The equation below can be used to approximate the world’s population in a simple, easily understood way.

In its simplest form, the equation below describes the world total population N(t) at time t:

(1) N(t) = N₀·e^rt

Where:

• N₀ (initial population) = The population at time t = 0.
• N(t) (future population) = The population at time t.
• r (rate) = The rate of population change as a function of t (a 1% increase is expressed as 0.01).
  • This variable is called the Malthusian Parameter.
  • In population studies, r is usually taken to mean births minus deaths.
• t (time) = The amount of time required to produce a growth in population proportional to N(t)/N₀.

Source: Wolfram Math World: Population Growth

Note that e in the above equations is the base of natural logarithms and its value is approximately 2.71828. It is also called Euler’s number.

Derived Forms

Here are the forms of equation (1) in terms of each of its variables:

(2) N₀ = N(t)·e^-rt (present population)

(3)  t = ln(N(t)/N₀)/r (time)

(4)  r = ln(N(t)/N₀)/t (rate)

Doubling time

At the present world population growth rate of 1.7% per year, how long will it take to double the world’s population?

The appropriate equation for this case is (3) above, with the following arguments:

t = ln(N(t)/N₀)/r = ln(2)/0.017 = 40.773 (years)

This equation shows that it will take about 41 years to double the world’s population at an annual growth rate of 1.7%. If this population growth rate is correct, there will be almost 14 billion people on Earth in 2055 and almost 30 billion people at the end of the 21^st century (based on current world population of 7 billion people).