Base seven one hundred forty-seven notation and pronunciation.

This article follows on from https://bartshmatthew.medium.com/an-easily-learned-algorithm-that-generates-names-for-all-powers-of-ten-79bd9804f758

We can do what we did with base ten with any other base with an integer radix. Let’s see what we can do with base seven hundred forty-seven.

With this base we (habitual base ten users) need about seven hundred thirty-seven new numerals, symbols and spoken names. Bigger bases would need more.

This sounds like a terrible obstacle until you realize that you can make the symbol readable, and the spoken name understandable by combining and modifying just enough the symbols and spoken names so base ten to make the new symbols and names we need.

Thus the biggest numeral in this base is pronounced ‘seven hundred forty-six’ and is written/drawn as ‘[746]’. Counting up from [740] in this base:

Base seven hundred forty seven: [740], [741], [742], [743], [744], [745], [746], 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1[10], 1[11], 1[12], 1[13], 1[14],…

Here’s a bigger series together how to pronounce the numbers:

Base seven hundred forty seven: 1,2,3,4,5,6,7,8,9,[10],[11], [12],[13],[14],[15], [16], [17],… [745], [746], 10, 11, 12, 13,… 1[746], 20,21, …[746][746], 100, 101…

Pronounced: ‘one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen,… seven hundred forty-five, seven hundred forty-six, one seven hundred forty-seven, one seven hundred forty seven one, one seven hundred forty-seven two, one seven hundred forty-seven three,… one seven hundred forty-seven seven hundred forty-six, two seven hundred forty-seven, two seven hundred forty-seven one,… seven hundred forty-six seven hundred forty-seven seven hundred forty-six, one five hundred fifty-eight thousand nine, one five hundred fifty-eight thousand nine one, …

100seven-hundred-forty-seven, if the student/reader has mastered what ‘squared’ means, could also be pronounced, ‘one seven hundred forty-seven squared’, and 101seven-hundred-forty-seven, could also be pronounced, ‘one seven hundred forty-seven squared one’,… The names of course start unwieldy and quickly become extremely unwieldy due to the base being so ludicrously high a number. But this method is valid and usable.

In base ten the biggest numeral is ‘9’ while in base seven hundred forty seven the biggest numeral is ‘[746]’. Base ten has 0.99, and base 747 has 0.[746][746]. Base ten has: 1 minus 0.01 = 0.99 while base 747 has 1 minus 0.01 = 0.[746][746].

Some possible variations on this notation system:

Instead of square brackets [], ordinary brackets could be used(), or braces {}, or any other way of surrounding the numerals.

Instead of numerals between brackets, a unique symbol could be used for each numeral. It might be tiny numerals inside a circle other shape, tiny numerals inside, as a sort of internal subscript (‘endoscript’, anyone?), a normal sized Unicode-sized symbol. Thus if you can’t recognize the symbol, you can zoom in on it, or use a magnifying glass, to examine the ‘endoscript’.

To make the numerals take up one place each in the positional notation, one could have the numerals written from top to bottom. Thus ‘[747]’ would be instead written as a ‘7’ above a ‘4’ above a ‘7’.

Another way to make the numbers line up, for example to make long division work properly when the conventional notation for that is used, would be to have each numeral occupy several spaces, e.g.

[746] [744]

[746] [745]

[746] [746]

____1____0 (The underscores represent empty space which I don’t know how type on Medium where I am publishing this).

____1____1

____1____2

____1____3

or

[746] [744]

[746] [745]

[746] [746]

Edit:

[001][000]

[001][001]

[001][002]

[001][003]

(Acknowledgement: I got the idea for using zeros as padding instead of blank spaces from this just now:

“As the comments have said, you have to define a symbol for it. Going to lower case letters is one possibility. A more general approach is to use two decimal digits for each base 37 digit so your base 37 digits range from 00 through 36. This has the advantage of being extendable to any base.” Ross Millikan “answered Dec 26 '20 at 16:41” which I saw here:https://math.stackexchange.com/questions/3962545/how-we-can-write-36-in-base-37 )

End of edit.

Edit: Wolfram Alpha, I just found out, uses ‘001 003 two digits’ where I would use ‘[001][003]’.

Another possibility is to use ‘0001’ to ‘0746’. Thus you’d count on paper like this:

0001

0002

…

0745

0746

0001 0000

0001 0001

That gets rid of the need for brackets and means you can type using only the numeric keypad.

End of edit.

Using the name generating algorithm (the naming scheme) on base 747.

Let’s see if we can make even better by using some new names generated by my algorithm.

With base ten, the radix is ten. With base seven hundred forty-seven, the radix is seven hundred forty-seven.

The name ‘tasam’ means ‘the radix raised to the power one hundred three (103 base ten). In base ten, tasam = 10¹⁰³ten, while in base 747, tasam = 10¹⁰³seven-hundred-forty-seven.

In binary ‘tasam’ means 10¹⁰³two ie 2¹⁰³ten. (There’s another name for 2¹⁰³ten, which is ‘tisim’ but I won’t go into that here. See my article : https://bartshmatthew.medium.com/how-a-math-teacher-should-pronounce-a-binary-number-1c41773df52f?source=your_stories_page------------------------------------- ).

In base ten, one hundred thousand is ‘one la’. Thirty-two converted to binary is also ‘one la’. That’s because one hundred thousand it ten to the power five, and in base ten the radix is ten. On the other hand thirty-two is two to the power five, and in base two the radix is two.

‘La’ means ‘the radix to the power five’ so the actual value of one la depends on the radix.

La means 100,000 in some base. 100,000ten is one hundred thousand. 100,000two is thirty-two. They are both one la in their respective contexts.

And 100,000seven-hundred-forty-seven is 747⁵ten. And so in base 747, ‘one la’ means seven hundred forty-seven to the power five,

which is in base ten: 232,596,410,928,507

and in base 747: 100,000.

Here’s my example counting in base 747 copypasted in from earlier in this article:

Base seven hundred forty seven: 1,2,3,4,5,6,7,8,9,[10],[11], [12],[13],[14],[15], [16], [17],… [745], [746], 10, 11, 12, 13,… 1[746], 20,21, …[746][746], 100, 101…

Pronounced: ‘one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen,… seven hundred forty-five, seven hundred forty-six, one seven hundred forty-seven, one seven hundred forty seven one, one seven hundred forty-seven two, one seven hundred forty-seven three,… one seven hundred forty-seven seven hundred forty-six, two seven hundred forty-seven, two seven hundred forty-seven one,… seven hundred forty-six seven hundred forty-seven seven hundred forty-six, one five hundred fifty-eight thousand nine, one five hundred fifty-eight thousand nine one, …

Here is the pronunciation using the new names for the powers of the radix :

‘one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen,… seven hundred forty-five, seven hundred forty-six, one ta, one ta one,one ta two, one ta three,… one ta seven hundred forty-six, two ta, two ta one,… seven hundred forty-six ta seven hundred forty-six, one na, one na one, …

Thus one na one means ‘one case of the radix squared plus one’. Na in a way means ‘square’, ma means ‘cube’, ra means ‘hypercube’, and ta means ‘unraised’, while sa means one.

I wonder whether we could say that ‘ra’ means ‘4-cube’ (four-cube), ‘la’ means ‘5-cube’, and so on?

How about ‘ra’ is replaced with ‘place four’ (referring to the place in the positional notation) and ‘la’ with ‘place five’, and so on.

Anyway it’s a lot more pronounceable, especially as the powers or 747 get bigger.