1. CLOCK ARITHMETIC

2. 2

3. 3
- 7
We define ordinary arithmetic by hopping up and down
a straight line.
Let’s define a new kind of arithmetic by hopping around
a circle!

4. 4 1 3 2 2
+ 4
= 1

5. 4 1 3 2
In this system, 4
+ 3
= 2

6. 4 1 3 2
and 2
- 3
= 4

7. 4 1 3 2 4
 3
= 2

8. 1 4 2 3
 3
= 2
We evaluate 43 by taking 12 hops around the clock. Because every
fifth hop is zero we land on the remainder when 12 is divided by 5.

9. 1 4 2 3 5 3 8
4 + 4 = 3

10. 1 4 2 3 10 2 12
4  3 = 2

11. 1 4 2 3 5 4 9
3  3 = 4

12. 15 10
16 etc 5 14 11 9 6 4 1 3 2 7 8 12 13
Numbers of the same color belong to the same modular class.

13. 1 4 2 3 ARITHMETIC MOD 5 Solve for x: 2 x + 4 = 21 4 2 3
ARITHMETIC MOD 5
Solve for x: 2 x = 2
Additive inverse of 4
associativity
2 x = 2
( )
+ 1
3( )
2 x =

14. 1 4 2 3 ARITHMETIC MOD 5 Solve for x: 2 x + 4 = 2 2 x + 4 = 2 ( ) + 11 4 2 3
ARITHMETIC MOD 5
Solve for x: 2 x = 2
2 x = 2
( )
+ 1
Multiplicative inverse of 2
3( )
2 x =

15. 1 4 2 3 ARITHMETIC MOD 5 Solve for x: 2 x + 4 = 2 2 x + 4 = 2 ( ) + 11 4 2 3
ARITHMETIC MOD 5
Solve for x: 2 x = 2
2 x = 2
( )
+ 1
(3• )
2 x =
3( )
1 x =