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Number Systems and Algebra Basics

Subject: Mathematics
Topic: 1
Cambridge Code: 0580 / 4024(D)

Number Types​

Natural Numbers​

N={1,2,3,4,...}\mathbb{N} = \{1, 2, 3, 4, ...\}

Positive whole numbers (sometimes includes 0)

Integers​

Z={...,−2,−1,0,1,2,...}\mathbb{Z} = \{..., -2, -1, 0, 1, 2, ...\}

Whole numbers (positive, negative, and zero)

Rational Numbers​

Q={pq:p,q∈Z,q≠0}\mathbb{Q} = \left\{\frac{p}{q} : p, q \in \mathbb{Z}, q \neq 0\right\}

Numbers expressible as fraction of integers

  • Includes: integers, fractions, terminating and repeating decimals

Irrational Numbers​

  • Cannot express as simple fraction
  • Non-repeating, non-terminating decimals
  • Examples: π\pi, ee, 2\sqrt{2}

Real Numbers​

R=Rational∪Irrational\mathbb{R} = \text{Rational} \cup \text{Irrational}

All numbers on number line

Basic Operations​

Order of Operations (BODMAS/PEDMAS)​

  1. Brackets/Parentheses
  2. Orders/Exponents
  3. Division and Multiplication (left to right)
  4. Addition and Subtraction (left to right)

Example: 3+4×2=3+8=113 + 4 \times 2 = 3 + 8 = 11 (not 14)

Fractions​

Operations​

Addition: ab+cd=ad+bcbd\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

Subtraction: ab−cd=ad−bcbd\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}

Multiplication: ab×cd=acbd\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

Division: ab÷cd=ab×dc=adbc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}

Simplification​

Find GCD (Greatest Common Divisor) and divide both numerator and denominator

Example: 1218=23\frac{12}{18} = \frac{2}{3}

Decimals​

Conversion​

Fraction to Decimal: Divide numerator by denominator 34=0.75\frac{3}{4} = 0.75

Decimal to Fraction: Write over appropriate power of 10 0.25=25100=140.25 = \frac{25}{100} = \frac{1}{4}

Significant Figures​

Digits that carry meaningful information

Rounding to n significant figures: Keep first n non-zero digits, round last digit

Examples:

  • 23.456 to 2 s.f. = 23
  • 0.001234 to 2 s.f. = 0.0012

Percentages​

Percentage=PartWhole×100%\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\%

Applications​

Percentage increase: New value=Original×(1+percentage100)\text{New value} = \text{Original} \times \left(1 + \frac{\text{percentage}}{100}\right)

Percentage decrease: New value=Original×(1−percentage100)\text{New value} = \text{Original} \times \left(1 - \frac{\text{percentage}}{100}\right)

Basic Algebra​

Simplifying Expressions​

Collect like terms: 3x+2y+5x−y=8x+y3x + 2y + 5x - y = 8x + y

Expanding Brackets​

a(b+c)=ab+aca(b + c) = ab + ac

Example: 2(3x+4)=6x+82(3x + 4) = 6x + 8

Factoring​

Reverse of expanding: find common factor and remove

6x+9y=3(2x+3y)6x + 9y = 3(2x + 3y)

Linear Equations​

ax+b=cax + b = c

Solving: Isolate variable using inverse operations

Example: Solve 2x+3=112x + 3 = 11 2x=82x = 8 x=4x = 4

Forming Equations​

From real-world situations: "5 more than 3 times a number equals 20"

3x+5=203x + 5 = 20

Key Points​

  1. Know number type classifications
  2. Follow order of operations
  3. Master fraction operations
  4. Convert between fractions/decimals/percentages
  5. Simplify algebraic expressions
  6. Solve linear equations

Practice Questions​

  1. Classify numbers: -3, 2.5, 2\sqrt{2}, 73\frac{7}{3}
  2. Calculate: 4+3×2−14 + 3 \times 2 - 1
  3. Simplify: 1824\frac{18}{24}
  4. Express 0.375 as fraction and percentage
  5. Expand: 3(2x+4)−2(x−1)3(2x + 4) - 2(x - 1)
  6. Solve: 5x−2=185x - 2 = 18

Revision Tips​

  • Know all number types
  • Practice BODMAS strictly
  • Master fraction operations
  • Percentage conversions
  • Algebraic simplification
  • Equation solving step-by-step