Maths Principles and Theorems Summarized

In mathematics, there a variety of important principles, laws, and theorems that form the foundation for further study in higher-level math. These concepts span number theory, algebra, geometry, probability, and more. Here’s an outline of the key principles, laws, and theorems that we should know:

1. Basic Arithmetic and Number Theory

Properties of Operations

  • Commutative Property:
    • Addition: a+b=b+aa + b = b + aa+b=b+a
    • Multiplication: a×b=b×aa \times b = b \times aa×b=b×a
  • Associative Property:
    • Addition: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c)(a+b)+c=a+(b+c)
    • Multiplication: (a×b)×c=a×(b×c)(a \times b) \times c = a \times (b \times c)(a×b)×c=a×(b×c)
  • Distributive Property: a(b+c)=ab+aca(b + c) = ab + aca(b+c)=ab+ac
  • Identity Property:
    • Addition: a+0=aa + 0 = aa+0=a
    • Multiplication: a×1=aa \times 1 = aa×1=a
  • Inverse Property:
    • Addition: a+(−a)=0a + (-a) = 0a+(−a)=0
    • Multiplication: a×1a=1a \times \frac{1}{a} = 1a×a1​=1 (for a≠0a \neq 0a=0)

Divisibility Rules:

  • Rules that determine whether a number is divisible by another without performing division (e.g., a number is divisible by 3 if the sum of its digits is divisible by 3).

Greatest Common Factor (GCF) and Least Common Multiple (LCM):

  • GCF: The largest number that divides two or more numbers.
  • LCM: The smallest number that is a multiple of two or more numbers.

2. Algebraic Principles

Properties of Equality

  • Addition/Subtraction Property of Equality: If a=ba = ba=b, then a+c=b+ca + c = b + ca+c=b+c and a−c=b−ca – c = b – ca−c=b−c.
  • Multiplication/Division Property of Equality: If a=ba = ba=b, then a×c=b×ca \times c = b \times ca×c=b×c and a÷c=b÷ca \div c = b \div ca÷c=b÷c (for c≠0c \neq 0c=0).

Solving Linear Equations

  • One-Step and Multi-Step Equations: Use inverse operations to isolate the variable and solve for its value.
  • Equations with Variables on Both Sides: Combine like terms and apply the properties of equality to solve.
  • Proportionality: If two ratios are equal, then the cross-products of the ratios are equal, often used in solving proportions.

Exponent Rules

  • Product of Powers: am×an=am+na^m \times a^n = a^{m+n}am×an=am+n
  • Power of a Power: (am)n=am×n(a^m)^n = a^{m \times n}(am)n=am×n
  • Quotient of Powers: aman=am−n\frac{a^m}{a^n} = a^{m-n}anam​=am−n (for a≠0a \neq 0a=0)
  • Zero Exponent Rule: a0=1a^0 = 1a0=1 (for a≠0a \neq 0a=0)
  • Negative Exponent Rule: a−n=1ana^{-n} = \frac{1}{a^n}a−n=an1​ (for a≠0a \neq 0a=0)

3. Geometry

Basic Geometric Principles

  • Point, Line, Plane: Basic undefined terms in geometry.
  • Types of Angles:
    • Acute Angle: Less than 90°.
    • Right Angle: Exactly 90°.
    • Obtuse Angle: More than 90° but less than 180°.
    • Straight Angle: Exactly 180°.
  • Congruent Figures: Figures that have the same size and shape.
  • Similar Figures: Figures that have the same shape but may have different sizes (proportional sides and equal angles).

Pythagorean Theorem (For Right Triangles):

  • Theorem: a2+b2=c2a^2 + b^2 = c^2a2+b2=c2, where ccc is the hypotenuse, and aaa and bbb are the legs of the triangle.
  • Used to calculate the length of a side in a right triangle when the other two sides are known.

Triangle Properties

  • Sum of Interior Angles: The sum of the interior angles of any triangle is 180°.
  • Types of Triangles:
    • Equilateral Triangle: All sides and angles are equal.
    • Isosceles Triangle: Two sides are equal, and the angles opposite those sides are equal.
    • Scalene Triangle: All sides and angles are different.

Parallel and Perpendicular Lines

  • Parallel Lines: Two lines that never intersect and have the same slope.
  • Perpendicular Lines: Two lines that intersect at a right angle (90°).

Area and Perimeter Formulas:

  • Rectangle:
    • Area = Length×Width\text{Length} \times \text{Width}Length×Width
    • Perimeter = 2(Length+Width)2(\text{Length} + \text{Width})2(Length+Width)
  • Triangle:
    • Area = 12×Base×Height\frac{1}{2} \times \text{Base} \times \text{Height}21​×Base×Height
    • Perimeter = Sum of the lengths of the sides.
  • Circle:
    • Area = πr2\pi r^2πr2
    • Circumference = 2πr2\pi r2πr, where rrr is the radius.
  • Pythagorean Theorem: As stated earlier, a2+b2=c2a^2 + b^2 = c^2a2+b2=c2.

4. Ratios, Proportions, and Percentages

Proportional Relationships:

  • If two quantities are proportional, they increase or decrease at the same rate.
  • Constant of Proportionality: The ratio between two proportional quantities (e.g., y=kxy = kxy=kx, where kkk is the constant).

Percentage Calculations:

  • Finding Percentages: Percent=PartWhole×100\text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100Percent=WholePart​×100
  • Finding the Part or Whole: Rearranging the percentage formula can be used to find either the part or the whole when the percentage is known.

5. Probability and Statistics

Basic Probability:

  • Probability of an Event: P(Event)=Number of favorable outcomesTotal number of possible outcomesP(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}P(Event)=Total number of possible outcomesNumber of favorable outcomes​
  • Complement Rule: The probability that an event does not occur is 1−P(Event)1 – P(\text{Event})1−P(Event).

Theoretical vs. Experimental Probability:

  • Theoretical Probability: Based on the possible outcomes.
  • Experimental Probability: Based on the outcomes of an actual experiment or trial.

Mean, Median, Mode, and Range:

  • Mean (Average): Mean=Sum of all data pointsNumber of data points\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}}Mean=Number of data pointsSum of all data points​
  • Median: The middle value in an ordered data set.
  • Mode: The value that appears most frequently in a data set.
  • Range: The difference between the highest and lowest values in a data set.

6. Functions and Graphs

Linear Functions:

  • Slope-Intercept Form: y=mx+by = mx + by=mx+b, where mmm is the slope and bbb is the y-intercept.
  • Slope: The steepness of a line, calculated as m=riserunm = \frac{\text{rise}}{\text{run}}m=runrise​ or m=y2−y1x2−x1m = \frac{y_2 – y_1}{x_2 – x_1}m=x2​−x1​y2​−y1​​.
  • Graphing Linear Functions: Plot points on the coordinate plane and draw a line through them.

Proportional Relationships:

  • A linear relationship between two variables where the graph passes through the origin (0, 0).
  • Represented by the equation y=kxy = kxy=kx, where kkk is the constant of proportionality.

7. Inequalities

  • Solving and Graphing Inequalities: Similar to solving equations but with special rules for reversing the inequality sign when multiplying or dividing by a negative number.
  • Graphing Inequalities: Use a number line or coordinate plane to represent solutions.

8. Set Theory

  • Union: The set of all elements that are in either set AAA, set BBB, or both (denoted A∪BA \cup BA∪B).
  • Intersection: The set of elements that are in both set AAA and set BBB (denoted A∩BA \cap BA∩B).
  • Complement: The set of elements not in set AAA (denoted AcA^cAc).

9. Theorems and Laws in Mathematics

  • Pythagorean Theorem: As stated in the geometry section, this theorem relates the sides of a right triangle.
  • Fundamental Theorem of Arithmetic: Every integer greater than 1 can be uniquely factored into prime numbers.
  • Euclid’s Theorem: There are infinitely many prime numbers.
  • Basic Theorem of Proportions: If ab=cd\frac{a}{b} = \frac{c}{d}ba​=dc​, then ad=bcad = bcad=bc.