Arithmetic Operations and Number Systems

Study Notes

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.

Description

Test your understanding of the fundamental arithmetic operations including addition, subtraction, multiplication, and division. Additionally, explore various number systems such as natural, whole, integers, rational, and irrational numbers. This quiz will help reinforce your knowledge in basic math concepts.