The easiest to learn way for base ten speakers/users to read, write, and pronounce numbers when counting in other bases.

This is what I think is the easiest way, not the ultimate way, the most efficient way, or the best possible way. It could be useful for communicating with laypersons in works of popular science, or teaching students in math class about counting in other bases.

Binary: (The conventional system used world wide for writing it) 1,10,11, 100, 101, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001,…

Pronounced ‘one, two, two one, four, four one, four two, four two one, eight, eight one, eight two, eight two one, eight four, eight four one, eight four two, eight four two one, sixteen, sixteen one, sixteen two, sixteen two one… sixteen eight four two one, thirty-two, thirty-two one, thirty-two two, thirty-two two one, …’.

As you can see the names get increasingly unwieldy. For my solution to this which is a system of easy short new names for the powers of two, see my article https://bartshmatthew.medium.com/how-a-math-teacher-should-pronounce-a-binary-number-1c41773df52f?source=your_stories_page-------------------------------------

Base twelve: 1,2,3,4,5,6,7,8,9,[10],[11], 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1[10], 1[11], 20, 21, …, [11]9, [11][10], [11][11],100,101,…[11][11][11], 1000, 1001,…

Pronounced:

Hexadecimal: 1,2,3,4,5,6,7,8,9,[10],[11],[12],[13],[14],[15], 10, 11,12,13,14,15,16,17,18,19,1[10],1[11], 1[12], 1[13], 1[14], 1[15], 20, 21,22,23,24,25, 26, 27, 28, 29, 2[10], 2[11], 2[12], 2[13], 2[14], 2[15], 30, 31…[15][15], 100, 101, 102,…

Pronounced: ‘one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, one sixteen, one sixteen one, one sixteen two, one sixteen three, … one sixteen fourteen, one sixteen fifteen, two sixteen. See my article for details of a more sophisticated version of this system: https://bartshmatthew.medium.com/how-to-pronounce-hexadecimal-155f4105725b?source=your_stories_page-------------------------------------

Base thirty-seven: 1,2,3,4,5,6,7,8,9,[10],[11], [12],[13],[14],[15], [16], [17], [18], [19], [20], [21], [22], [23], …[35], [36], 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1[10], 1[11], 1[12], 1[13], 1[14], 1[15], 1[16], 1[17], 1[18], 1[19], 1[20], 1[21], …1[35], 1[36], 20, 21, 22, 23, 24,… [36][34], [36][35], [36][36], 100, 101,…

Pronounced: See my article on a general purpose way to pronounce any number in any base: https://bartshmatthew.medium.com/concise-and-easy-ways-to-pronounce-numbers-in-any-base-5ecba4342557?source=your_stories_page-------------------------------------

Base sixty-four: 1,2,3,4,5,6,7,8,9,[10],[11], [12],[13],[14],[15], [16], [17], [18], [19], … [59], [60], [61], [62], [63], 10, 101…

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. For a way to cut this problem down to size see my article which I am linking to for the second time in this article: https://bartshmatthew.medium.com/concise-and-easy-ways-to-pronounce-numbers-in-any-base-5ecba4342557?source=your_stories_page-------------------------------------

Some possible variations on this 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]

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My favorite activity is learning new things.