# 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

1. 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.
2. 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 2: 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.

1. McAdams, David E.. All Math Words Dictionary, modular arithmetic. 2nd Classroom edition 20150108-4799968. pg 119. Life is a Story Problem LLC. January 8, 2015. Buy the book

McAdams, David E. Modular Arithmetic. 4/25/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/m/modulararithmetic.html.

### Revision History

4/25/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)