Number Systems and Arithmetic Operations

Number Systems

• Natural numbers are the counting numbers (1, 2, 3,...).
• Whole numbers include zero and all natural numbers (0, 1, 2, 3,...).
• Integers are whole numbers and their negative counterparts (..., -3, -2, -1, 0, 1, 2, 3,...).
• Rational numbers can be expressed as a fraction p/q, where p and q are integers and q is not zero. Examples include fractions, terminating decimals (e.g., 0.5), and repeating decimals (e.g., 0.333...).
• Irrational numbers cannot be expressed as a fraction of two integers. Examples include √2 and π.
• Real numbers encompass all rational and irrational numbers.
• Imaginary numbers are numbers containing the imaginary unit 'i', where i² = -1.
• Complex numbers are numbers in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit.

Arithmetic Operations

• Addition (+) combines two or more numbers to find a sum.
• Subtraction (-) finds the difference between two numbers.
• Multiplication (× or *) combines numbers repeatedly.
• Division (÷ or /) separates a number into equal parts.
• Order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Algebra

• Variables represent unknown quantities.
• Equations state that two expressions are equal.
• Inequalities show that one expression is greater than or less than another.
• Linear equations have the form y = mx + b, where 'm' is the slope and 'b' is the y-intercept.
• Quadratic equations have the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants.
• Polynomials are expressions consisting of variables and coefficients.
• Factoring is expressing a polynomial as a product of simpler polynomials.
• Solving equations involves finding the value(s) of the variable(s) that make the equation true.
• Systems of equations involve multiple equations with multiple variables.

Geometry

• Points, lines, and planes are fundamental geometric objects.
• Angles are formed by two rays sharing a common endpoint.
• Polygons are closed figures with straight sides.
• Triangles are polygons with three sides.
• Quadrilaterals are polygons with four sides.
• Circles are sets of points equidistant from a center point.
• Perimeter is the distance around a two-dimensional shape.
• Area is the space enclosed by a two-dimensional shape.
• Volume is the amount of space occupied by a three-dimensional object.
• Congruent figures have the same shape and size.
• Similar figures have the same shape but not necessarily the same size.

Trigonometry

• Trigonometry deals with relationships between angles and sides of triangles.
• Trigonometric ratios (sine, cosine, tangent) relate angles to ratios of sides.
• Trigonometric identities are equations that are true for all values of the variables.
• Applications of trigonometry include solving triangles, navigation, and surveying.
• Understanding of right angled triangles is important.

Calculus

• Calculus deals with continuous change.
• Limits determine the behavior of a function as its input approaches a specific value.
• Derivatives measure the rate of change of a function.
• Integrals calculate accumulated change.
• Applications include finding slopes, areas, volumes, and rates of growth and decay.

Statistics

• Statistics involves collecting, organizing, analyzing, interpreting, and presenting data.
• Measures of central tendency (mean, median, mode) describe the center of a data set.
• Measures of dispersion (variance, standard deviation) describe the spread of a data set.
• Probability measures the likelihood of events occurring.
• Data visualization tools (graphs, charts) help in understanding data patterns.

Discrete Mathematics

• Discrete mathematics focuses on countable objects.
• Topics include set theory, logic, graph theory, combinatorics, and discrete probability.
• Set theory deals with collections of objects.
• Combinatorics deals with counting and arranging objects.
• Graph theory involves studying relationships between objects using graphs.
• Propositional logic deals with statements and their truth values.