Mathematics Key Concepts Summary

Basic Mathematical Concepts

Natural Numbers
  • Consecutive Numbers: Numbers that follow each other in order. Example: 4, 5
  • Even Numbers: Form 2k2k (e.g., 0, 2, 4, 6, 8, …)
  • Odd Numbers: Form 2k−12k - 1 (e.g., 1, 3, 5, 7, …)
Operations and Properties
  • Arithmetic Operations: Addition, subtraction, etc.
  • Gauss's Formula: Used for summing the first nn natural numbers: Sn=n(n+1)2S_n = \frac{n(n+1)}{2}
  • Perfect Squares: A perfect square can have last digit 0, 1, 4, 5, 6, or 9. Example: 42=164^2 = 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., x2−9=(x−3)(x+3)x^2 - 9 = (x - 3)(x + 3)).
  • Absolute Value: ∣x∣|x| represents the distance from zero.
Geometry
  • Angles:
    • Complementary Angles: Sum equals 90ext°90^ ext{°} (e.g., 30ext°30^ ext{°} and 60ext°60^ ext{°}).
    • Supplementary Angles: Sum equals 180ext°180^ ext{°} (e.g., 110ext°110^ ext{°} and 70ext°70^ ext{°}).
  • Triangles:
    • The sum of angles in any triangle is 180ext°180^ ext{°}.
    • Isosceles triangles have two equal angles.
Theorems
  • Pythagorean Theorem: In a right triangle, a2+b2=c2a^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+bf(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.