# 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.

### References

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.)