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The number system is the foundational pillar of all arithmetic and mathematics. It is a structured way of representing and understanding numbers which we use daily-from counting household items to managing money, measuring quantities, and solving problems in competitive exams.
Understanding the classification of numbers, their operations, and conversions between different forms is essential to master arithmetic topics such as percentages, profit-loss calculations, ratios, and statistics. This chapter will guide you through these concepts step-by-step, making sure you grasp the basics as well as their practical applications relevant for entrance exams and daily life.
Numbers are grouped into different categories based on their properties and the types of values they represent. Let's explore each category with clear definitions and examples.
These are the numbers you first learn for counting objects: 1, 2, 3, 4, 5, .... They are also called positive integers. Natural numbers do not include zero or any negative values.
Example: Counting books on a shelf: 1, 2, 3...
Whole numbers include all natural numbers and zero. In other words, they are 0, 1, 2, 3, 4, 5, ...
Example: Number of cars parked (can be zero) would be a whole number.
Integers expand whole numbers to include negative numbers. They can be positive, negative or zero:
..., -3, -2, -1, 0, 1, 2, 3, ...
Example: Temperature readings which can be below zero (negative) or above zero (positive).
A rational number is any number that can be written as a fraction \(\frac{p}{q}\), where p and q are integers and q \neq 0. This includes integers (since 5 = 5/1) and fractions such as 3/4 or -7/2.
All decimals that end or repeat periodically can be expressed as rational numbers.
Example: 0.75 = \(\frac{3}{4}\), 0.333... = \(\frac{1}{3}\)
These numbers cannot be expressed as a simple fraction. Their decimal expansions never end and never settle into a repeating pattern.
Example: \(\sqrt{2}\) ≈ 1.414213..., \(\pi\) ≈ 3.14159...
The set of all rational and irrational numbers combined is called the real numbers. These represent any value along the infinite number line.
Once we recognize types of numbers, we perform arithmetic operations: addition, subtraction, multiplication, and division. Important properties govern these operations which you will encounter frequently.
Addition joins two quantities; subtraction finds the difference.
Example: \(7 + 3 = 10\), \(10 - 4 = 6\)
Multiplication is repeated addition and division splits into equal parts.
Example: \(5 \times 4 = 20\) (add 5 four times), \(20 \div 5 = 4\)
| Property | Addition | Multiplication | Notes |
|---|---|---|---|
| Closure | Yes (e.g. 3 + 5 = 8, an integer) | Yes (e.g. 4 x 6 = 24, an integer) | Sum or product of integers is an integer |
| Commutative | \(a + b = b + a\) | \(a \times b = b \times a\) | Order does not affect result |
| Associative | \((a + b) + c = a + (b + c)\) | \((a \times b) \times c = a \times (b \times c)\) | Grouping does not affect result |
| Distributive | \(a \times (b + c) = a \times b + a \times c\) | Multiplication distributes over addition | |
| Identity Element | 0 (\(a + 0 = a\)) | 1 (\(a \times 1 = a\)) | Elements that don't change the value |
Numbers can be represented in various forms-decimals, fractions, and percentages. Converting between these forms helps in solving problems easily.
graph TD A[Decimal Number] --> B[Convert to Fraction] B --> C[Simplify Fraction] A --> D[Convert to Percentage] E[Fraction] --> F[Convert to Decimal] F --> A D --> G[Express as %]
Step 1: Count the number of digits after the decimal point.
Step 2: Write the decimal without the decimal point as numerator.
Step 3: Write denominator as 1 followed by zeros equal to the digits counted.
Step 4: Simplify the fraction.
Divide numerator by denominator to get decimal form. If it repeats, note the repeating digits.
Multiply the decimal by 100 and add % sign.
Divide the percentage by 100.
Step 1: Count digits after decimal point. There are 2 digits (7 and 5).
Step 2: Write 75 as numerator and denominator as 100 (1 followed by 2 zeros): \(\frac{75}{100}\).
Step 3: Simplify by dividing numerator and denominator by their greatest common divisor (GCD) 25:
\(\frac{75 \div 25}{100 \div 25} = \frac{3}{4}\).
Step 4: To convert decimal 0.75 to percentage, multiply by 100:
\(0.75 \times 100 = 75\%\).
Answer: \(0.75 = \frac{3}{4} = 75\%\).
Step 1: Calculate profit:
\(\text{Profit} = \text{Selling Price} - \text{Cost Price} = 1650 - 1500 = 150\) INR.
Step 2: Calculate profit percentage using formula:
\[ \text{Profit \%} = \left(\frac{\text{Profit}}{\text{Cost Price}}\right) \times 100 = \left(\frac{150}{1500}\right) \times 100 = 10\% \]
Answer: Profit = INR 150, Profit percentage = 10%.
Step 1: Calculate total volume:
Total = 3 liters + 2 liters = 5 liters.
Step 2: Confirm the ratio of solutions in mixture remains 3:2.
Ratio of solution A to B in mixture = \(\frac{3}{2}\).
Answer: Total mixture = 5 liters; ratio = 3:2 as given.
Step 1: Arrange data in ascending order:
3, 4, 5, 6, 7, 8, 8
Step 2: Calculate mean (average):
\(\bar{x} = \frac{3 + 4 + 5 + 6 + 7 + 8 + 8}{7} = \frac{41}{7} \approx 5.86\)
Step 3: Calculate median (middle value):
7 data points, median is 4th value -> 6
Step 4: Find mode (most frequent value):
Value 8 appears twice; others less. Mode = 8.
Answer: Mean ≈ 5.86, Median = 6, Mode = 8.
Step 1: Calculate discount percentage using formula:
\[ \text{Discount \%} = \left(\frac{\text{Discount}}{\text{Marked Price}}\right) \times 100 = \left(\frac{300}{2000}\right) \times 100 = 15\% \]
Step 2: Calculate final selling price:
\(\text{Selling Price} = \text{Marked Price} - \text{Discount} = 2000 - 300 = 1700\) INR.
Answer: Discount = 15%, Final price = INR 1700.
When to use: When converting fractions to decimals or reducing fractions to simplest form.
When to use: Whenever facing profit and loss questions.
When to use: When applying percentage values in equations or calculations.
When to use: Throughout all calculations involving monetary or measurement values.
When to use: To quickly solve mixture or partnership problems.
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