﻿ Modular Arithmetic: Adding numbers that wrap around and begin again at 1.

# Modular Arithmetic

Pronunciation: /ˈmɒdʒ ə lər əˈrɪθ mə tɪk/ ?

 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) 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

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 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.

 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) 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.

Modular Arithmetic. 2009-12-18. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/m/modulararithmetic.html.

### Revision History

2008-09-17: Initial version (McAdams, David.)

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