Foundations of Mathematics

The real number system brings together the number types used for counting, measuring, and comparing quantities. It includes natural numbers, integers, rational numbers, and irrational numbers. Each set adds new kinds of numbers to the same system.

Natural numbers are the basic counting numbers: 1, 2, 3, and so on. Integers extend that set by adding zero and negative whole numbers, such as ..., -3, -2, -1, 0, 1, 2, .... Rational numbers are numbers that can be written as a ratio of two...

Real numbers are the values that can be placed on a continuous number line. The system grew from counting numbers used for enumeration. It later expanded to include zero, which shows the absence of quantity, and negative numbers, which are opposites of counting numbers.

The real number system also includes fractions and decimals that represent parts of a whole or equal divisions. When written in decimal form, these numbers may end or repeat in a regular pattern. Together, these values describe...

Exponents show repeated multiplication in a short form. They are important in algebra and in many other math topics, including scientific computation, scaling laws, and dimensional analysis. For a nonzero real number a and an integer n, a^n means a multiplied by itself n times.

Exponent rules help simplify expressions with powers. These properties make it easier to work with exponential expressions in both symbolic and numerical form. They also support common calculations that use exponents in...

Radicals show the root of a number and the exponent needed to get that value. A radical expression has two main parts. The radicand is the number inside the root symbol, and the index tells which root to take. The notation n√a means the principal nth root of a.

The index changes the type of root. When n equals 2, the expression is a square root. When n equals 3, it is a cube root. The sign of the radicand also matters. If the index is even, a negative radicand does not give a real result. If...

Algebraic expressions use variables, constants, and operations to show mathematical relationships. They help describe patterns and solve problems across many areas of mathematics. Knowing the parts of an expression makes it easier to simplify and work with it correctly.

Each algebraic expression has separate pieces that work together. A coefficient is the number attached to a variable, and an exponent shows repeated multiplication. Some expressions have a single part, while others include...

Rational expressions are algebraic fractions with polynomials in the numerator and denominator. They follow the same basic rules as numerical fractions, but variables make them more delicate to work with. A key step is finding values that make the expression undefined, usually because of division by zero or an undefined radical.

To determine the domain, start by identifying the values excluded from the denominator. The domain includes all real numbers except those that make the denominator...

Linear equations are a basic algebra tool for modeling constant rates and everyday relationships. They use constants and a single variable, and each term is either a constant or a product of a constant and that variable. When graphed on a Cartesian coordinate plane, they form a straight line that shows a constant rate of change between two quantities.

A linear equation in one variable often has the form ax + b = c, where a, b, and c are constants and x is the variable. Solving the equation...

Quadratic equations are algebraic equations with a variable squared, a first-power term, and a constant term. They are set equal to zero. In high school math, they often help model area, motion, and optimization problems.

The general form of a quadratic equation uses real numbers a, b, and c, with a not equal to zero. The nonzero a value keeps the squared term in the equation. One common way to solve a quadratic is to rewrite it as a product of two linear expressions.

In factored form, the...

Complex numbers extend the real number system so equations with negative square roots can still be solved. The real numbers cannot represent the square root of a negative number. That limits some quadratic equations, especially those with negative discriminants. To solve this problem, mathematicians introduced the imaginary unit i, where i = √(-1).

A complex number is written as x + yi, where x and y are real numbers. In this form, x is the real part and y is the imaginary part. The key rule...

Quadratic equations can have real or complex roots, depending on the discriminant. For an equation in the form ax² + bx + c = 0, the coefficients a, b, and c are the numbers that multiply x², x, and the constant term. The discriminant, b² − 4ac, tells us what kind of solutions to expect.

When the discriminant is negative, the equation has no real number solutions. In that case, complex numbers provide a way to continue solving the problem. The imaginary unit i is defined by i = √−1, so a...

Radical equations are equations with a variable inside a radical, most often a square root or cube root. These equations appear in science, engineering, and real-world measurement problems that involve nonlinear relationships. Solving them starts by isolating the radical expression. Then the radical is removed by raising both sides to the power that matches the radical’s index.

This process can create extraneous solutions, which are values that work in the transformed equation but not in the...

Inequalities show how two values compare when they are not equal. They use symbols such as , ≤, and ≥. These statements often describe a range of possible answers instead of one exact value.

Interval notation is a shorter way to write these solution sets. An open interval, written as (a, b), leaves out the endpoints. A closed interval, written as [a, b], includes both endpoints. Half-open intervals, such as (a, b] and [a, b), include only one endpoint.

Intervals can also extend forever in one...

Absolute value inequalities show how far a number can be from zero or from a target value. Absolute value means distance on the number line, so the sign of the number does not matter. In inequalities, this distance helps define a valid range of values for a variable.

An inequality of the form |x| ≤ a, where a ≥ 0, includes every x between −a and a. This can also be written as a compound inequality. On a number line, the solution is shown as a solid segment from −a to a with closed circles at...

Mathematical modeling turns a real-life situation into an equation that can be solved. It starts with defining the problem, choosing measurable quantities, and assigning variables to them. Then a matching model is selected and the equation is solved for the unknown value.

This process is especially useful in finance, where models help evaluate investments, loans, and repayment plans. One common example is finding the fixed monthly payment on a loan. That calculation uses the standard annuity...