Arithmetic Operations and Number Systems

Arithmetic Operations

• Addition involves combining two or more numbers to find their total. The result is called the sum.
• Subtraction involves finding the difference between two numbers. The result is the difference.
• Multiplication involves repeated addition of the same number. The result is the product. Factors multiplied together produce a product.
• Division involves splitting a number into equal parts. The result is the quotient. A dividend divided by a divisor produces a quotient.

Number Systems

• Natural numbers are the positive integers (1, 2, 3,...) often used for counting.
• Whole numbers include zero and all natural numbers (0, 1, 2, 3,...).
• Integers include zero, positive and negative whole numbers (... -3, -2, -1, 0, 1, 2, 3...).
• Rational numbers can be expressed as fractions (a/b) where 'a' and 'b' are integers, and 'b' is not zero. Examples include 1/2, 3/4, -2/5, and any terminating or repeating decimal.
• Irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal representations. Examples include √2 and π.
• Real numbers encompass all rational and irrational numbers. They can be plotted on a number line.
• Complex numbers extend the real number system to include the imaginary unit 'i', where i² = -1. They have the form a + bi, where 'a' and 'b' are real numbers.

Basic Algebra

• Variables represent unknown quantities.
• Expressions combine numbers, variables, and operators (+,-, x, /).
• Equations state that two expressions are equal.
• Solving an equation means finding the value(s) of the variable(s) that make the equation true.
• Inequalities describe relationships where one expression is greater than, less than, or not equal to another.
• Order of operations (PEMDAS/BODMAS) dictates the sequence for evaluating expressions: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (left to right), Addition and Subtraction (left to right).

Geometry

• Basic shapes include points, lines, angles, triangles, quadrilaterals, and circles.
• Properties of shapes describe their characteristics (e.g. angles, sides).
• Perimeter is the distance around a two-dimensional shape.
• Area is the space enclosed within a two-dimensional shape.
• Volume is the space occupied by a three-dimensional shape.
• Pythagorean theorem relates the sides of a right-angled triangle (a² + b² = c²).
• Congruent figures have the same shape and size.
• Similar figures have the same shape but not necessarily the same size.

Measurement

• Units of measurement are used to quantify physical quantities.
• Common units include meters, centimetres, kilograms, grams, litres, seconds, minutes, hours.
• Converting between units involves using conversion factors.
• Significant figures indicate the precision of a measurement.
• Absolute error calculates the difference between the measured value and the true value.
• Relative error expresses the absolute error as a percentage of the true value.

Data Handling

• Data collection involves gathering information about a topic.
• Data representation includes various methods such as charts, graphs (bar charts, histograms, pie charts, line graphs, scatter diagrams), tables to display data.
• Measures of central tendency (mean, median, mode) summarize data.
• Measures of dispersion (range, standard deviation) describe the spread of data.
• Probability quantifies the likelihood of an event occurring.
• Statistical inference involves drawing conclusions about a population based on a sample.

Set Theory

• Sets are collections of objects.
• Set operations include union, intersection, and complement.
• Venn diagrams are visual representations of sets.

Functions

• Functions are relationships between inputs and outputs.
• Notation for functions typically uses f(x) to represent the output for an input x.
• Linear functions have a constant rate of change and produce a straight line graph.
• Quadratic functions involve squares of variables and produce parabolic graphs.
• Other function types include exponential, logarithmic, trigonometric, and more complex models.

Ratio and Proportion

• Ratio compares two quantities.
• Proportion relates equal ratios.
• Similar shapes have corresponding sides in the same ratio.

Percentages

• Percentages are ratios that compare a part to a whole, using 100 as the reference.
• Percentage increase/decrease calculations involve comparing a change from an original value.