Pronunciation: /ˈmæg nɪˌtud/ ?

The magnitude of a point in a coordinate system is the distance of that point from zero.[1] For real numbers, the magnitude is also called the absolute value. Magnitude (and absolute value) are written using a vertical line '|'.

The magnitude of x is written |x|. The magnitude of -7 is written |-7|.

Number line showing that 3 and -3 are a distance of three units from zero.
Figure 1: Number line showing that 3 and -3 are both a distance of 3 from zero.

The number line in figure 1 shows that both the numbers 3 and -3 are a distance of 3 from zero.


The magnitude of a vector <-3, 4> is calculated using the distance formula: square root(3^2+4^2)=5.

Complex Numbers

The definition of magnitude is given as the distance of a number from zero. For complex numbers, the definition holds. Use the distance formula |a+bi|=square root(a^2+b^2), where a is the length of one leg from zero and b is the length of the other leg from zero.

Take the complex number 4+3i. Figure 2 shows what this point looks like when plotted on the complex plain.

Complex number 4+3i plotted on a rectangular grid showing that the magnitude is sqrt(4^2+3^2)=5
Figure 2: The complex number 4 + 3i plotted on the complex plane

We can also use the distance formula to calculate the magnitude of 4+3i:
|4+3i|=square root(4^2+3^2)=square root(16+9)=square root(25)=5
We can write this in an equation as |4 + 3i| = 5.

Note that magnitude is always positive or zero. It can never be negative.


  1. Fine, Henry B., Ph. D.. Number-System of Algebra Treated Theoretically and Historically, 2nd edition, pp 28-51 (modulus). D. C. Heath & Co., Boston, U.S.A., 1907. (Accessed: 2009-12-19).

Cite this article as:

Magnitude. 2009-12-19. All Math Words Encyclopedia. Life is a Story Problem LLC.


Image Credits

Revision History

2009-12-19: Added "References" (McAdams, David.)
2009-12-18: Added more information. (McAdams, David.)
2008-03-11: Added magnitude of a vector (McAdams, David.)
2007-07-12: Initial version (McAdams, David.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2005-2011 Life is a Story Problem LLC. All rights reserved.
Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License