Pronunciation: /ˈmæg.nɪˌtud/ Explain

The magnitude of a point in a coordinate system is the distance of that point from zero.[2] 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 is . Example: The magnitude of < -3, 4 > is calculated using the distance formula: Square root of -3 squared plus 4 squared equals 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. McAdams, David E.. All Math Words Dictionary, magnitude. 2nd Classroom edition 20150108-4799968. pg 112. Life is a Story Problem LLC. January 8, 2015. Buy the book
  2. Fine, Henry B., Ph. D.. Number-System of Algebra Treated Theoretically and Historically. 2nd edition. pp 28-51. Last Accessed 9/2/2018. Buy the book

Cite this article as:

McAdams, David E. Magnitude. 4/24/2019. All Math Words Encyclopedia. Life is a Story Problem LLC.

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Revision History

4/24/2019: Changed equations and expressions to new format. (McAdams, David E.)
3/23/2019: Clarified magnitude of a vector. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
9/2/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
12/19/2009: Added "References". (McAdams, David E.)
12/18/2009: Added more information. (McAdams, David E.)
3/11/2008: Added magnitude of a vector. (McAdams, David E.)
7/12/2007: Initial version. (McAdams, David E.)

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