Unit 1: Number System and Commercial Arithmetic

Number System

This unit covers the foundational concepts of different types of numbers and their properties.

Place Value and Face Value

• Face Value: The inherent value of a digit, regardless of its position in a number.
  • Example: In the number 739, the face value of the digit '3' is simply 3.
• Place Value: The value of a digit based on its position (ones, tens, hundreds, etc.) in the number.
  • Example: In the number 739, the place value of the digit '3' is 3 × 10 = 30.

Types of Numbers

1. Natural Numbers (N): The set of counting numbers. N = {1, 2, 3, 4, ...}
2. Integers (Z): The set of all whole numbers and their negative counterparts. Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
3. Rational Numbers (Q): Any number that can be expressed as a fraction p/q, where 'p' and 'q' are integers and 'q' is not zero.
   • Includes all integers (e.g., 5 = 5/1).
   • Includes terminating decimals (e.g., 0.5 = 1/2).
   • Includes repeating decimals (e.g., 0.333... = 1/3).
4. Real Numbers (R): The set of all numbers on the number line. This includes all rational numbers and irrational numbers (numbers that cannot be expressed as a simple fraction, e.g., √2, π).

Basic Operations

HCF (Highest Common Factor)

Also known as the Greatest Common Divisor (GCD). It is the largest positive integer that divides two or more numbers without leaving a remainder.

Example: Find the HCF of 12 and 18.

1. Prime factors of 12: 2 × 2 × 3
2. Prime factors of 18: 2 × 3 × 3
3. Common factors: 2, 3
4. HCF = Product of common factors = 2 × 3 = 6

LCM (Lowest Common Multiple)

It is the smallest positive integer that is a multiple of two or more numbers.

Example: Find the LCM of 12 and 18.

1. Prime factors of 12: 2² × 3¹
2. Prime factors of 18: 2¹ × 3²
3. Take the highest power of each prime factor present in either number: 2² and 3²
4. LCM = 2² × 3² = 4 × 9 = 36

Commercial Arithmetic

Ratio and Proportion

• Ratio: A comparison of two quantities, written as `a:b` or `a/b`.
• Proportion: An equality of two ratios. If `a:b` is proportional to `c:d`, we write `a:b = c:d` or `a/b = c/d`.
  • This implies the "product of extremes equals product of means": ad = bc.

Percentage

The word "percent" means "per hundred". It's a way to express a number as a fraction of 100.

Formula: Percentage = (Part / Whole) × 100%

Example: If you score 30 out of 40 on a test, your percentage is (30 / 40) × 100% = 0.75 × 100% = 75%.

Profit and Loss

• Cost Price (CP): The price at which an item is purchased.
• Selling Price (SP): The price at which an item is sold.
• Profit: If SP > CP, then Profit = SP - CP.
• Loss: If CP > SP, then Loss = CP - SP.

Formulas:
Profit % = (Profit / CP) × 100
Loss % = (Loss / CP) × 100

Simple Interest (SI)

Interest calculated only on the original principal amount.

• P = Principal (the initial amount of money)
• R = Rate of interest (per year)
• T = Time (in years)

Formulas:
Simple Interest (SI) = (P × R × T) / 100
Total Amount (A) = Principal + Simple Interest = P + SI

Compound Interest (CI)

Interest calculated on the initial principal and also on the accumulated interest from previous periods ("interest on interest").

Formula (compounded annually):
Amount (A) = P × (1 + R/100)ᵀ
Compound Interest (CI) = A - P = P × [ (1 + R/100)ᵀ - 1 ]

Example: Find the CI on $1000 for 2 years at 10% p.a. compounded annually.

1. A = 1000 × (1 + 10/100)²
2. A = 1000 × (1.1)²
3. A = 1000 × 1.21 = $1210
4. CI = A - P = $1210 - $1000 = $210