Septenary

And hours of sitting down actually doing math to derive the conversion formula....

Septenary, A Base-7 System

By Jett Shutter

Wait, when you were counting in base-7 I didn't see any 7's, 8's, or 9's....

Exactly! This is because those numbers do not even exist in the base-7 number vocabulary. Every time you would reach a multiple of 7 when counting in base-10, you are actually hitting an interval in a base-7 system without knowing it. Base-7 just means that 7's equal 10's and the numbers between vanish.

What is a Base-7 system?

A Base-7 system means that the numbers repeat in cycles of 7 as opposed to what we know as the decimal system where numbers repeat in cycles of ten.

Exhibit's

Science Center of Ithaca "How many is 1million?"

Websites

DeVlieger, Michael T. "The Dozenal Society Of America: Multiplication Tables of Various Bases." The Dozenal Society Of America. The Dozenal Society Of Americ, 6 Feb. 2011. Web. 14 Apr. 2013. <http://www.dozenal.org/articles/DSA-Mult.pdf>.

Books

Paulos, John Allen. Innumeracy: Mathematical Illiteracy and Its Consequences. New York: Hill and Wang, 1988. Print.

Jay, Alison. 1 2 3: A Child's First Counting Book. New York: Dutton Children's, 2007. Print.

For example instead of counting as;

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ......

We would count as;

1, 2, 3, 4, 5, 6, 10, 11, 12, .......

Sources:

This just goes to show that counting with any base system just means that the numbers repeat in intervals of whatever base the system is in. If it is base-2 the cycle will repeat every 2 numbers, and if it is base 3, they will repeat every 3 numbers. (One place value up of course.)

This chart shows the side by side order of both base-7 and base-10 number systems. It shows good comparison of how the numbers work.

The idea of a base number system outside of base-10 is difficult for us to wrap our heads around because we have only known, and have always been taught with the base-10 system. Using anything else makes us second guess ourselves and confuses us because we know 5+5 equals 10, but what we don't know is why and how 5+5= 10. However, for a child who has not learned math yet they can still learn why and how 5+5=10 and what this means for other base systems.

A comfortable way to think about numbers is that every place value, ones, tens, hundreds, etc, are each wheels. Each wheel can only move one unit every time the previous wheel makes a full rotation. The units just change depending on the base system value.

Why does it seem complicated for us?

Is there a formula for converting one base to another?

Yes! There is!

In the formula we let X stand for the number in base 10 we are converting, we then divide this number by 7 and we will name this number Y for the sake of clarity. We then round down Y to the nearest whole number for Yr. We can then take this number Yr, multiply it by three, and add our result to X to get our base 7 equivalent. Below there is an example problem using the number 48.

Formula:

X/7=Y

Round Down Y to the nearest whole for Yr

(Yr*3) + X = the Base 7 Equivalent of X.

Using 48 as X

48/7=6.857...

Round Down to 6.

(6*3) + 48 = 18 + 48 = 66

This tells us that 48 in base-10 is equal to 66 in base-7.

5

Here we also have a times table to show how multiplication looks when using base-7.

5

1, 2, 3, 4, 5, 6, 10, 11, 12, 13

5

Base-7: 5+5=13

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Base-10: 5+5=10

5

The first thing we must understand is that all mathematical functions are done in the same fashion; counting. Every mathematical function can be broken down in to a form of counting, from squaring a number, to basic addition. All that we really do when changing a base system is changing the names and look of the numbers we use, not the function.

Not only this, but numerical strength is key for success in an ever changing technological world.

The sooner a student is numerically literate, the better they will be able to adapt to this ever changing world.

If certain numbers don't exist, how do we do math problems?

Elementary students, possibly as early as kindergarten, should be introduced to other base systems besides the standard base-10 decimal system. This is so students can more clearly understand the principles of cardinality and how numbers actually work with each other. If I child is raised knowing how numbers work and that it is all as simple as counting, and not strict memorization there should be a drop in math class failure rates across the board for all ages.

So in essence, the main 4 operations are all equally simple more or less.

Subtraction is counting backwards so many from one number to get an answer.

Multiplication is counting several numbers many times.

Division is counting out a certain number as many times as you can out of another number.

Why Elementary students should be introduced to other base systems other than base-10.

Multiplication, Subtraction, and Division are all forms of counting one, by one.