Modular Arithmetic
Pronunciation: /ˈmɒdʒ ə lər əˈrɪθ mə tɪk/ Explain
Click on the orange and red points on the sliders and drag them to change the figure.
What is 0+0 mod 12? What is 11+11 mod 12?
 Manipulative 1  Modular Arithmetic Created with GeoGebra. 

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.
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.
Click on the orange and red points on the sliders and drag them to change the figure.
What is 1 +5 mod 5? What is 3+4 mod 5?
 Manipulative 2  Modular Arithmetic Created with GeoGebra. 

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:
McAdams, David E. Modular Arithmetic. 9/4/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/m/modulararithmetic.html.
Image Credits
Revision History
9/4/2018: Removed broken links, updated license, implemented new markup. (
McAdams, David E.)
12/18/2009: Added revision. (
McAdams, David E.)
9/17/2008: Initial version. (
McAdams, David E.)