Pronunciation: /groʊθ/ ?

Growth is an increase in quantity. When one says, "I've grown one inch in the last month," one is stating that their height has increased by one inch. Mathematicians also use the term negative growth to mean a decrease in quantity.

Mathematics is often used to study and predict growth. Models of growth that are often used by mathematicians are arithmetic growth, geometric growth and exponential growth.

Arithmetic Growth

Snow Crystal
Figure 2: Snow crystal growth.
Click on picture to see movie.

Arithmetic growth occurs at a constant rate each time period. Arithmetic growth is also called constant growth and linear growth. For example, if a crystal grows in length 1 inch every year the constant rate of growth is 1 inch/year. The crystal grows 1 inch the first year, and 1 inch the second year, and so on.

YearCrystal Size
02 inches
13 inches
24 inches
35 inches
46 inches
57 inches
Table 1: Arithmetic growth of a crystal.

The general formula for arithmetic growth is y = ax + i. a represents the growth rate, and i represents the initial quantity. Using the data from Table 1, the formula for this crystal growth is y = x + 2.

Note that the equation is a linear equation in slope-intercept form (y = mx + b). When we graph arithmetic growth, the result is a line where the growth rate is the slope, and the initial quantity is the y-intercept.

Graph of f(x)=1000*2^x
Figure 1: Graph showing
arithmetic growth

Geometric growth

Geometric growth occurs at an increasing or decreasing rate each time period. The general formula for geometric growth is y = a·tb where y is the quantity at time t, a is the initial quantity and b is the growth rate.

Exponential growth

For more on exponential growth, see Exponential Function.


  1. growth. WordNet. Princeton University. (Accessed: 2011-01-08).
  2. Linda Almgren Kime, Judy Clark, Beverly K. Michael. Explorations in College Algebra, 4th ed., pp 266-276,298-309. Wiley, November 16, 2007.

More Information

  • McAdams, David. Exponential Function. All Math Words Encyclopedia. Life is a Story Problem LLC. 2010-02-08. Function.
  • McAdams, David. Constant of variation. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-03-12.
  • McAdams, David. Slope. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-03-12. of a Line.

Cite this article as:

Growth. 2010-02-08. All Math Words Encyclopedia. Life is a Story Problem LLC.


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Revision History

2010-02-08: Added text on geometric and exponential growth. Added "References" (McAdams, David.)
2009-03-10: Clarified definition of exponential growth to be a 'fixed multiple' (McAdams, David.)
2008-08-07: Added 'calculation' to exponential growth of bacteria table (McAdams, David.)
2008-06-07: Corrected link errors. Corrected spelling (McAdams, David.)
2008-05-05: Initial version (McAdams, David.)

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