Formula or algorithm and variables.
N(t) = N₀ × 2⁻ᵗ/ᵗ¹ᐟ²; t = t¹ᐟ² × log₂(N₀/N)N₀ is initial amount, N(t) is remaining amount, t is elapsed time, and t¹ᐟ² is the positive half-life.
Formula or algorithm and variables
N(t) = N₀ × 2⁻ᵗ/ᵗ¹ᐟ²; t = t¹ᐟ² × log₂(N₀/N). N₀ is initial amount, N(t) is remaining amount, t is elapsed time, and t¹ᐟ² is the positive half-life. Remaining amount after time t is initial amount × 2 raised to −t divided by half-life; inverse time is half-life × log₂(initial divided by remaining amount). Start by identifying the mathematical object represented by each input rather than treating every number as interchangeable. Keep the chosen mode visible when a tool has more than one interpretation. The calculation preserves exact integer arithmetic where the model is discrete and uses controlled numeric rounding where the result is real-valued. This makes the path from the input to the displayed answer inspectable and gives you a repeatable way to check a hand calculation.
Worked example
After three half-lives, 100 units become 12.5 units because 100 × 2⁻³ equals 12.5. Reproduce the example by writing the formula, substituting the values, and retaining units or place-value labels through each step. For a list-based tool, sort or group the values before checking the result; for a base-conversion tool, expand each digit by its place value. When the result is a large integer, compare the exact string rather than a rounded approximation. A worked example is a verification aid, not a default recommendation, and it should be replaced with the values from the problem you are actually solving.
How to interpret the result
Forward mode predicts remaining quantity. Inverse mode finds the elapsed time needed to reach a chosen positive remaining quantity. Read the primary result together with its supporting details and the selected mode. A mathematically correct result can still answer the wrong question if the inputs describe a different population, representation, unit, ordering rule, or sample space. Compare scenarios only when the definitions and assumptions remain the same. Use exact notation for factors, matrices, repeating decimals, and integer bases; use the displayed precision for real-number summaries and avoid implying more certainty than the inputs support.
Validation and edge cases
Initial amount and half-life must be positive; time cannot be negative; inverse remaining amount must be greater than zero and no greater than the initial amount. Blank fields, malformed tokens, non-finite values, incompatible dimensions, invalid divisors, impossible probability ranges, and out-of-domain operations are rejected explicitly. Boundary values such as zero, one, a prime, a singular matrix, an empty remainder cycle, or a one-term sequence are tested according to the selected operation. If the output seems surprising, first verify the mode, the input order, the representation width, and whether a quantity was intended to be a count or a measurement.
Limitations and responsible use
The model assumes a constant half-life and exponential decay. It does not model replenishment, multiple decay chains, changing conditions, or measurement noise. This page is an explanatory calculator, not a substitute for a statistical design, numerical-analysis package, software type specification, laboratory model, or professional review. Results are processed locally in the browser and no entered values are sent to a calculation service. Keep the original inputs, mode, and assumptions with any copied result. For decisions involving health, finance, safety, experiments, or production systems, verify the model with current authoritative guidance and an independent method before acting.
Check assumptions and sources.
The model assumes a constant half-life and exponential decay. It does not model replenishment, multiple decay chains, changing conditions, or measurement noise.
Common questions.
What does the Half-Life Calculator calculate?
Calculate remaining amount after elapsed half-lives or solve backward for the time to reach a chosen amount.
Which formula or algorithm does it use?
N(t) = N₀ × 2⁻ᵗ/ᵗ¹ᐟ²; t = t¹ᐟ² × log₂(N₀/N). N₀ is initial amount, N(t) is remaining amount, t is elapsed time, and t¹ᐟ² is the positive half-life.
How should the result be interpreted?
Forward mode predicts remaining quantity. Inverse mode finds the elapsed time needed to reach a chosen positive remaining quantity.
What limitations or edge cases matter?
Initial amount and half-life must be positive; time cannot be negative; inverse remaining amount must be greater than zero and no greater than the initial amount. The model assumes a constant half-life and exponential decay. It does not model replenishment, multiple decay chains, changing conditions, or measurement noise.