Expression Calculator Glossary: Every Maths Term Defined

A value passed into a function. In sqrt(25), the argument is 25. In log(1000, 10), the two arguments are 1000 and 10. Some functions take one argument; others take several.

See also: Function, Parameter

The distance of a number from zero, always non-negative. Written |x| in maths; in an expression calculator, use abs(x). abs(-7) and abs(7) both return 7.

Has two meanings. (1) In a power expression like 2^8, the base is the number being raised to a power (here, 2). (2) In a logarithm like log₁₀(x), the base is the number whose powers are counted (here, 10).

See also: Exponent, Logarithm

The universally agreed order in which operations are evaluated. BODMAS: Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction. PEMDAS is the American equivalent. Every expression calculator follows this automatically.

See also: Operator Precedence, Parentheses

A named value that never changes. Common constants in expression calculators include pi (≈ 3.14159…), e (≈ 2.71828…), and phi (the golden ratio, ≈ 1.61803…). You can use the name directly in any expression.

The numerical factor multiplying a variable or expression. In 3x², the coefficient is 3. In -5sin(x), the coefficient is -5. Coefficients are evaluated as plain numbers before the rest of the expression is computed.

A number expressed in base 10 with a decimal point separating the whole part from the fractional part. Expression calculators work in decimals by default. Very large or small results may be shown in scientific notation (e.g. 1.5e-8).

The number indicating how many times a base is multiplied by itself. In 2^8, the exponent is 8. Also called a power or index. In expression calculators, the ^ symbol is used to write exponents. Fractional exponents express roots: x^(1/2) is the square root of x.

See also: Base

Any combination of numbers, operators, variables and functions that can be evaluated to produce a single value. Unlike an equation, an expression has no equals sign — it simply represents a value waiting to be computed. 3 + 4 * 2 is an expression; its value is 11.

The way computers store decimal numbers — as a sign, a significand and an exponent. Most expression calculators use 64-bit floating-point, which gives about 15–17 significant decimal digits of precision. This is why 0.1 + 0.2 may return 0.30000000000000004.

A whole number with no fractional part: …−3, −2, −1, 0, 1, 2, 3… Expression calculators treat all numbers as decimals internally, but functions like floor(), ceil() and round() return integer values.

A real number whose decimal expansion never terminates and never repeats. π, e, and √2 are classic examples. Expression calculators store these as floating-point approximations, but symbolic calculators can represent them exactly.

The size of a value, ignoring its sign or direction. For real numbers, the magnitude is the absolute value. For complex numbers, the magnitude (or modulus) is sqrt(a² + b²). Used extensively in physics and signal processing.

A compact way to write very large or small numbers as a decimal between 1 and 10 multiplied by a power of 10. In expression calculators, type 1.5e6 to mean 1,500,000, or 3.2e-9 for 0.0000000032. The calculator will display large results in this form automatically.

When two operators have the same precedence, associativity determines which side is evaluated first. Most operators are left-associative: 8 / 4 / 2 is (8/4)/2 = 1, not 8/(4/2) = 4. The exponent ^ is right-associative: 2^3^2 means 2^(3^2) = 512.

An operator that takes exactly two operands — one on each side. All the common arithmetic operators (+, −, ×, ÷, ^) are binary operators. Contrast with unary operators, which take only one operand.

See also: Unary Operator

The standard way of writing operators between their operands: 3 + 4. This is what people naturally write and what expression calculators accept. Internally, parsers often convert infix to a tree structure before evaluation.

The remainder after integer division. Written mod(a, b) or a % b in many calculators. 17 mod 5 = 2 because 17 = 3×5 + 2. Essential in number theory, cryptography, and cyclic problems like clock arithmetic.

A value that an operator acts upon. In 3 + 4, both 3 and 4 are operands. Operands can be literals (plain numbers), variables, or the results of nested expressions like (2 * 5).

A symbol that specifies an operation to perform on one or more operands. Arithmetic operators include +, -, *, /, and ^. The calculator has rules for which operators are evaluated first (precedence) and in which direction (associativity).

The ranking that determines which operations are performed first when an expression contains multiple operators. Higher precedence operators bind more tightly to their operands. Precedence from highest to lowest: exponentiation ^ → ×÷ → +−. Parentheses override all precedence.

The symbols ( and ) used to group parts of an expression and force a specific evaluation order. Everything inside parentheses is evaluated first, regardless of operator precedence. Parentheses can be nested to any depth.

An operator that takes a single operand. The unary minus -x negates a value; the unary plus +x is a no-op but valid. Functions like sqrt() and abs() are unary in the sense that they take one argument.

A named operation that takes one or more arguments and returns a single value. Expression calculators include built-in functions for maths (sin, cos, sqrt, log…) and you type them as name(argument). The calculator evaluates the argument first, then applies the function to the result.

A function that reverses the effect of another. If sin(x) takes an angle and returns a ratio, then arcsin(x) takes the ratio and returns the angle. Inverse functions are written with the prefix arc (or a) in most calculators: arcsin, arccos, arctan.

The inverse of exponentiation. log_b(x) asks: "to what power must I raise b to get x?" log10(1000) = 3 because 10³ = 1000. ln(x) is the natural logarithm (base e). Logarithms turn multiplication into addition, making them essential in science and engineering.

The named placeholder in a function's definition for the value it will receive. When you call sin(x), x is the parameter — the slot that the argument fills. In use, "argument" and "parameter" are often used interchangeably, though technically parameters belong to the definition, arguments to the call.

An expression or function definition that refers to itself. Factorials are the classic example: n! = n × (n−1)! — the definition of n! uses the definition of factorial again at n−1. Expression calculators with a built-in fact(n) handle this internally using a loop or a recursive algorithm.

The value a function produces after evaluating its argument(s). In an expression, the return value of a function takes the place of the entire function call. In 1 + sqrt(9), sqrt(9) returns 3, so the expression becomes 1 + 3, which returns 4.

A function whose derivative is the given function. The antiderivative of 2x is x² + C (where C is an arbitrary constant). Indefinite integration finds the antiderivative; definite integration evaluates the area between two bounds.

The instantaneous rate of change of a function. Written d/dx f(x) or f′(x). The derivative of x² is 2x — at any point x, the slope of the curve is 2x. Symbolic expression calculators compute derivatives by applying differentiation rules to the expression tree.

The value a function or expression approaches as its input gets arbitrarily close to a point. Written lim(x→a) f(x). Limits are the foundation of calculus — derivatives and integrals are both defined as limits. Expression calculators can evaluate limits symbolically or numerically.

A compact notation for the sum of many terms. Σ(k=1 to n) k means 1 + 2 + 3 + … + n. Expression calculators that support summation let you write this as sum(k, 1, n, k) or similar. Infinite series — where n→∞ — are evaluated as limits.

A value that is either true or false (1 or 0). Named after mathematician George Boole. In expression calculators, comparison operators (>, <, ==) produce boolean results. These are often used inside conditional expressions like if(condition, true_value, false_value).

An expression that evaluates to one value if a condition is true and another if it's false. In expression calculators: if(condition, a, b). The condition is evaluated first; only the matching branch (a or b) is then evaluated. This is how calculators implement decision-making within a single expression.

An operator that combines or modifies boolean values. The three primary logical operators are AND (true only if both operands are true), OR (true if either operand is true), and NOT (inverts a boolean). In expressions: and(a, b), or(a, b), not(a).

An approximate, rule-based evaluation strategy used when an exact algorithm is too costly. Symbolic calculators use heuristics to decide whether to simplify, expand, or factor an expression. For example, a heuristic might try factoring before expansion when the expression looks polynomial.

A formal system for expressing computation using only function abstraction and application. Every computable function can be expressed as a lambda expression. The notation λx. x+1 defines a function that adds 1 to its argument. Modern functional programming languages are essentially elaborated lambda calculi.

A canonical, fully-simplified form of an expression that cannot be reduced further. In lambda calculus, an expression in normal form has no remaining beta-reductions (substitutions) to apply. In algebra, normal forms include expanded polynomial form and factored form — different normal forms for different purposes.

The component of an expression calculator that reads the raw text input and converts it into a structured form (usually an abstract syntax tree) that the evaluator can process. The parser enforces the grammar of valid expressions — it rejects 3 + * 4 as a syntax error while accepting 3 + 4.

A rule that replaces one pattern of expression with a simpler or equivalent one. Symbolic calculators apply thousands of rewrite rules: x + 0 → x, x * 1 → x, sin²(x) + cos²(x) → 1. Simplification is the process of applying rewrite rules until no more apply.

Manipulating mathematical expressions as data structures rather than evaluating them to numbers. A symbolic calculator can differentiate x³·sin(x) or factor x²−1 without substituting any numerical values for x. The expression tree is both input and output, transformed by algebraic rules.

An Abstract Syntax Tree — the internal tree structure a parser produces from an expression. Each node is an operation; its children are the operands or sub-expressions. The tree encodes precedence structurally: deeper nodes are always evaluated before shallower ones. The evaluator traverses the tree bottom-up to compute the result.

The first stage of parsing: splitting the raw input string into typed units called tokens. sin(pi/2) becomes [FUNC:sin][LPAREN][CONST:pi][OP:/][NUM:2][RPAREN]. Each token has a type and a value. The tokeniser (lexer) runs before the parser, which then assembles tokens into a tree.

A named symbol that represents a value which may be unknown or may change. In expressions, variables are typically single letters: x, y, n. A numeric calculator resolves all variables to known values before evaluating. A symbolic calculator keeps variables as symbols and manipulates the expression algebraically.

An expression that has no valid numerical result. Common cases: division by zero (1/0), the square root of a negative number in the reals (sqrt(-1)), and 0^0 (mathematically indeterminate). Calculators return Infinity, NaN, or an error message depending on the case.