### Carry On

September 8, 2010 Imagine for a moment that you're a young child again, and you're just being taught how to add large numbers. Sure, you can do 5 + 3 = 8, but now they hit you with 5 + 6 = ?. You have just ten fingers, so counting on your fingers doesn't work, and the teacher might object to your taking your shoes off. Then they teach you about carrying digits, so you can compute 5 + 6 = 11, just like the big boys do. How simpler life would be if you didn't need to carry! Well, that's the idea that Marc LeBrun of Fixpoint Inc.,Novato, CA, and David Applegate and Neil J. A. Sloane of AT&T Shannon Labs decided to investigate.[1] Neil J. A. Sloane may be known to many of you as the originator of the The On-Line Encyclopedia of Integer Sequences, something he started on index cards while a graduate student at Cornell University in 1965. These integer sequences have gone through publication twice in book form, finally developing a life of their own on the internet with 175,000 entries. If still printed in book form, the integer sequences would fill 750 volumes; and these volumes would not be very useful when it came to searching for your particular sequence. In mathspeak, Addition and multiplication of single-digit numbers in a carryless arithmetic system are performed "reduction mod 10, meaning simply that the carry digits are ignored. Take, for example, this simple addition7 8 5and this simple multiplication

+3 7 6

0 5 1

6 4 3You get the idea. The authors investigated the manifold consequences (pun intended) of this modulus type of arithmetic. We'll just summarize the idea of prime numbers. First, there can be no prime numbers in the usual sense, since all numbers in the carryless arithmetic system are divisible by nine. Here's the logic behind that

x5 9

4 6 7

0 0 5

0 4 1 7

- Note that 9 = 9 x 1, 8 = 9 x 2, 7 = 9 x 3, etc.
- This means we can replace all nines in a number by nine time one, all eights in a numbers by nine times two, etc.
- But every digit in the number is now known to be divisible by nine
- So, the number is divisible by nine.

*Advocate of a base-twelve number system. Illustration from the Nuremberg Chronicle by Hartmann Schedel (1440-1514).*

### Reference:

*Permanent Link to this article*

Linked Keywords: Decimal; Novato, CA; Neil J. A. Sloane; AT&T Shannon Labs; The On-Line Encyclopedia of Integer Sequences; Cornell University; modular arithmetic; manifold; prime numbers; number theorists; integer sequence A169887; twelve fingers.

RSS Feed

### Google Search

Latest Books by Dev Gualtieri

**Previews Availableat Tikalon Press**

*STEM-themed novel for middle-school students*

*Mathematics-themed novel for middle-school students*

*Complete texts of LGM, Mother Wode, and The Alchemists of Mars*

Other Books

- Quantum Year 2025, August 12, 2024

- Plant Sounds, August 5, 2024

- Diophantine Equations, July 29, 2024

- Green Ammonia, July 22, 2024

- Cold Brew Coffee, July 15, 2024

- US Independence Day, July 4, 2024

- Copper and Green Energy, July 1, 2024

- Baseball Mud, June 24, 2024

- Wiedemann-Franz Law, June 17, 2024

- A.I., Wine, and Beer, June 10, 2024

- Bergmann's rule, June 3, 2024

- 60 Hertz Standard, May 27, 2024

- Droplets, May 20, 2024

- Levitation, May 13, 2024

- 100 Years of IBM, May 6, 2024

- Cicada Apocalypse, April 29, 2024

- Critical Materials, April 22, 2024

- Eclipse 2024, April 15, 2024

- Reverse Sprinkler, April 8, 2024

- T Coronae Borealis, April 1, 2024

- Coffee and Tea, March 25, 2024

- Arno Penzias (1933-2024), March 18, 2024

- Wave-Particle Duality, March 11, 2024

- Offshore Wind, March 4, 2024

- High Entropy Ceramics, February 26, 2024

- Methane Emissions, February 19, 2024

- Thales, Heron, and Experimental Mathematics, February 12, 2024

- Mica, February 5, 2024

- Disk Packing, January 29, 2024

- Is e
^{π}> π^{e}?, January 22, 2024

- Brownian Motion Energy Harvesting, January 15, 2024

- Coin Flip, January 8, 2024

### Deep Archive

Deep Archive 2006-2008

**Blog Article Directory on a Single Page**