Basic Math Concepts

Numbers

• Types of numbers:
  • Natural numbers: 1, 2, 3, ...
  • Whole numbers: 0, 1, 2, ...
  • Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
  • Rational numbers: fractions (e.g. 1/2, 3/4)
  • Irrational numbers: non-repeating decimals (e.g. π, e)
  • Real numbers: rational and irrational numbers

Algebra

• Equations:
  • Linear equations: ax + by = c (e.g. 2x + 3y = 7)
  • Quadratic equations: ax^2 + bx + c = 0 (e.g. x^2 + 4x + 4 = 0)
• Graphs:
  • Linear graphs: straight lines (e.g. y = 2x - 3)
  • Quadratic graphs: parabolas (e.g. y = x^2 + 2x + 1)

Geometry

• Points, Lines, and Planes:
  • Points: locations in space (e.g. A, B, C)
  • Lines: sets of points extending infinitely (e.g. AB, CD)
  • Planes: flat surfaces (e.g. ABC, DEF)
• Angles and Shapes:
  • Angles: formed by two lines or planes (e.g. ∠AOB, ∠ABC)
  • Triangles: three-sided shapes (e.g. ABC, DEF)
  • Quadrilaterals: four-sided shapes (e.g. ABCD, EFGH)

Calculus

• Limits:
  • Concept of a limit: a value approached as x approaches a certain point
• Derivatives:
  • Rate of change of a function with respect to x
  • Geometric interpretation: slope of the tangent line
• Integrals:
  • Area under a curve
  • Accumulation of a quantity over an interval

Numbers

• Natural numbers are positive integers starting from 1, such as 1, 2, 3, and so on.
• Whole numbers include natural numbers and 0, making them 0, 1, 2, 3, and so on.
• Integers are whole numbers including negative numbers, resulting in ..., -3, -2, -1, 0, 1, 2, 3, and so on.
• Rational numbers are fractions, such as 1/2, 3/4, and can be expressed as a finite decimal or a ratio of integers.
• Irrational numbers are non-repeating decimals, such as π and e, which cannot be expressed as a finite decimal or a ratio of integers.
• Real numbers comprise both rational and irrational numbers.

Algebra

• Linear equations are of the form ax + by = c, where a, b, and c are constants, and x and y are variables, such as 2x + 3y = 7.
• Quadratic equations are of the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable, such as x^2 + 4x + 4 = 0.
• Linear graphs are straight lines, represented by the equation y = 2x - 3, where m is the slope and b is the y-intercept.
• Quadratic graphs are parabolas, represented by the equation y = x^2 + 2x + 1, where a determines the shape of the parabola.

Geometry

• Points are locations in space, represented by capital letters, such as A, B, C.
• Lines are sets of points extending infinitely in two directions, represented by two capital letters, such as AB, CD.
• Planes are flat surfaces, represented by three capital letters, such as ABC, DEF.
• Angles are formed by two lines or planes, represented by the angle symbol, such as ∠AOB, ∠ABC.
• Triangles are three-sided shapes, represented by three capital letters, such as ABC, DEF.
• Quadrilaterals are four-sided shapes, represented by four capital letters, such as ABCD, EFGH.

Calculus

• Limits are a value approached as x approaches a certain point, used to define continuity and differentiability of functions.
• Derivatives measure the rate of change of a function with respect to x, and can be interpreted geometrically as the slope of the tangent line.
• Integrals calculate the area under a curve or the accumulation of a quantity over an interval, and are used to solve problems in physics, engineering, and economics.

Rational Numbers: Addition and Subtraction

Adding Rational Numbers

• To add two rational numbers, find a common denominator (LCD) for both fractions.
• Convert both fractions to have the LCD, then add the numerators (numbers on top) and keep the same denominator.
• Simplify the resulting fraction, if possible, to ensure the answer is in its simplest form.

Subtracting Rational Numbers

• To subtract one rational number from another, follow the same steps as adding rational numbers.
• Find a common denominator (LCD) for both fractions, convert both fractions to have the LCD, and then subtract the numerators.
• Simplify the resulting fraction, if possible, to ensure the answer is in its simplest form.

Key Concepts

• The denominators must be the same when adding or subtracting rational numbers.
• The least common multiple (LCM) of the denominators can be used to find the common denominator.
• Simplifying the resulting fraction is important to ensure the answer is in its simplest form.

Examples

• Adding 1/4 and 1/6 involves finding the LCD (12), converting fractions to 3/12 and 2/12, adding numerators to get 5/12, and simplifying to 5/12.
• Subtracting 2/3 from 1/4 involves finding the LCD (12), converting fractions to 8/12 and 3/12, subtracting numerators to get 5/12, and simplifying to 5/12.