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±

Arithmetic & Basic Operators

The four fundamental operations — the building blocks of every expression

a + bAddition
Addition
Adds two numbers together. Works with integers, decimals, and any valid expression as operands.
3 + 4→7
1.5 + 2.75→4.25
100 + 0.05 * 100→105
a - bSubtraction
Subtraction
Subtracts b from a. Can produce negative results. Use unary minus for negating a single value.
10 - 3→7
5 - 8→-3
-4 - (-2)→-2
a * bMultiplication
Multiplication
Multiplies a by b. Uses the asterisk (*) symbol. Always performed before addition and subtraction unless parentheses override.
6 * 7→42
1.5 * 4→6
2 + 3 * 4→14 (not 20)
Multiplication is performed before addition due to PEMDAS order of operations.
a / bDivision
Division
Divides a by b. Returns a decimal result when the division is not exact. Division by zero is undefined and will produce an error.
20 / 4→5
7 / 2→3.5
1 / 3→0.33333…
Dividing by zero returns an error. Use floor(a/b) for integer division.
a % bModulo
Modulo (Remainder)
Returns the remainder after dividing a by b. Useful for checking divisibility, cycling through values, and working with time.
10 % 3→1
7 % 2→1 (odd)
217 % 60→37 (37 mins)
If result is 0, a is exactly divisible by b. Great for checking if a number is odd or even.
()

Order of Operations & Parentheses

PEMDAS / BODMAS — how the calculator decides what to evaluate first

(expression)Grouping
Parentheses — Force Evaluation Order
Anything inside parentheses is evaluated first, regardless of the natural order of operations. Parentheses can be nested to any depth.
(2 + 3) * 4→20
2 + 3 * 4→14 (no brackets)
(2 + (3 * 4)) / 2→7
When in doubt, add parentheses. They never hurt and prevent many common mistakes.
PEMDAS RuleReference
PEMDAS Evaluation Order
Parentheses → Exponents → Multiplication & Division (left to right) → Addition & Subtraction (left to right). This is applied automatically by ExpressionCalculator.com.
2 + 3^2 * 4→38
10 - 2 * 3 + 1→5
8 / 2 * 4→16 (left to right)
((a + b) * c)Nesting
Nested Parentheses
Parentheses can be nested inside each other. The innermost pair is evaluated first, then working outward. Each opening bracket must have a matching closing bracket.
((3 + 4) * (2 + 1))→21
(10 / (2 + 3)) * 6→12
xⁿ

Powers, Roots & Exponents

Raising numbers to powers and extracting roots of any degree

a ^ bExponent
Power / Exponent
Raises a to the power b. Evaluated right-to-left when chained. Works with negative and fractional exponents.
2^10→1024
3^(-2)→0.1111… (1/9)
16^(0.5)→4 (square root)
sqrt(x)Square Root
Square Root
Returns the positive square root of x. Equivalent to x^(1/2). Only defined for non-negative values of x in real numbers.
sqrt(25)→5
sqrt(2)→1.41421…
sqrt(3^2 + 4^2)→5 (Pythagoras)
x^(1/n)nth Root
Cube Root & nth Root
The nth root of x is written as x^(1/n). The cube root is x^(1/3). Any root can be expressed this way.
27^(1/3)→3 (cube root)
32^(1/5)→2 (5th root)
1000000^(1/6)→10
a * 10^nScientific
Scientific Notation
Express very large or very small numbers using powers of 10. Some calculators support the shorthand 1e6 for 1×10⁶.
6.022 * 10^23→6.022e+23
1.6 * 10^(-19)→1.6e-19
⌊⌉

Rounding, Absolute Value & Remainder

Control precision, handle negative values, and work with integer parts

round(x, n)Round
Round to n Decimal Places
Rounds x to n decimal places using standard rounding rules (round half up). If n is omitted, rounds to the nearest integer.
round(3.14159, 2)→3.14
round(2.675, 2)→2.68
round(1/3, 4)→0.3333
floor(x)Floor
Floor — Round Down
Returns the largest integer less than or equal to x. Always rounds towards negative infinity, even for negative numbers.
floor(4.9)→4
floor(-2.3)→-3
floor(217 / 60)→3 (hours)
ceil(x)Ceiling
Ceiling — Round Up
Returns the smallest integer greater than or equal to x. Always rounds towards positive infinity. Useful for working out how many boxes, pages, or trips are needed.
ceil(4.1)→5
ceil(-2.7)→-2
ceil(50 / 12)→5 (boxes needed)
abs(x)Absolute Value
Absolute Value
Returns the magnitude of x — always a non-negative number. Strips the sign. Geometrically, the distance of x from zero on the number line.
abs(-7)→7
abs(3.5)→3.5
abs(-15 + 8)→7
sign(x)Sign
Sign Function
Returns 1 if x is positive, -1 if x is negative, and 0 if x is zero. Useful for extracting direction without magnitude.
sign(-42)→-1
sign(7.3)→1
sign(0)→0
min(a, b)Minimum
Minimum of Two Values
Returns the smaller of a and b. Can be used with any numeric expressions as arguments.
min(3, 7)→3
min(-5, -2)→-5
max(a, b)Maximum
Maximum of Two Values
Returns the larger of a and b. Combine with min() to clamp a value to a given range: min(max(x, lo), hi).
max(3, 7)→7
max(-5, -2)→-2
∿

Trigonometric Functions

Sine, cosine, tangent — angles, triangles, waves and oscillation. Default unit: radians.

sin(x)Sine
Sine
Returns the sine of angle x (in radians). Represents the ratio of the opposite side to the hypotenuse in a right triangle. Range: -1 to 1.
sin(pi/2)→1
sin(pi/6)→0.5
sin(30 * pi/180)→0.5 (30°)
Use x * pi/180 to convert degrees to radians before passing to sin().
cos(x)Cosine
Cosine
Returns the cosine of angle x (in radians). Represents the ratio of the adjacent side to the hypotenuse. Range: -1 to 1. cos(0) = 1.
cos(0)→1
cos(pi)→-1
cos(pi/3)→0.5
tan(x)Tangent
Tangent
Returns sin(x)/cos(x). Represents the slope of a line at angle x. Undefined at pi/2 + n*pi (vertical tangent). Useful for height/distance problems.
tan(pi/4)→1
50 * tan(32 * pi/180)→31.24m
tan(pi/2) is undefined — the calculator may return a very large number.
sin(x)^2 + cos(x)^2Identity
Pythagorean Identity
sin²(x) + cos²(x) = 1 for every value of x. This is one of the most fundamental identities in all of mathematics, derived from the Pythagorean theorem.
sin(1)^2 + cos(1)^2→1
sin(pi/7)^2 + cos(pi/7)^2→1
deg * pi/180Convert
Degrees to Radians Conversion
Multiply degrees by pi/180 to convert to radians. Multiply radians by 180/pi to convert to degrees. Essential before using trig functions with degree values.
90 * pi/180→1.5708 (π/2)
pi/4 * 180/pi→45°
∠

Inverse Trigonometric Functions

Given a ratio, find the angle — solving triangles and working backwards

asin(x)Arcsin
Arcsine — Inverse Sine
Returns the angle (in radians) whose sine is x. Input must be in [-1, 1]. Output is in [-π/2, π/2]. Multiply result by 180/pi for degrees.
asin(0.5)→0.5236 (π/6)
asin(1) * 180/pi→90°
asin(7/25) * 180/pi→16.26°
acos(x)Arccos
Arccosine — Inverse Cosine
Returns the angle (in radians) whose cosine is x. Input in [-1, 1]. Output is in [0, π]. Useful for finding angles in triangles given two sides.
acos(0.5)→1.0472 (π/3)
acos(-1) * 180/pi→180°
atan(x)Arctan
Arctangent — Inverse Tangent
Returns the angle (in radians) whose tangent is x. Output is in (-π/2, π/2). Useful for finding angles from slopes or gradients.
atan(1) * 180/pi→45°
atan(0)→0
atan2(y, x)Arctan2
Two-Argument Arctangent
Returns the full-circle angle (in radians) from the origin to point (x, y). Output is in (-π, π]. Unlike atan(), this correctly identifies the quadrant of the angle.
atan2(1, 1) * 180/pi→45°
atan2(-1, -1) * 180/pi→-135°
atan2(4, -3) * 180/pi→126.87°
㏒

Logarithmic Functions

Inverses of exponentials — the mathematics of scales, decibels, and pH

log(x) / ln(x)Natural Log
Natural Logarithm (base e)
Returns the power to which e must be raised to produce x. The inverse of e^x. Only defined for x > 0. log(e) = 1 exactly.
log(e)→1
log(1)→0
log(e^5)→5
log10(x)Log base 10
Base-10 Logarithm
Returns the power to which 10 must be raised to produce x. Used in decibel calculations, pH, Richter scale, and any base-10 scale. log10(1000) = 3.
log10(1000)→3
log10(0.01)→-2
10 * log10(50)→16.99 dB
log2(x)Log base 2
Base-2 Logarithm
Returns the power to which 2 must be raised to produce x. Used in computer science (number of bits), information theory, and binary contexts.
log2(8)→3
log2(1024)→10
log(x) / log(b)Change of Base
Logarithm in Any Base
To compute log base b of x, use log(x)/log(b). This is the change-of-base formula. Works with any positive base b ≠ 1.
log(125) / log(5)→3
log(343) / log(7)→3
log(a*b) = log(a)+log(b)Log Laws
Laws of Logarithms
Three key laws: (1) log(a*b) = log(a) + log(b); (2) log(a/b) = log(a) - log(b); (3) log(a^n) = n*log(a). These work for any base.
log10(4) + log10(25)→2
10 * log10(2)→3.0103
eˣ

Exponential Functions

Growth, decay, compound interest, and Euler's remarkable number e

e^xExponential
Exponential Function e^x
Raises e (≈2.71828) to the power x. Models continuous growth and decay. Its own derivative — the only function unchanged by differentiation. e^0 = 1.
e^1→2.71828…
e^0→1
1000 * e^(0.05 * 10)→1648.72
A * e^(k*t)Growth/Decay
Continuous Growth & Decay
A₀ × e^(k×t) models continuous change. k > 0 for growth, k < 0 for decay. A₀ is the initial amount, t is time, k is the rate constant.
500 * e^(0.03 * 24)→1027.2
100 * e^(-0.000121 * 2000)→78.4g
log(2) / kHalf-Life
Half-Life Calculation
The time for a decaying quantity to halve is T = ln(2)/k, where k is the decay rate constant. Used in radioactive decay, pharmacology, and finance.
log(2) / 0.000121→5728 yrs (C-14)
log(2) / 0.0693→10 hours
e^(i*pi) + 1 = 0Euler's Identity
Euler's Identity
The most famous equation in mathematics. Euler's formula e^(ix) = cos(x) + i·sin(x) at x=π gives e^(iπ) = -1. Connects e, i, π, 1, and 0 in one elegant statement.
cos(pi)→-1 (real part)
sin(pi)→≈0 (imag part)
i

Complex Numbers

Imaginary numbers, the complex plane, and polar form

sqrt(-1) = iImaginary
The Imaginary Unit i
i is defined as √(-1), so i² = -1. Complex numbers have the form a + bi where a is the real part and b is the imaginary part. Every polynomial has complex roots.
sqrt(4^2 + 3^2)→5 (modulus of 3+4i)
sqrt(a^2 + b^2)Modulus
Modulus of a + bi
The modulus (magnitude) of complex number a + bi is √(a²+b²). It measures the distance from the origin in the complex plane. Also written |z|.
sqrt(3^2 + 4^2)→5
sqrt(5^2 + 12^2)→13
atan2(b, a)Argument
Argument (Phase Angle)
The argument of complex number a + bi is the angle θ = atan2(b, a) in radians. Together with the modulus, this gives the polar form r·e^(iθ).
atan2(1, 1) * 180/pi→45° (1+i)
atan2(0, -1) * 180/pi→180° (-1+0i)
r^n, n*thetaDe Moivre
De Moivre's Theorem — Powers
To raise a complex number to the power n: raise the modulus to n and multiply the argument by n. The result has modulus r^n and argument n×θ.
sqrt(2)^8→16 (|1+i|⁸)
8 * (pi/4) * 180/pi→360° (full circle)
sqrt(R^2 + X^2)Impedance
AC Circuit Impedance Magnitude
In AC circuits, impedance Z = R + jX where R is resistance and X is reactance. The magnitude is √(R²+X²) and the phase angle is atan2(X, R).
sqrt(120^2 + 135.45^2)→181.7 Ω
σ

Statistics & Probability

Descriptive statistics, z-scores, normal distribution and expected values

sum / countMean
Arithmetic Mean (Average)
Add all values together and divide by the count. Type the full sum expression directly into ExpressionCalculator.com for a transparent, verifiable calculation.
(4+7+13+2+8+11+6) / 7→7.286
(85+92+78+96+88) / 5→87.8
sqrt(sum(x-mean)^2 / n)Std Dev
Standard Deviation
Measures how spread out the data is. Variance is the mean of squared deviations. Standard deviation is the square root of variance.
sqrt(((4-7.286)^2+(7-7.286)^2+(13-7.286)^2)/3)→4.73
(x - mean) / stdZ-Score
Z-Score (Standard Score)
Measures how many standard deviations a value lies from the mean. Z = (x − μ) / σ. A z-score of 2 means the value is 2 standard deviations above average.
(83 - 65) / 12→1.5
(189 - 175) / 7→2.0
v1*w1 + v2*w2 + …Weighted Avg
Weighted Average
Multiply each value by its weight and sum the products. Used for grade calculations, index construction, and portfolio returns.
74*0.40 + 81*0.60→78.2
75*0.20 + 82*0.30 + 91*0.50→84.1
1/sqrt(2*pi) * e^(-x^2/2)Normal PDF
Standard Normal Probability Density
The bell curve density function for a standard normal distribution (mean 0, std 1). Peaks at x = 0 with value 1/√(2π) ≈ 0.3989. The 68-95-99.7 rule applies.
1/sqrt(2*pi) * e^0→0.3989 (peak)
1/sqrt(2*pi) * e^(-2)→0.0540
sum(x * P(x))Expected Value
Expected Value
The probability-weighted average of all possible outcomes. Multiply each outcome by its probability and sum the results. The long-run average result per trial.
10*0.3 + 2*0.5 + (-5)*0.2→£3.00
C(n,r)

Combinatorics — Counting & Arrangements

Factorials, permutations, combinations and the binomial distribution

1*2*3*…*nFactorial n!
Factorial
n! is the product of all positive integers from 1 to n. It counts the number of ways to arrange n distinct objects. Grows extremely quickly.
1*2*3*4*5→120 (5!)
1*2*3*4*5*6*7→5040 (7!)
n! / (n-r)!Permutation
Permutations P(n,r)
The number of ways to choose r items from n and arrange them in order. Order matters. P(n,r) = n!/(n-r)! = n × (n-1) × … × (n-r+1).
10 * 9 * 8→720 = P(10,3)
8 * 7 * 6→336 = P(8,3)
n! / (r! * (n-r)!)Combination
Combinations C(n,r)
The number of ways to choose r items from n where order does not matter. C(n,r) = n!/(r!×(n-r)!). Used in probability, lotteries, and Pascal's triangle.
(10*9*8*7) / (4*3*2*1)→210 = C(10,4)
(59*58*57*56*55*54) / (6*5*4*3*2*1)→45057474
C(n,k) * p^k * (1-p)^(n-k)Binomial
Binomial Probability
Probability of exactly k successes in n independent trials, each with probability p. P(X=k) = C(n,k) × p^k × (1-p)^(n-k).
(10*9*8*7*6*5)/(6*5*4*3*2*1) * 0.5^6 * 0.5^4→0.2051
Example: probability of exactly 6 heads in 10 fair coin flips ≈ 20.5%
∫

Calculus — Limits, Derivatives & Integrals

Numerical approaches to rates of change and areas using ExpressionCalculator.com

f(x+h) near h=0Limit
Numerical Limit Estimation
Estimate limits by evaluating the expression at values of x progressively closer to the target. Observe whether the outputs converge to a single value.
sin(0.001) / 0.001→0.9999998…
(1 + 1/1000000)^1000000→2.71828… (e)
lim(x→0) sin(x)/x = 1, and lim(n→∞) (1+1/n)^n = e are two classic limits.
(f(x+h) - f(x-h)) / 2hDerivative
Numerical Derivative (Central Difference)
Approximates f'(x) using the central difference formula. Use a small h (e.g. 0.001). More accurate than the one-sided formula (f(x+h)-f(x))/h.
(sin(1.001) - sin(0.999)) / 0.002→0.5403…
cos(1)→0.5403… (exact)
The derivative of sin(x) is cos(x) — both values match, confirming the approximation.
sum of f(midpoints) * hIntegral
Numerical Integration — Midpoint Riemann Sum
Divide the interval into n strips of width h, evaluate the function at each midpoint, sum the results and multiply by h. More strips = more accurate.
(sin(pi/20)+sin(3*pi/20)+sin(5*pi/20)+sin(7*pi/20)+sin(9*pi/20)) * (pi/5)→≈2.0
∫₀^π sin(x)dx = 2 exactly. The Riemann sum above approximates this with 5 strips.
(h/3)*(f0 + 4f1 + 2f2 + … + fn)Simpson's Rule
Simpson's Rule — Accurate Numerical Integration
More accurate than Riemann sums. Requires an even number of strips. Error is proportional to h⁴. Formula: (h/3)[f₀ + 4f₁ + 2f₂ + 4f₃ + … + fₙ].
(pi/4)/3 * (sin(0) + 4*sin(pi/4) + 2*sin(pi/2) + 4*sin(3*pi/4) + sin(pi))→2.0046
x - f(x)/f'(x)Newton's Method
Newton's Method — Root Finding
Starting from an initial guess x₀, iterate: x_{n+1} = x_n - f(x_n)/f'(x_n). Converges rapidly. Type each iteration directly into ExpressionCalculator.com.
2 - (2^3 - 2*2 - 5) / (3*2^2 - 2)→2.1 (1st iter)
2.0946 - (2.0946^3 - 2*2.0946 - 5) / (3*2.0946^2 - 2)→2.09455 ✓
π

Mathematical Constants

Built-in constants available at ExpressionCalculator.com

piConstant
Pi (π)
The ratio of a circle's circumference to its diameter. Approximately 3.14159265358979. Used in area of circles, trigonometry, and countless formulas.
pi→3.14159265…
pi * 5^2→78.5398… (area)
2 * pi * 7→43.98… (circumference)
eConstant
Euler's Number (e)
The base of the natural logarithm. Approximately 2.71828182845904. The unique number where the function e^x equals its own derivative. Central to growth and decay models.
e→2.71828182…
e^1→2.71828182…
log(e)→1
↔

Unit Conversion Expressions

Common conversions you can type directly into ExpressionCalculator.com

(C * 9/5) + 32Temperature
Celsius to Fahrenheit
Multiply by 9/5 and add 32. Reverse: subtract 32 then multiply by 5/9 for Fahrenheit to Celsius.
(100 * 9/5) + 32→212°F
(98.6 - 32) * 5/9→37°C
miles * 1.60934Distance
Miles to Kilometres
Multiply miles by 1.60934 to get kilometres. Divide km by 1.60934 (or multiply by 0.62137) for km to miles.
26.2 * 1.60934→42.16 km
100 / 1.60934→62.14 miles
lbs * 0.453592Weight
Pounds to Kilograms
Multiply pounds by 0.453592 to get kilograms. Divide kg by 0.453592 for the reverse. 1 stone = 14 lbs = 6.35 kg.
160 * 0.453592→72.57 kg
11 * 14 * 0.453592→69.85 kg (11st)
feet * 0.3048Length
Feet to Metres
Multiply feet by 0.3048 to get metres. For inches to centimetres, multiply by 2.54.
6 * 0.3048→1.829 m (6ft)
5*12 + 10→70 inches (5'10")
litres * 0.21997Volume
Litres to Gallons (UK)
Multiply litres by 0.21997 for UK gallons, or by 0.26417 for US gallons. Divide UK gallons by 0.21997 to go back to litres.
40 * 0.21997→8.8 UK gal
10 / 0.21997→45.46 litres
$

Financial & Business Formulas

Interest, investment, mortgages and discounting — all expressible at ExpressionCalculator.com

P * (1 + r)^nCompound Interest
Compound Interest — Future Value
P × (1 + r)^n gives the future value of principal P after n periods at rate r per period. For monthly compounding, use r = annual_rate/12 and n = years×12.
1000 * (1 + 0.05)^10→£1628.89
5000 * (1 + 0.03/12)^(12*5)→£5808.08
P * e^(r*t)Continuous Comp.
Continuous Compounding
When interest is compounded continuously (at every instant), the future value is P × e^(r×t). Always gives a slightly higher result than discrete compounding.
10000 * e^(0.06 * 20)→£33201.17
CF/(1+r)^tPresent Value
Present Value / NPV
The present value of a future cash flow CF received in t years, discounted at rate r per year, is CF/(1+r)^t. Sum multiple terms for NPV of a project.
3000/1.08 + 3000/1.08^2 + 3000/1.08^3 + 3000/1.08^4 + 3000/1.08^5 - 10000→£1978.13
P*r*(1+r)^n / ((1+r)^n - 1)Mortgage
Mortgage Monthly Payment
Monthly payment for a loan of P at monthly rate r (= annual_rate/12) over n months. The full amortisation formula — type it directly to get your exact monthly payment.
250000 * (0.045/12) * (1+0.045/12)^300 / ((1+0.045/12)^300 - 1)→£1389.54/mo
log(2) / log(1+r)Doubling Time
Doubling Time
How many periods until an investment doubles at rate r? Use log(2)/log(1+r) for discrete compounding, or log(2)/r for continuous. The Rule of 72 approximates: 72/rate%.
log(2) / log(1.07)→10.24 years (7%)
72 / 7→10.3 years (Rule 72)
(new - old) / old * 100% Change
Percentage Change
The percentage change from an old value to a new value. Positive = increase, negative = decrease. Multiply by 100 to express as a percentage.
(650 - 500) / 500 * 100→30%
(85 - 100) / 100 * 100→-15%
10 * log10(I / I0)Decibels
Decibels (Sound Intensity)
The decibel level is 10×log₁₀(I/I₀) for intensity ratios, or 20×log₁₀(V_out/V_in) for voltage ratios. Every 10 dB is a 10× increase in intensity.
10 * log10(1000)→30 dB
20 * log10(50)→33.98 dB
-log10([H+])pH
pH (Acidity/Alkalinity)
pH = -log₁₀([H⁺]) where [H⁺] is hydrogen ion concentration in mol/L. pH 7 = neutral, below 7 = acidic, above 7 = alkaline. Each pH unit = 10× change in [H⁺].
-log10(1e-7)→7.0 (water)
-log10(1e-3)→3.0 (acidic)