Math

Which Math Calculator Do You Need?

Math calculations span from basic arithmetic through advanced statistics and number theory. Whether you're a student checking algebra homework, a teacher preparing problems, or a professional running statistical analysis, these tools handle the calculations so you can focus on understanding the results — not grinding through arithmetic.

Arithmetic and Number Theory

Number theory operations form the foundation of higher mathematics and appear constantly in practical computing. The greatest common divisor (GCD) of two integers is the largest number that divides both without remainder — computed efficiently using the Euclidean algorithm: GCD(48, 18) = GCD(18, 12) = GCD(12, 6) = 6. The least common multiple (LCM) is LCM(a, b) = (a × b) / GCD(a, b).

Fraction arithmetic follows strict rules: to add a/b + c/d, find a common denominator (typically LCM(b, d)), convert each fraction, then add numerators. To multiply a/b × c/d, multiply numerators and denominators directly: (ac)/(bd), then simplify by dividing by GCD(ac, bd). Prime factorization decomposes any integer into its prime factors — for example, 360 = 2³ × 3² × 5. Knowing the prime factorization makes GCD, LCM, and divisibility problems straightforward. The Fraction Calculator handles all four operations with step-by-step explanations, and the Prime Factorization Calculator shows the complete factor tree.

Algebra and Equation Solving

Algebra problems range from simple linear equations to complex polynomial systems. A linear equation ax + b = c solves directly: x = (c - b) / a. A quadratic equation ax² + bx + c = 0 has solutions x = (-b ± √(b² - 4ac)) / (2a). The discriminant b² - 4ac determines the nature of roots: positive means two real solutions, zero means one repeated solution, negative means two complex conjugates.

Systems of linear equations can be solved by substitution, elimination, or matrix methods. For a 2×2 system with equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂, Cramer's rule gives x = (c₁b₂ - c₂b₁) / (a₁b₂ - a₂b₁) and y = (a₁c₂ - a₂c₁) / (a₁b₂ - a₂b₁), provided the determinant a₁b₂ - a₂b₁ ≠ 0. Polynomial long division is the standard method for dividing polynomials — the Quadratic Formula Calculator shows all steps including discriminant analysis, and the Equation Solver handles linear, quadratic, and higher-degree polynomials.

Geometry and Shapes

Geometry calculations cover areas, perimeters, volumes, and surface areas of standard shapes. Key area formulas: circle = πr², triangle = ½bh, rectangle = lw, trapezoid = ½(a+b)h, regular polygon = ½ × perimeter × apothem. Volume formulas: sphere = (4/3)πr³, cylinder = πr²h, cone = (⅓)πr²h, rectangular prism = lwh, pyramid = (⅓) × base area × height.

The Pythagorean theorem (a² + b² = c²) connects the sides of a right triangle. For any triangle, the law of cosines (c² = a² + b² - 2ab cos C) generalizes the Pythagorean theorem to non-right triangles. The law of sines (a/sin A = b/sin B = c/sin C) handles the ambiguous case and angle-finding problems. Arc length of a circle sector is s = rθ (where θ is in radians), and sector area is A = ½r²θ. The Circle Calculator computes area, circumference, arc length, and sector area from any input, while the Triangle Calculator solves all triangle types from any three given values.

Statistics and Probability

Descriptive statistics summarize datasets through measures of center and spread. The mean (arithmetic average) is the sum divided by count. The median is the middle value when sorted — for even-count datasets, average the two middle values. The mode is the most frequent value. Variance is the average squared deviation from the mean: σ² = Σ(xᵢ - μ)² / N for population, s² = Σ(xᵢ - x̄)² / (n-1) for sample. Standard deviation is the square root of variance.

Probability calculations use counting principles and distribution functions. For independent events, P(A and B) = P(A) × P(B). For mutually exclusive events, P(A or B) = P(A) + P(B). The binomial distribution models k successes in n trials: P(X = k) = C(n,k) × pᵏ × (1-p)^(n-k), where C(n,k) = n! / (k! × (n-k)!). Normal distribution probabilities require z-scores: z = (x - μ) / σ. Hypothesis testing involves t-statistics, p-values, and significance levels — the Statistics Calculator computes all descriptive stats from raw data, and the Probability Calculator covers binomial, normal, and Poisson distributions.

How to Pick the Right Starting Point

If you're doing algebra homework, the quadratic formula calculator and equation solver handle 80% of common problems. For geometry, identify the shape first (circle, triangle, rectangle, polygon) and use the corresponding shape calculator — all include step-by-step working. For statistics coursework, the statistics calculator handles descriptive stats in one step, while the probability distribution tools cover hypothesis testing.

For number theory problems (GCDs, LCMs, modular arithmetic), the arithmetic tools give step-by-step Euclidean algorithm traces that are useful for understanding the method, not just getting the answer. All tools support both decimal and fraction input where applicable, and most can display answers in multiple forms (decimal, fraction, exact radical). All calculations run locally in your browser — no data is sent anywhere and no account is required.