Basics of Mathematics Explained: Arithmetic, Algebra, Geometry & More

Let’s break down the basics of mathematics in detail, with examples for each key area. I’ll cover arithmetic, algebra, geometry, and measurement, step-by-step, to make it clear and concrete.

1. Arithmetic

Arithmetic is about working with numbers using four basic operations. Here’s each one explained with examples:

• Addition: Combines two or more numbers into a sum.
  • Example: You have 4 apples, and a friend gives you 3 more. How many do you have?
    4 + 3 = 7. So, you have 7 apples.
  • How it works: Start with 4, count up 3 more (5, 6, 7), and you land on 7.
• Subtraction: Takes one number away from another to find the difference.
  • Example: You have 9 cookies and eat 2. How many are left?
    9 – 2 = 7. You’re left with 7 cookies.
  • How it works: Start at 9, count down 2 (8, 7), and you get 7.
• Multiplication: Repeated addition of the same number.
  • Example: You buy 3 packs of pens, each with 4 pens. How many pens total?
    3 × 4 = 12. You have 12 pens.
  • How it works: Think 4 + 4 + 4 = 12. Or, 3 groups of 4 make 12.
• Division: Splits a number into equal parts.
  • Example: You have 12 candies and want to share them equally among 3 friends. How many does each get?
    12 ÷ 3 = 4. Each friend gets 4 candies.
  • How it works: 12 split into 3 groups means each group has 4 (since 3 × 4 = 12).

Numbers also have categories:

• Natural: 1, 2, 3, … (used for counting).
• Integers: …, -2, -1, 0, 1, 2, … (include negatives).
• Example: If it’s -3°C outside and warms up by 5°C, the new temperature is -3 + 5 = 2°C.

2. Algebra

Algebra uses letters (variables) to stand for unknown numbers and solves equations.

• Variables and Equations: An equation is like a balance scale—it must stay equal on both sides.
  • Example: Solve 2x + 5 = 11.
    • Step 1: Subtract 5 from both sides (to isolate the term with x):
      2x + 5 – 5 = 11 – 5 → 2x = 6.
    • Step 2: Divide both sides by 2 (to solve for x):
      2x ÷ 2 = 6 ÷ 2 → x = 3.
    • Check: 2(3) + 5 = 6 + 5 = 11. It works!
• Real-World Example: You’re buying shirts. Each costs $5, and you spend $20 total, including a $5 shipping fee. How many shirts did you buy?
  • Equation: 5x + 5 = 20 (where x is the number of shirts).
  • Subtract 5: 5x = 15.
  • Divide by 5: x = 3. You bought 3 shirts.
• Properties: Rules like the distributive property help simplify.
  • Example: 2 × (3 + 4) = (2 × 3) + (2 × 4) = 6 + 8 = 14.

3. Geometry

Geometry studies shapes and their properties.

• Points, Lines, Angles:
  • A point is a dot with no size (e.g., point A).
  • A line goes forever in both directions (e.g., line AB).
  • An angle is formed when two lines meet.
    • Example: A right angle is 90°. Think of a corner of a square.
• Shapes:
  • Triangle: 3 sides, 3 angles.
    • Example: A triangle with sides 3 cm, 4 cm, and 5 cm. Its perimeter (total side length) is 3 + 4 + 5 = 12 cm.
  • Square: 4 equal sides, 4 right angles.
    • Example: A square with side 4 cm has an area of 4 × 4 = 16 cm².
  • Circle: Defined by its radius (distance from center to edge).
    • Example: A circle with radius 3 cm has a circumference of 2π × 3 ≈ 18.85 cm (using π ≈ 3.14).
• Real-World Example: You’re fencing a square garden with 10-meter sides. How much fencing?
  Perimeter = 4 × 10 = 40 meters.

4. Measurement

Measurement applies numbers to physical things.

• Length: How long something is.
  • Example: A pencil is 15 cm long.
• Area: Space inside a 2D shape.
  • Example: A rectangle 5 m long and 3 m wide has an area of 5 × 3 = 15 m².
• Volume: Space inside a 3D object.
  • Example: A box 2 m long, 3 m wide, 1 m high has a volume of 2 × 3 × 1 = 6 m³.
• Time: Duration.
  • Example: If you walk 1 km in 15 minutes, your speed is 1 ÷ (15/60) = 4 km/hour.

5. Logic and Patterns

Math follows consistent rules and often involves spotting patterns.

• Commutative Property: Order doesn’t matter in addition or multiplication.
  • Example: 5 + 7 = 7 + 5 = 12. Or 2 × 6 = 6 × 2 = 12.
• Pattern Example: The sequence 1, 3, 5, 7, … adds 2 each time. The 5th term is 9 (1 + 2 + 2 + 2 + 2).

Putting It Together

Imagine you’re planning a party:

• Arithmetic: 20 guests, each gets 3 cookies. Total cookies = 20 × 3 = 60.
• Algebra: Budget is $50, decorations cost $20. How much for food? 20 + f = 50 → f = 30.
• Geometry: Round table, radius 1 m. Circumference = 2π × 1 ≈ 6.28 m of tablecloth edge.
• Measurement: Party lasts 3 hours, from 2 PM to 5 PM.