Mathematics Key Concepts Summary

Basic Mathematical Concepts

Natural Numbers
  • Consecutive Numbers: Numbers that follow each other in order. Example: 4, 5
  • Even Numbers: Form $2k$ (e.g., 0, 2, 4, 6, 8, …)
  • Odd Numbers: Form $2k - 1$ (e.g., 1, 3, 5, 7, …)
Operations and Properties
  • Arithmetic Operations: Addition, subtraction, etc.
  • Gauss's Formula: Used for summing the first $n$ natural numbers: $S_n = rac{n(n+1)}{2}$
  • Perfect Squares: A perfect square can have last digit 0, 1, 4, 5, 6, or 9. Example: $4^2 = 16$.
  • Divisibility Rules:
    • Divisible by 2: Last digit is even.
    • Divisible by 3: Sum of digits divisible by 3.
    • Divisible by 5: Last digit is 0 or 5.
Fractions
  • Types:
    • Proper: Numerator < Denominator (e.g., 34\frac{3}{4})
    • Improper: Numerator ≥ Denominator (e.g., 53\frac{5}{3})
    • Equivalent: Two fractions that simplify to the same value (e.g., 24\frac{2}{4} = 12\frac{1}{2}).
Algebraic Expressions
  • Factoring: Breaking down expressions into simpler factors (e.g., $x^2 - 9 = (x - 3)(x + 3)$).
  • Absolute Value: $|x|$ represents the distance from zero.
Geometry
  • Angles:
    • Complementary Angles: Sum equals $90^ ext{°}$ (e.g., $30^ ext{°}$ and $60^ ext{°}$).
    • Supplementary Angles: Sum equals $180^ ext{°}$ (e.g., $110^ ext{°}$ and $70^ ext{°}$).
  • Triangles:
    • The sum of angles in any triangle is $180^ ext{°}$.
    • Isosceles triangles have two equal angles.
Theorems
  • Pythagorean Theorem: In a right triangle, $a^2 + b^2 = c^2$.
  • Thales's Theorem: Points on a circumference subtended by the same arc are equal.
Units of Measurement
  • Length: 1 m = 100 cm = 1000 mm.
  • Area: Area of rectangle = length × width.
  • Volume: Volume of cylinder V=πr2hV = \pi r^2 h.
Functions
  • Definition: A relation from set A to B where each element in A corresponds to exactly one element in B.
  • Linear Function: Form $f(x) = mx + b$ likes to form a straight line.
Probability
  • The likelihood of an event occurring; calculated as the number of favorable outcomes divided by the total number of outcomes.