Introduction to Number System and Basic Operations

Mathematics is the language of numbers, and understanding how numbers work is essential not only for exams like the Meghalaya Police Sub Inspector (SI) Written Examination but also for everyday life. In this chapter, we focus on three important types of numbers: whole numbers, decimal fractions, and integers. These numbers form the foundation for many mathematical concepts and practical applications.

Whole numbers are the numbers we use to count objects, decimal fractions help us express parts of a whole precisely, and integers include both positive and negative numbers, which are useful in many real-world situations such as temperature changes, bank balances, and elevations.

By mastering the basic operations - addition, subtraction, multiplication, and division - on these numbers, you will develop strong problem-solving skills that are crucial for the exam and daily calculations involving measurements (like meters and kilograms) or money (like Indian Rupees).

In this chapter, you will learn:

• What whole numbers, decimal fractions, and integers are, with clear definitions and examples.
• How to perform basic operations on these numbers step-by-step.
• How to avoid common mistakes and use smart strategies to solve problems quickly and accurately.

Whole Numbers and Their Operations

Whole numbers are the numbers starting from 0 and going upwards: 0, 1, 2, 3, 4, and so on. They do not include fractions, decimals, or negative numbers. Whole numbers are used for counting and ordering.

Some important properties of whole numbers help us understand how operations work:

These properties make calculations easier and allow us to rearrange and group numbers without changing the answer.

Addition and Subtraction of Whole Numbers

Adding whole numbers means combining their values. Subtraction means finding the difference between two numbers.

Example: If you have 7 apples and get 5 more, how many apples do you have in total? You add 7 + 5 = 12 apples.

When subtracting, if you have 12 apples and give away 5, you subtract 12 - 5 = 7 apples left.

Remember, subtraction is only defined for whole numbers when the first number is greater than or equal to the second (no negative results here).

Multiplication and Division of Whole Numbers

Multiplication is repeated addition. For example, 4 x 3 means adding 4 three times: 4 + 4 + 4 = 12.

Division is splitting into equal parts or groups. For example, 12 / 3 means dividing 12 into 3 equal parts, each part having 4.

Division by zero is undefined, so always ensure the divisor is not zero.

Decimal Fractions and Operations

Decimal fractions (or simply decimals) are numbers that have a whole number part and a fractional part separated by a decimal point. They represent parts of a whole in a base-10 system.

For example, 3.75 means 3 whole units and 75 hundredths.

Each digit after the decimal point has a place value:

Understanding place value is key to performing operations on decimals correctly.

Operations on Decimals

Addition and Subtraction: Always align the decimal points vertically before adding or subtracting. This ensures digits of the same place value are added or subtracted.

Multiplication: Multiply as if there are no decimal points, then count the total number of decimal places in both numbers and place the decimal point in the product accordingly.

Division: To divide decimals, multiply both the divisor and dividend by the same power of 10 to make the divisor a whole number, then divide as usual.

Conversion Between Fractions and Decimals

Decimal fractions can be converted to ordinary fractions by expressing the decimal part over a power of 10. For example, 0.75 = \(\frac{75}{100}\) = \(\frac{3}{4}\).

Similarly, fractions can be converted to decimals by dividing the numerator by the denominator.

Integer Operations

Integers are whole numbers that include positive numbers, zero, and negative numbers. For example, -5, 0, 7 are all integers.

Integers are useful for representing situations like temperature below zero, bank overdrafts, or elevations below sea level.

Positive and Negative Numbers

Positive integers are greater than zero and are written without a sign or with a plus sign (+). Negative integers are less than zero and are written with a minus sign (-).

Integer Addition and Subtraction Rules

Adding and subtracting integers requires careful attention to their signs:

graph TD    A[Start] --> B{Are both integers positive?}    B -- Yes --> C[Add magnitudes, result positive]    B -- No --> D{Are both integers negative?}    D -- Yes --> E[Add magnitudes, result negative]    D -- No --> F[Subtract smaller magnitude from larger]    F --> G{Which magnitude is larger?}    G -- First number larger --> H[Result sign same as first number]    G -- Second number larger --> I[Result sign same as second number]

Subtraction of integers is converted to addition by adding the opposite. For example, \(a - b = a + (-b)\).

Integer Multiplication and Division Rules

Multiplying or dividing integers depends on their signs:

• Positive x Positive = Positive
• Negative x Negative = Positive
• Positive x Negative = Negative
• Negative x Positive = Negative

The same rules apply for division.

This number line helps visualize integer operations: moving right means adding positive numbers, moving left means adding negative numbers.

Worked Examples