Mathematics · Differential Equations
Half-Life Differential Equation Calculator
Model exponential decay from an initial amount, half-life and elapsed time.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Half-lives elapsed = 18÷6=3.
- Decay factor = 0.5^3=0.125.
- Remaining amount = 80×0.125=10.
Understand Half-life decay
One idea, three depths
Choose how deeply to explain Half-life decay
Half-life decay: Model exponential decay from an initial amount, half-life and elapsed time.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Half-life decay to answer this question: model exponential decay from an initial amount, half-life and elapsed time? Enter Initial amount N₀, Half-life H, Elapsed time t; the calculator shows Remaining amount. For example: An initial 80 units with half-life 6 after 18 time units leaves 10 units. The answer tells you Remaining amount.
Age 15Explain it to a 15-year-oldConnect it to the formula
Every complete half-life multiplies the remaining amount by one half; fractional half-lives follow the same exponential law. The rule is N(t)=N₀(1/2)ᵗ⁄ᴴ. Its input values are Initial amount N₀, Half-life H, Elapsed time t, and the main result is Remaining amount. For example: An initial 80 units with half-life 6 after 18 time units leaves 10 units.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated half-life decay relation over the valid real-number domain stated below. The implemented relation is N(t)=N₀(1/2)ᵗ⁄ᴴ, evaluated from Initial amount N₀, Half-life H, Elapsed time t to produce Remaining amount. Every complete half-life multiplies the remaining amount by one half; fractional half-lives follow the same exponential law. Half-life is not the time required for the whole amount to disappear.
Inputs and valid domain
- Initial amount N₀ must be a finite real number, at least 0.
- Half-life H must be a finite real number, at least 0.
- Elapsed time t must be a finite real number, at least 0.
Important boundary: Half-life is not the time required for the whole amount to disappear.
The formula
N(t)=N₀(1/2)ᵗ⁄ᴴ
How the calculator works through it
It substitutes Initial amount N₀, Half-life H, Elapsed time t into the formula and exposes every numerical step above. The main output is Remaining amount, accompanied by Half-lives elapsed, Fraction remaining.
Read the result correctly
The Remaining amount is the direct answer to “model exponential decay from an initial amount, half-life and elapsed time.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
An initial 80 units with half-life 6 after 18 time units leaves 10 units.
Where this model stops being reliable
Half-life is not the time required for the whole amount to disappear.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Half-life decay works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Half-life decay uses N(t)=N₀(1/2)ᵗ⁄ᴴ. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Derivatives and changing systems
A derivative describes the changing quantity that Half-life decay models or approximates.
Review this foundation about 7 min
Optional enrichment
- Exponential solution behaviour
Exponential behaviour helps you recognise common growth, decay and response patterns related to Half-life decay.
Review this foundation about 7 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Initial amount N₀, Half-life H, Elapsed time t.
- Evaluate the principal relationship: N(t)=N₀(1/2)ᵗ⁄ᴴ.
- Return Remaining amount and check the domain conditions described above.
Python
from math import *
def half_life_ode(a, b, x) -> float:
return (a * pow(0.5, (x / b)))
assert abs(half_life_ode(80, 6, 18) - 10) < 1e-6 * max(1.0, abs(10))
C
#include <assert.h>
#include <math.h>
double half_life_ode(double a, double b, double x) {
return (a * pow(0.5, (x / b)));
}
int main(void) {
const double expected = 10;
const double actual = half_life_ode(80, 6, 18);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double half_life_ode(double a, double b, double x) {
return (a * std::pow(0.5, (x / b)));
}
int main() {
constexpr double expected = 10;
const double actual = half_life_ode(80, 6, 18);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double half_life_ode(double a, double b, double x)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern pow
global half_life_ode
section .text
half_life_ode:
push rbp
mov rbp, rsp
sub rsp, 64
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
mov rax, 0x3fe0000000000000
movq xmm0, rax
movsd [rbp-48], xmm0
movsd xmm0, [rbp-24]
divsd xmm0, [rbp-16]
movsd [rbp-56], xmm0
movsd xmm0, [rbp-48]
movsd xmm1, [rbp-56]
call pow wrt ..plt
movsd [rbp-40], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-40]
movsd [rbp-32], xmm0
movsd xmm0, [rbp-32]
leave
ret
MATLAB
function result = half_life_ode(a, b, x)
result = (a * (0.5 ^ (x / b)));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_, x_] := (a * (0.5 ^ (x / b)));
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Calculus Volume 1
Read OpenStax Calculus: Derivatives and integrationCite this book
- APA 7
- Strang, G., & Herman, E. (2016). Calculus volume 1. OpenStax. https://openstax.org/books/calculus-volume-1/pages/1-introduction
- MLA 9
- Strang, Gilbert, and Edwin Herman. Calculus Volume 1. OpenStax, 2016, https://openstax.org/books/calculus-volume-1/pages/1-introduction.
- Chicago author-date
- Strang, Gilbert, and Edwin Herman. 2016. Calculus Volume 1. Houston, TX: OpenStax. https://openstax.org/books/calculus-volume-1/pages/1-introduction.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Half-Life Differential Equation Calculator. MW SysArc Tools. https://math.mwsysarc.com/differential-equations/half-life-decay
MLA 9
MW SysArc. “Half-Life Differential Equation Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/differential-equations/half-life-decay. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Half-Life Differential Equation Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/differential-equations/half-life-decay.
Harvard
MW SysArc (2026) ‘Half-Life Differential Equation Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/differential-equations/half-life-decay (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_half_life_ode_2026,
author = {{MW SysArc}},
title = {Half-Life Differential Equation Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/differential-equations/half-life-decay},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Half-Life Differential Equation Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/differential-equations/half-life-decay
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Half-life decay do?
Model exponential decay from an initial amount, half-life and elapsed time.
How does the Half-life decay work?
The calculator applies N(t)=N₀(1/2)ᵗ⁄ᴴ. Every complete half-life multiplies the remaining amount by one half; fractional half-lives follow the same exponential law.
What can I learn from the Half-life decay?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .