### 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

*Mathematics-themed novel for middle-school students*

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

Other Books

- Georges Lemaitre, May 22, 2023

- Reading Old Manuscripts, May 15, 2023

- The IQ Flynn Effect, May 8, 2023

- Cosmic Water, May 1, 2023

- Warm Whales, April 24, 2023

- Jumping Beans, April 17, 2023

- Gordon Moore (1929-2023), April 10, 2023

- Speed of Light, April 3, 2023

- Amorphous Ice, March 27, 2023

- Curly Hair, March 20, 2023

- Optical Communication, March 13, 2023

- Hygroscopic Energy, March 6, 2023

- Ionocaloric Refrigeration, February 27, 2023

- Roman Concrete, February 20, 2023

- Nuclear Fusion Energy, February 13, 2023

- Super-Tough CrCoNi Alloy, February 6, 2023

- Squeeze Bottles, January 30, 2023

- A Grammar of Cooking, January 23, 2023

- Cloud Seeding, January 16, 2023

- Drawing Lots, January 9, 2023

- Tree Rings and Cosmic Rays, January 2, 2023

- Season's Greetings - Holidays, 2022

- The Great Filter - December 12, 2022

- Plant Crystals - December 5, 2022

- Gregor Mendel (1822-1884) - November 28, 2022

- Passive Thermal Management - November 21, 2022

- Hominin Evolution - November 14, 2022

- Bird Nest Mechanics - November 7, 2022

### Deep Archive

Deep Archive 2006-2008

**Blog Article Directory on a Single Page**