Grade 6
  1. Number System  1.1. Whole Numbers  1.1.1. Place Value and Face Value  1.1.2. Comparing and Ordering Numbers  1.1.3. Rounding Numbers  1.1.4. Properties of Whole Numbers (Associative, Commutative, Distributive)  1.2. Integers  1.2.1. Introduction to Integers  1.2.2. Addition and Subtraction of Integers  1.2.3. Multiplication and Division of Integers  1.2.4. Properties of Integer Operations  1.3. Fractions  1.3.1. Types of Fractions  1.3.2. Equivalent Fractions  1.3.3. Simplification of Fractions  1.3.4. Addition and Subtraction of Fractions  1.3.5. Multiplication and Division of Fractions  1.3.6. Comparing and Ordering Fractions  1.4. Decimals  1.4.1. Understanding Decimals  1.4.2. Conversion between Fractions and Decimals  1.4.3. Addition and Subtraction of Decimals  1.4.4. Multiplication and Division of Decimals  1.4.5. Comparing and Ordering Decimals
  2. Algebra  2.1. Basics of Algebra  2.1.1. Variables and Constants  2.1.2. Expressions and Equations  2.1.3. Algebraic Terms and Coefficients  2.2. Solving Simple Equations  2.2.1. Balancing Equations  2.2.2. Solving One-Step Equations  2.2.3. Solving Two-Step Equations  2.3. Patterns and Sequences  2.3.1. Identifying Patterns  2.3.2. Arithmetic Sequences
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Equivalent Ratios  3.1.3. Simplifying Ratios  3.2. Proportion  3.2.1. Introduction to Proportion  3.2.2. Direct and Inverse Proportions  3.3. Percentages  3.3.1. Converting Ratios to Percentages  3.3.2. Percentage Increase and Decrease  3.3.3. Discount and Profit Calculations  3.3.4. Simple Interest Calculations in Percentages in Ratio and Proportion
  4. Geometry  4.1. Basic Geometric Shapes  4.1.1. Points, Lines, and Angles  4.1.2. Types of Angles  4.1.3. Intersecting and Parallel Lines  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Properties of Triangles  4.2.3. Sum of Angles in a Triangle  4.2.4. Congruence of Triangles  4.3. Quadrilaterals  4.3.1. Types of Quadrilaterals  4.3.2. Properties of Quadrilaterals  4.4. Circles  4.4.1. Radius and Diameter  4.4.2. Circumference and Area  4.4.3. Arcs and Chords  4.4.4. Sector and Segment
  5. Mensuration  5.1. Perimeter  5.1.1. Perimeter of Simple Shapes  5.2. Area  5.2.1. Area of Rectangles and Squares  5.2.2. Area of Triangles and Parallelograms  5.2.3. Area of Composite Figures  5.2.4. Area of Trapezium  5.3. Volume  5.3.1. Volume of Cubes and Cuboids  5.3.2. Volume of Cylinders  5.3.3. Volume of Cones  5.3.4. Surface Area of Simple Solids
  6. Data Handling  6.1. Introduction to Data  6.1.1. Types of Data  6.1.2. Organizing Data  6.1.3. Frequency Distribution  6.2. Graphs and Charts  6.2.1. Bar Graphs  6.2.2. Line Graphs  6.2.3. Pictographs  6.2.4. Pie Charts  6.3. Mean, Median, and Mode  6.3.1. Calculating Mean  6.3.2. Finding Median  6.3.3. Identifying Mode  6.3.4. Range of Data in Mean, Median, and Mode
  7. Probability  7.1. Basics of Probability  7.1.1. Understanding Probability  7.1.2. Probability of Simple Events  7.1.3. Experimental and Theoretical Probability
  8. Practical Geometry  8.1. Construction of Shapes  8.1.1. Constructing Triangles  8.1.2. Constructing Quadrilaterals  8.1.3. Constructing Circles and Arcs  8.2. Symmetry  8.2.1. Line of Symmetry  8.2.2. Rotational Symmetry  8.2.3. Translational Symmetry

Grade 6 → Number System → Integers ↓


Properties of Integer Operations


Integers are a set of numbers that includes zero, positive whole numbers, and negative whole numbers. In mathematics, we often perform various operations on integers such as addition, subtraction, multiplication, and division. Understanding the properties of integer operations is important in simplifying expressions and solving equations.

Addition of integers

When we add integers, we follow some simple rules based on the signs of the numbers:

  • If both integers are positive, the result will be positive. For example, 3 + 5 = 8.
  • If both integers are negative, the result will be negative. For example, (-3) + (-5) = -8.
  • If one integer is positive and the other is negative, subtract the smaller absolute value from the larger absolute value, and give the result the sign of the integer with the larger absolute value. For example, 5 + (-3) = 2 and -5 + 3 = -2.
-3 0 3 +5

Subtraction of integers

Subtraction of integers can be turned into an addition problem. Subtracting an integer is the same as adding its opposite. For example:

  • 7 - 3 can be written as 7 + (-3) which equals 4.
  • -4 - 2 can be written as -4 + (-2) which equals -6.
  • -3 - (-5) can be written as -3 + 5 which equals 2.
-7 , 0 +2 7

Multiplication of integers

Consider the following rules when multiplying integers:

  • The product of two integers with the same sign is positive. For example, 2 × 3 = 6 and (-2) × (-3) = 6.
  • The product of two integers with different signs is negative. For example, 2 × (-3) = -6 and (-2) × 3 = -6.
  • The product of any integer and zero is zero. For example, 7 × 0 = 0.
-3 0 3 6 -2

Division of integers

Division of integers follows the same rules as for multiplication:

  • The quotient of two integers with the same sign is positive. For example, 8 ÷ 2 = 4 and (-8) ÷ (-2) = 4.
  • The quotient of two integers with different signs is negative. For example, 8 ÷ (-2) = -4 and (-8) ÷ 2 = -4.
  • Division by zero is undefined. We cannot divide any number by zero.
-8 -6 0 4 2

Properties of integer operations

In addition to these basic rules, there are also some properties that integer operations obey:

1. Exchangeable assets

The commutative property states that changing the order of numbers in an operation does not change the result.

  • Addition: a + b = b + a. For example, 3 + 5 = 5 + 3 = 8.
  • Multiplication: a × b = b × a. For example, 4 × 6 = 6 × 4 = 24.

2. Associative property

The associative property states that the way numbers are grouped in an operation has no effect on the result.

  • Addition: (a + b) + c = a + (b + c). For example, (1 + 2) + 3 = 1 + (2 + 3) = 6.
  • Multiplication: (a × b) × c = a × (b × c) For example, (2 × 3) × 4 = 2 × (3 × 4) = 24.

3. Distributive property

The distributive property connects the operations of multiplication and addition. It states that multiplying a sum by a number gives the same result as multiplying each sum by that number and then adding the products.

a × (b + c) = a × b + a × c

For example, 2 × (3 + 4) = 2 × 3 + 2 × 4 = 6 + 8 = 14.

4. Identity property

The identity property states that there are certain numbers which when used in operations with any number do not change the value of that number.

  • Additive identity: The number 0 is the additive identity because a + 0 = a.
  • Multiplication Identity: The number 1 is a multiplication identity because a × 1 = a.

5. Inverse property

The inverse property states that every number has an opposite, and when they are added together, they provide the identity element.

  • Additive Inverse: The additive inverse of any number a is -a, since a + (-a) = 0.
  • Multiplicative Inverse: The multiplicative inverse of a number is 1/a, which is mainly found in rational numbers.

6. Zero property of multiplication

The zero property states that multiplying any number by 0 will give the result 0.

a × 0 = 0

For example, 7 × 0 = 0

Conclusion

Understanding the properties of integer operations helps us simplify and solve mathematical problems more efficiently. The commutative, associative, distributive, identity, inverse, and zero properties each have unique characteristics that apply to the addition and multiplication of integers, making them fundamental concepts in mathematics.


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Properties of Integer Operations

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