### 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

Thanks to Cory Doctorow of BoingBoing for his favorable review of Secret Codes!

Other Books

- Fizzy Graphene - September 25, 2017

- The First Angiosperm - September 18, 2017

- Noise Thermometry and the Boltzmann Constant - September 11, 2017

- Walking in the Rain - September 4, 2017

- Agitated Atoms - August 28, 2017

- Partial Solar Eclipse at New Jersey - August 24, 2017

- Magnetocapacitive Tunnel Junctions - August 21, 2017

- Tardigrades - August 14, 2017

- Roman Concrete - August 7, 2017

- Solar Spicules - July 31, 2017

- Schroeder Diffuser - July 24, 2017

- Rough Microparticles - July 17, 2017

- Robot Musicians - July 10, 2017

- Walter Noll (1925-2017) - July 6, 2017

- cosmogony - July 3, 2017

- Crystal Prototypes - June 29, 2017

- Voice Synthesis - June 26, 2017

- Refining Germanium - June 22, 2017

- Granular Capillarity - June 19, 2017

- Kirchhoff–Plateau Problem - June 15, 2017

- Self-Assembly - June 12, 2017

- Physics, Math, and Sociology - June 8, 2017

- Graphene from Ethylene - June 5, 2017

- Crystal Alignment Forces - June 1, 2017

- Martian Brickwork - May 29, 2017

- Carbon Nanotube Textile - May 25, 2017

- The Scent of Books - May 22, 2017

- Patterns from Randomness - May 18, 2017

- Terpene - May 15, 2017

- The Physics of Inequality - May 11, 2017

- Asteroid 2015 BZ509 - May 8, 2017

- Fuzzy Fibers - May 4, 2017

- The Sofa Problem - May 1, 2017

- The Wisdom of Composite Crowds - April 27, 2017

- J. Robert Oppenheimer and Black Holes - April 24, 2017

- Modeling Leaf Mass - April 20, 2017

- Easter, Chicks and Eggs - April 13, 2017

- You, Robot - April 10, 2017

- Collisions - April 6, 2017

- Eugene Garfield (1925-2017) - April 3, 2017

- Old Fossils - March 30, 2017

- Levitation - March 27, 2017

- Soybean Graphene - March 23, 2017

- Income Inequality and Geometrical Frustration - March 20, 2017

- Wireless Power - March 16, 2017

- Trilobite Sex - March 13, 2017

- Freezing, Outside-In - March 9, 2017

- Ammonia Synthesis - March 6, 2017

- High Altitude Radiation - March 2, 2017

### Deep Archive

Deep Archive 2006-2008

**Blog Article Directory on a Single Page**