Doubling time

The notion of doubling time dates to interest on loans in Babylonian mathematics. Clay tablets from circa 2000 BCE include the exercise "Given an interest rate of 1/60 per month (no compounding), come the doubling time." This yields an annual interest rate of 12/60 = 20%, and hence a doubling time of 100% growth/20% growth per year = 5 years.[2][3] Further, repaying double the initial amount of a loan, after a fixed time, was common commercial practice of the period: a common Assyrian loan of 1900 BCE consisted of loaning 2 minas of gold, getting back 4 in five years,[2] and an Egyptian proverb of the time was "If wealth is placed where it bears interest, it comes back to you redoubled."[2][4]

Examining the doubling time can give a more intuitive sense of the long-term impact of growth than simply viewing the percentage growth rate.

For a constant growth rate of r% within time t, the formula for the doubling time T_d is given by

${\displaystyle T_{d}=t{\frac {\ln(2)}{\ln(1+{\frac {r}{100}})}}\approx t{\frac {70}{r}}}$

Some doubling times calculated with this formula are shown in this table.

Simple doubling time formula:

${\displaystyle N(t)=N_{0}2^{t/T_{d}}}$

• N(t) = the number of objects at time t
• T_d = doubling period (time it takes for object to double in number)
• N₀ = initial number of objects
• t = time

For example, with an annual growth rate of 4.8% the doubling time is 14.78 years, and a doubling time of 10 years corresponds to a growth rate between 7% and 7.5% (actually about 7.18%).

When applied to the constant growth in consumption of a resource, the total amount consumed in one doubling period equals the total amount consumed in all previous periods. This enabled U.S. President Jimmy Carter to note in a speech in 1977 that in each of the previous two decades the world had used more oil than in all of previous history (The roughly exponential growth in world oil consumption between 1950 and 1970 had a doubling period of under a decade).

Given two measurements of a growing quantity, q₁ at time t₁ and q₂ at time t₂, and assuming a constant growth rate, you can calculate the doubling time as

${\displaystyle T_{d}=(t_{2}-t_{1})\cdot {\frac {\ln(2)}{\ln({\frac {q_{2}}{q_{1}}})}}.}$

A constant relative growth rate means simply that the increase per unit time is proportional to the current quantity, i.e. the addition rate per unit amount is constant. It naturally occurs when the existing material generates or is the main determinant of new material. For example, population growth in virgin territory, or fractional-reserve banking creating inflation. With unvarying growth the doubling calculation may be applied for many doubling periods or generations.

In practice eventually other constraints become important, exponential growth stops and the doubling time changes or becomes inapplicable. Limited food supply or other resources at high population densities will reduce growth, or needing a wheel-barrow full of notes to buy a loaf of bread will reduce the acceptance of paper money. While using doubling times is convenient and simple, we should not apply the idea without considering factors which may affect future growth. In the 1950s Canada's population growth rate was over 3% per year, so extrapolating the current growth rate of 0.9% for many decades (implied by the doubling time) is unjustified unless we have examined the underlying causes of the growth and determined they will not be changing significantly over that period.

The equivalent concept to doubling time for a material undergoing a constant negative relative growth rate or exponential decay is the half-life.

The equivalent concept in base-e is e-folding.

Graphs comparing doubling times and half lives of exponential growths (bold lines) and decay (faint lines), and their 70/t and 72/t approximations. In the SVG version, hover over a graph to highlight it and its complement.

Cell doubling time can be calculated in the following way using growth rate (amount of doubling in one unit of time)

Growth rate:

${\displaystyle N(t)=N(0)e^{gr*t}}$

or

${\displaystyle gr={\frac {\ln \left(N(t)/N(0)\right)}{t}}}$

• ${\displaystyle N(t)}$ = the number of cells at time t
• ${\displaystyle N(0)}$ = the number of cells at time 0
• ${\displaystyle gr}$ = growth rate
• ${\displaystyle t}$ = time (usually in hours)

Doubling time:

${\displaystyle {\text{doubling time}}={\frac {\ln(2)}{\text{growth rate}}}}$

The following is the known doubling time for the following cells:

• Albert Allen Bartlett
• Binary logarithm
• e-folding
• Exponential decay
• Exponential growth
• Half-life
• Relative growth rate
• Rule of 72

• Doubling Time Calculator
• http://geography.about.com/od/populationgeography/a/populationgrow.htm