Beyond Counting to 10

Introduction

Counting

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Introduction

We use ten digits—'0' through '9'—mostly because we have ten fingers/digits. If we had eight or twelve fingers, we’d count using only eight or twelve digits. When we base our counting on ten digits, that’s called base 10. It’s so natural to us that we hardly ever question it, but it’s not the only way to count.

What if our number system was shaped differently? Imagine a world of eight-fingered shopkeepers, pricing their wares and calculating their profits in base 8.

Or twelve-fingered artists weaving patterns in base 12.

The choice of base 10 isn't a necessity—it's a result of our biology, the physical structure of our hands. And now it’s a filter through which we view the world. Changing that filter can reveal surprising possibilities.

Alternative number bases unlock patterns and relationships hidden in the familiar. Base 2 powers modern technology, while base 12 once guided ancient commerce and design. (And base 12 still guides our clocks.) By stepping beyond our ten-fingered norm, we can uncover fresh ways of thinking, creating, and understanding.

In the Scriptorium, we’ll explore these alternate number systems — not to master mathematics, but to marvel at how they can shape art, music, and the very way we think. Each system is a new horizon, and each number holds secrets waiting to be discovered.

Counting

Note: for number bases greater than 10, we'll run out of digits. So after '9' we start using the upper-case letters of the alphabet:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, ...

Occasionally computers use base 64 (!!). To have enough 'digits' we add the lower case alphabet (a-z) and two symbols:

ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/

Decimal	Base 2	Base 3	Base 4	Base 5	Base 6	Base 7	Base 8	Base 9	Base 10	Base 11	Base 12	Base 13	Base 14	Base 15	Base 16
1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
2	10	2	2	2	2	2	2	2	2	2	2	2	2	2	2
3	11	10	3	3	3	3	3	3	3	3	3	3	3	3	3
4	100	11	10	4	4	4	4	4	4	4	4	4	4	4	4
5	101	12	11	10	5	5	5	5	5	5	5	5	5	5	5
6	110	20	12	11	10	6	6	6	6	6	6	6	6	6	6
7	111	21	13	12	11	10	7	7	7	7	7	7	7	7	7
8	1000	22	20	13	12	11	10	8	8	8	8	8	8	8	8
9	1001	100	21	14	13	12	11	10	9	9	9	9	9	9	9
10	1010	101	22	20	14	13	12	11	10	A	A	A	A	A	A
11	1011	102	23	21	15	14	13	12	11	10	B	B	B	B	B
12	1100	110	30	22	20	15	14	13	12	11	10	C	C	C	C
13	1101	111	31	23	21	16	15	14	13	12	11	10	D	D	D
14	1110	112	32	24	22	20	16	15	14	13	12	11	10	E	E
15	1111	120	33	30	23	21	17	16	15	14	13	12	11	10	F
16	10000	121	100	31	24	22	20	17	16	15	14	13	12	11	10
17	10001	122	101	32	25	23	21	18	17	16	15	14	13	12	11
18	10010	200	102	33	30	24	22	20	18	17	16	15	14	13	12
19	10011	201	103	34	31	25	23	21	19	18	17	16	15	14	13
20	10100	202	110	40	32	26	24	22	20	19	18	17	16	15	14
21	10101	210	111	41	33	30	25	23	21	1A	19	18	17	16	15
22	10110	211	112	42	34	31	26	24	22	20	1A	19	18	17	16
23	10111	212	113	43	35	32	27	25	23	21	1B	1A	19	18	17
24	11000	220	120	44	40	33	30	26	24	22	20	1B	1A	19	18
25	11001	221	121	100	41	34	31	27	25	23	21	1C	1B	1A	19

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Note: This base conversion tool handles fractions. For example:

12.34 base 10 ≈ 15.22 base 7.

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Famous Numbers

Transcendental Numbers

In mathematics, a 'transcendental number' is one for which no finite combination of integers, addition, subtraction, multiplication, division, and exponents can equal its value exactly. Transcendental numbers are infinite, non-repeating decimals. They transcend our ability to calculate them exactly.

One of the most famous equations in math — "Euler's Identity" — uses two of these transcendental constants: π and e:

\[ e^{i\pi} + 1 = 0 \]

See Euler's Identity

The Constant Pi (π)

Pi (π) to 200 places in bases 2 through 16. Pi (π) represents the ratio of a circle's circumference to its diameter, and appears all over mathematics and science — sometimes in rather surprising places... More...

The Constant e

e to 200 places in bases 2 through 16. The mathematical constant e, approximately 2.718, is the base of the natural logarithm and is widely used to model real-world phenomena involving growth and change. More...

The Constant Phi (φ)

φ to 200 places in bases 2 through 16. Without getting into the math of it, φ — the golden ratio (approximately 1.681) — is a special proportion which shows up in nature, art, and design, making things feel balanced and beautiful. It’s found in the way flowers arrange their petals, the spiral of a seashell, and in great works of art and architecture like the Mona Lisa and the Parthenon. More...

Series

Fibonacci Numbers

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