However, whole numbers are not closed under subtraction andunder division. Division by zerois not defined. Zero is theidentity for addition of whole numbers. The whole number 1 is theidentity for multiplication of whole numbers. You can addtwo whole numbers in any order. You can multiply two whole numbers in anyorder. We saythat addition andmultiplication are commutative for whole numbers. Addition and multiplication, both, are associative for whole numbers. Multiplication isdistributive over addition for whole numbers. Commutativity, associativity and distributivity properties of whole numbers are useful in simplifying calculations and we use them without being aware of them.Patterns with numbers are not onlyinteresting, but areuseful especially forverbal calculations and helpus to understand properties of numbers better.
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