Tolvtallsystemet

Tolvtallsystemet (også kalt duodesimalsystemet) er et tallsystem med grunntall tolv. Tolvtallsystemet trenger tolv sifre: 0 til 9 og ytterligere to sifre som representerer ti og elleve. Ofte brukes A for ti og B for elleve. For å indikere at et tall er skrevet i tolvtallsystemet kan man, der det er hensiktsmessig, føye til 12 med senket skrift etter tallet: 10₁₂ = 12 (underforstått 12₁₀). De første 24 positive heltallene skrives på følgende måte:

Titallsystemet (n₁₀)	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
Tolvtallsystemet (n₁₂)	0	1	2	3	4	5	6	7	8	9	A	B	10	11	12	13	14	15	16	17	18	19	1A	1B	20

Tolvtallsystemets lille multiplikasjonstabell
×	1	2	3	4	5	6	7	8	9	A	B	10
1	1	2	3	4	5	6	7	8	9	A	B	10
2	2	4	6	8	A	10	12	14	16	18	1A	20
3	3	6	9	10	13	16	19	20	23	26	29	30
4	4	8	10	14	18	20	24	28	30	34	38	40
5	5	A	13	18	21	26	2B	34	39	42	47	50
6	6	10	16	20	26	30	36	40	46	50	56	60
7	7	12	19	24	2B	36	41	48	53	5A	65	70
8	8	14	20	28	34	40	48	54	60	68	74	80
9	9	16	23	30	39	46	53	60	69	76	83	90
A	A	18	26	34	42	50	5A	68	76	84	92	A0
B	B	1A	29	38	47	56	65	74	83	92	A1	B0
10	10	20	30	40	50	60	70	80	90	A0	B0	100

En fordel med tolvtallsystemet i forhold til titallsystemet er at brøker oftere kan skrives ut med et endelig antall desimaler:

Titall → tolvtall og tolvtall → titall
Titall	Tolvtall	Tolvtall	Titall
1	1	1	1
10	A	10	12
100	84	100	144
1 000	6B4	1 000	1 728
10 000	5 954	10 000	20 736
100 000	49 A54	100 000	248 832
1 000 000	402 854	1 000 000	2 985 984

Titallsystemet			Tolvtallsystemet
1⁄2	0,5	10/2 = 5	1⁄2	0;6	10/2 = 6
1⁄3	0,333333333333... ≈ 0,333	10/3 = 31⁄3	1⁄3	0;4	10/3 = 4
1⁄4	0,25	10/4 = 21⁄2	1⁄4	0;3	10/4 = 3
1⁄5	0,2	10/5 = 2	1⁄5	0;249724972497... ≈ 0;24A	10/5 = 22⁄5
1⁄6	0,166666666666... ≈ 0,167	10/6 = 12⁄3	1⁄6	0;2	10/6 = 2
1⁄7	0,142857142857… ≈ 0,143	10/7 = 13⁄7	1⁄7	0;186A35186A35… ≈ 0;187	10/7 = 15⁄7
1⁄8	0,125	10/8 = 11⁄4	1⁄8	0;16	10/8 = 11⁄2
1⁄9	0,111111111111… ≈ 0,111	10/9 = 11⁄9	1⁄9	0;14	10/9 = 11⁄3
1⁄10	0,1	10/10 = 1	1⁄A	0;1249724972497… ≈ 0;125	10/A = 11⁄5
1⁄11	0,090909090909… ≈ 0;091	10/11 = 10⁄11	1⁄B	0;111111111111… ≈ 0;111	10/B = 11⁄B
1⁄12	0,083333333333… ≈ 0;083	10/12 = 5⁄6	1⁄10	0;1	10/10 = 1
	«,» er desimalt skilletegn.			«;» er duodesimalt skilletegn.
	5 desimalbrøker med et endelig antall desimaler, opp til 3.	3 heltallige kvotienter.		7 duodesimalbrøker med et endelig antall duodesimaler, opp til 2.	5 heltallige kvotienter.

Omregninger

For å konvertere et tall fra tolvtallsystemet til titallsystemet multipliserer man hvert siffer med en potens av tolv og adderer, som vist i eksempelet med tallet 314₁₂ nedenfor:

3×12² + 1×12¹ + 4×12⁰ = 3×144 + 1×12 + 4×1 = 448.

Tallet 314₁₂ i tolvtallsystemet blir altså 448 i titallsystemet.

For å konvertere et tall fra titallsystemet til tolvtallsystemet må man gjentatte ganger utføre heltallsdivisjon med grunntallet 12 og merke seg resten, som vist i eksempelet med tallet 448 nedenfor:

Heltalldivisjon	Rest
448/12 = 37	4
37/12 = 3	1
3/12 = 0	3
	↑

Så begynner man med restene nedenfra. Tallet 448 blir dermed 314₁₂ i tolvtallsystemet.

Mengde- og måleenheter

Flere mengde- og måleenheter er relatert til tolvtallsystemet:

dusin / tylft = 12
skokk = 5 dusin = 60
storhundre = 10 tylfter = 120
gross = 12 dusin = 144 = 100₁₂
tomme = 1⁄12 fot
linje = 1⁄12 tomme
skruppel = 1⁄12 linje
punkt = 1⁄12 skruppel^[1]^[2]

Referanser

^ Sizes.com - Punkt
^ Gamle norske lengdemål Arkivert 26. juni 2009 hos Wayback Machine.

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