Modular Arithmetic
Pronunciation: /ˈmɒdʒ ə lər əˈrɪθ mə tɪk/ ?
 Manipulative 1: Modulo 12 arithmetic using a clock. Created with GeoGebra. Download GeoGebra file mod12.ggb. 

A clock is a common example of modular arithmetic.
What time is it when it is 3 hours later than 11:00? It is 2:00. In regular
arithmetic 3+11 = 14, but in modulo 12 arithmetic, 3+11 = 2.
Click on the sliders
in manipulative 1 and drag them to change the clock. The thick black line points
to the answer.
Understanding Check
 Use the sliders in manipulative 1 to represent 5+8. What is 5+8 in modulo 12
arithmetic?
Click for answer
5+8 in modulo 12 is 1.
 Use the sliders in manipulative 1 to represent 3+2. What is 3+2 in modulo 12
arithmetic?
Click for answer
3+2 in modulo 12 is 5.

Modular arithmetic can be done without a clock. To add 11+5 in modulo 12, first find
11+5 in normal arithmetic. 11+5 = 16. Now find the remainder when 16 is divided by 12.

Figure 1: 16÷12 
Since 4 is the remainder, 4 is the answer. So 11+5 = 4 in modulo 12 arithmetic. Verify
the answer using the modulo 12 gadget in manipulative 1.
Now let's look at modulo 5 arithmetic. What is 3+3 in modulo 5? First add using regular
arithmetic: 3+3 = 6. Now find the remainder when 6 is divided by 5.

Figure 1: 6÷5 
1 is the remainder, so 3+3 in modulo 5 arithmetic is 1. Verify the answer using the
modulo 5 gadget in manipulative 2.
What is 3+2 in modulo 5? 3+2 in regular arithmetic is 5. When we divide 5 by 5, what is
the remainder? Since the remainder is 0, 3+2 modulo 5 is 0.
 Manipulative 2: Modulo 5 arithmetic gadget. Created with GeoGebra. Download GeoGebra file mod5.ggb. 

Manipulative 2 is a modulo 5 adding device. Click on the sliders and drag them to change
the addends. The thick black line points to the answer.

Cite this article as:
Modular Arithmetic. 20091218. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/m/modulararithmetic.html.
Translations
Image Credits
Revision History
20091218: Added revision (
McAdams, David.)
20080917: Initial version (
McAdams, David.)