# Base-7 Number System (Septenary)

The base-7 number system represents integers with the digits 0, 1, 2, 3, 4, 5 and 6. Base-7, also called septenary, has no practical use, but yet many wonderful and curious mathematical properties. Converting a regular base-10 number to septenary is easy using the remainder method.

## Understanding Place Values in Septenary

The basic idea behind the usual base-10 number system is that the strings "1," "10," "100," "1000," etc. represent powers of ten.That is, "1" is the 0th power of ten, "10" is the 1st power of ten, "100" is the 2nd power of ten, "1000" is the 3rd power of ten, etc. The digit places of a number represent how many of each power there are. For example, the base-10 number "849" is equal to the sum eight times one hundred, plus four times ten, plus nine times one.

In base-7, the strings "1," "10," "100," "1000," etc. represent the powers of seven: one, seven, forty-nine, three hundred forty-three, etc. The base-7 number "4261" is equal to the sum four times three hundred forty-three, plus two times forty-nine, plus six times seven, plus one times one. Working this out in base-10 gives you the base-10 number "1513." Thus, the base-7 integer "4261" is the same as one thousand five hundred thirteen in the base-10 system.

## How to Convert a Base-10 Integer to Base-7 Via Remainders

To convert a familiar decimal integer to septenary, you successively divide the quotients by seven, noting the remainder at each step. The process ends when you reach a quotient of zero. The sequence of remainders from last to first gives the septenary representation of the number. Let's start with the base-10 number "98372" as an example.

- 98372 / 7 = 14053 remainder
**1** - 14053 / 7 = 2007 remainder
**4** - 2007 / 7 = 286 remainder
**5** - 286 / 7 = 40 remainder
**6** - 40 / 7 = 5 remainder
**5** - 5 / 7 = 0 remainder
**5**

When you read the list from bottom to top you get "556541." This is how you write the base-10 number "98372" in base-7.

## How to Convert a Base-7 Number to Base-10

To perform the reverse conversion, we need apply the concept of place values as powers. Each place value in this number represents a power of seven, read in increasing order from right to left starting with the zeroth power. (Any positive integer raised to the zeroth power is equal to one.)

For example, let's take the base-7 integer "20035." The right-most place is the ones place, the next is the sevens place, the next is forty-nines place, followed by three hundred forty-threes place, and ending with two thousand four hundred ones place in the left-most position. Each of these powers of seven is multiplied by the digit in that place. And finally these products are all added to obtain the equivalent number in base-10.

Working out the example above we have

5*1 + 3*7 + 0*49 + 0*343 + 2*2401 = 4828.

Therefore, the base-7 integer "20035" is equivalent to the base-10 integer "4828."

## Fun Facts About Base-7 Numbers

- The last digit of a base-7 number is 0 if and only if the number is a multiple of seven.
- The digits of a base-7 number add up to a multiple of six if and only if the number itself is a multiple of six. For example, in the base-7 number "102504" the digits add up to twelve, so "102504" is a multiple of six. You can double check this; "102504" in base-7 is the same as "17742" in base-10, which a calculator will prove is a multiple of six.
- The divisibility test above holds for checking if a base-7 number is divisible by two or three, that is, you add the digits and check if the sum is divisible by two or three.
- Alternately subtract and add the digits of a base-7 number. The number you get is a multiple of eight if and only if the number itself is a multiple of eight. (Zero counts as a multiple of eight) For example, consider the base-7 integer "66521301." The alternating sum of the digits is 6 - 6 + 5 - 2 + 1 - 3 + 0 - 1 = 0, which means "66521301" is a multiple of eight.
- The divisibility test above holds for checking if a base-7 number is divisible by four, that is, you alternately subtract and add the digits and check if your result is a multiple of four.
- The first sixteen Fibonacci numbers written in base-7 are 1, 1, 2, 3, 5, 11, 16, 30, 46, 106, 155, 264, 452, 1046, 1531, 2610. The pattern of last digits repeats with a cycle length of sixteen. In comparison, the pattern of Fibonacci last digits in base-10 repeats with a cycle-length of sixty.
- A square number written in base-7 will never have 3, 5, or 6 as the last digit.
- A cube number written in base-7 will never have 2, 3, 4, or 5 as the last digit.
- A triangular number written in base-7 will never have 2, 4, or 5 as the last digit.

## Non-Integers in Base-7

Just as irrational numbers and non-integer rational numbers can be written in base-10 as decimals, so can these numbers be written in base-7. In the list below, the part of the decimal in braces { } is the repeating part. Ellipses ... represent a non-repeating non-terminating decimal, aka an irrational number.

- one half = 0.{3}
- one third = 0.{2}
- one fourth = 0.{15}
- one fifth = 0.{1254}
- one sixth = 0.{1}
- one seventh = 0.1
- one eight = 0.{06}
- one ninth = 0.{053}
- one tenth = 0.{0462}
- thirteen seventeenths = 0.5{2320261143464055}
- pi = 3.066365143203613411026340224...
- e = 2.50124106542265043353530003006...
- phi, the golden ratio = 1.42166203646016035532250514...
- square root of seven = 2.434330665424603323332613624661566...

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## Comments 5 comments

Very interesting but as you said not practical. You can have any base as a number system but the binary system is most useful for computers and the decimal system is best for humans.

I don't see that. Try doing long division or multiplication in base 12 or any base for that matter. The decimal system is the easiest to work with. That is why the metric system won out over the American way of measurements around the world. It is also the standards with all scientific disciplines.

However, I do seem to remember in college science class especially in Chemistry where the calculations are done using metric nomenclature because they are inter related. I will have to go back to refresh my memory... Anyway, good hub to make us think.

5