Arithmetic and Algebra Basics

Arithmetic

• Arithmetic is the branch of mathematics that deals with the basic operations of numbers: addition, subtraction, multiplication, and division.
• These operations form the foundation for more advanced mathematical concepts.
• Addition involves combining quantities.
• Subtraction involves finding the difference between quantities.
• Multiplication involves repeated addition.
• Division involves distributing a quantity into equal parts.
• Order of operations (PEMDAS/BODMAS) is crucial for evaluating expressions with multiple operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Algebra

• Algebra is a branch of mathematics that uses symbols to represent numbers and quantities in equations and formulas.
• Variables are symbols used to represent unknown values.
• Equations involve an equal sign (=) and represent relationships between variables and numbers.
• Solving equations involves finding the value of the variable that makes the equation true.
• Linear equations relate variables with a degree of 1, forming straight lines when graphed.
• Quadratic equations involve variables raised to the power of 2, forming parabolas when graphed.

Geometry

• Geometry is the branch of mathematics that deals with shapes, sizes, positions, and properties of figures in space.
• Basic shapes include points, lines, angles, triangles, quadrilaterals, circles, and polygons.
• Properties of shapes include length, width, height, area, perimeter, and volume.
• Geometric figures can be 2-dimensional (e.g., polygons, circles) or 3-dimensional (e.g., cubes, spheres).
• Geometric transformations involve modifying shapes (e.g., rotations, reflections, translations).

Calculus

• Calculus involves the study of change and motion.
• Differential calculus deals with rates of change and tangents to curves.
• Derivatives represent instantaneous rates of change.
• Integral calculus deals with accumulation of quantities over intervals.
• Integrals represent areas bounded by curves.
• Applications of calculus are present in numerous fields, including physics, engineering, and economics.

Number Systems

• Number systems constitute a structured way to represent quantities.
• Natural numbers (counting numbers) are 1, 2, 3...
• Whole numbers include natural numbers and 0.
• Integers include zero, positive and negative natural numbers.
• Rational numbers can be expressed as fractions (p/q), where p and q are integers (and q is not zero).
• Irrational numbers cannot be expressed as fractions and their decimal representations are non-repeating and non-terminating (e.g., π, √2).
• Real numbers include both rational and irrational numbers.
• Complex numbers include real and imaginary numbers combined.

Statistics

• Statistics involve collecting, organizing, analyzing, and interpreting data.
• Data can be represented using tables, charts, or graphs.
• Measures of central tendency (mean, median, mode) describe the typical value in a dataset.
• Measures of dispersion (range, standard deviation) show the spread of the data around the central tendency.
• Probability describes the likelihood of an event occurring.
• Statistical inferences are techniques used to draw conclusions about a population based on a sample.