What is exponential growth in mathematics?

Exponential growth in mathematics is when a quantity increases by a fixed percentage over equal time intervals.

In more detail, exponential growth occurs when the rate of increase of a quantity is proportional to its current size. This means that as the quantity grows, the rate at which it grows also increases. A common example of exponential growth is the way populations of organisms, like bacteria, can double in size over regular intervals of time if conditions are ideal.

The general formula for exponential growth is \( P(t) = P_0 \times (1 + r)^t \), where:
- \( P(t) \) is the amount of the quantity at time \( t \),
- \( P_0 \) is the initial amount of the quantity,
- \( r \) is the growth rate (expressed as a decimal),
- \( t \) is the time period.

For instance, if you start with 100 bacteria and they double every hour, after 1 hour you would have 200 bacteria, after 2 hours you would have 400, and so on. This rapid increase is characteristic of exponential growth.

In graphs, exponential growth is represented by a curve that starts off slowly and then rises steeply. This is different from linear growth, where the increase is constant and the graph is a straight line.

Understanding exponential growth is important in various fields, including biology, finance, and computer science, as it helps to predict how quickly things can change over time.