Showing posts with label Math Base Numbering Systems. Show all posts
Showing posts with label Math Base Numbering Systems. Show all posts

How to Do Duodecimal, Dozenal, Base 12 Number System Conversions — Examples, Math Problems

Latest update: October 10, 2023

If you understand the everyday decimal (base 10) number system, then you already understand the duodecimal, base 12, dozenal counting and numbering system. You just don’t know you know yet. The complete lesson immediately follows the short introduction.


Timekeeping is heavily reliant on the number 12 and its composites (evenly divisible numbers: 2, 3, 4, 6).


How to Learn the Duodecimal, Dozenal, Base 12 Numbering System


An Interesting "Political" Side Note.

The base 12 numerical system, also known as the duodecimal or dozenal system, is just like all the other base numbering and counting systems. However, this is the only base numbering system which has a "political" aspect to it. This has to do with the number 12 being a very useful number and as to which symbols to use for the base 10 numbers "10" and "11".

If one wishes to remain within the standardized structure of hexadecimal and the other base numbering and counting systems up to and including base 36, then the use of sequential numbers and letters should be used. Thus, as in hexadecimal, the base 10 number "10" is equal to the base 12 number "A", and the base 10 number "11" is equal to the base 12 number "B".

Others advocate the use of different symbols, some examples being:
  • 10 = T
  • 10 = X
  • 11 = E
This how-to tutorial will stick with the base 2 through base 36 mathematical "standard" of 10 being designated by "A" and 11 being designated by "B".

Complete Lesson and Examples

A quick review of the decimal, base 10 structure...

Base 10, Decimal Orders of Magnitude
1 · 10 · 100 · 1,000 · 10,000 · 100,000
Positional
100,000 · 10,000 · 1,000 · 100 · 10 · 1

We use the base 10 numbering/counting system in our day-to-day living. Base 10 has ten numbers (0-9) and orders of magnitude that are times ten.
  • The lowest order number represents itself times one.
  • The next order number represents itself times 10.
  • The next order number represents itself times 10 x 10, or itself times 100.
  • The next order of magnitude would be 10 x 10 x 10, or 1000.
And so on.

A base 10, decimal example would be the number 7824. This number means there are:
  • Four 1’s,
  • Two 10’s,
  • Eight 100’s,
  • And seven 1000's.
Which represents 4 + 20 + 800 + 7000 for a total of 7824.

The duodecimal, base 12, dozenal numbering system...

...uses the same structure, the only difference being the orders of magnitude. Base 12 aka duodecimal has twelve numbers (0 thru B). The numbers are:
  • 0 = 0, 1 = 1, 2 = 2, 3 = 3, 4 = 4, 5 = 5, 6 = 6, 7 = 7, 8 = 8, 9 = 9
  • A = 10
  • B = 11
The orders of magnitude are times twelve.
  • The lowest order number represents itself times one.
  • The next order number represents itself times 12. The next order number represents itself times 12 x 12, or itself times 144.
  • The next order number represents itself times 12 x 12 x 12, or itself times 1728.
  • The next order number represents itself times 12 x 12 x 12 x 12, or itself times 20736.
And so on.

Duodecimal, Base 12 Orders of Magnitude

1 · 12 · 144 · 1728 · 20736 · 248832

Positional

248832 · 20736 · 1728 · 144 · 12 · 1

A basic, first example of a duodecimal number would be the base 12 number 11111. This would mean there is:
  • one 1,
  • one 12,
  • one 144,
  • one 1728,
  • and one 20736.
Which represents 1 + 12 + 144 + 1728 + 20736 for a total of 22621 in base 10 decimal.

Another base 12 example would be the number 2B9A. This number means there are:
  • Ten 1’s,
  • Nine 12’s,
  • Eleven 144’s,
  • And two 1728’s.
Which represents 10+108+1584+3456 for a total of 5158 in base 10 decimal.

Another base 12 example would be the number A51B. This number means there are:
  • Eleven 1’s,
  • One 12,
  • Five 144’s,
  • And ten 1728’s.
Which represents 11+12+720+17280 for a total of 18023 in base 10 decimal.

Convenience relist...

Base 12, Duodecimal Orders of Magnitude
1 · 12 · 144 · 1728 · 20736 · 248832
Positional
248832 · 20736 · 1728 · 144 · 12 · 1

Side notes...

  • Latitude and longitude are heavily reliant on the number 12 and its multiples and composites.
  • Dice probability theory loves the number 12 composites.
  • Astrology, the zodiac, and ancient cultures recognized the uniqueness of the number 12.

More Duodecimal, Dozenal, Base 12 to Base 10 Conversion Examples

0=0
92=110
B00=1584
1=1
100=144
BBB=1727
5=5
101=145
1000=1728
9=9
110=156
1001=1729
A=10
200=288
1010=1740
B=11
202=290
1100=1872
10=12
20A=298
1111=1885
11=13
20B=299
2000=3456
18=20
210=300
42BB=7343
20=24
7B6=1146
AB2B=18899
5A=70
A00=1440
B460=19656
5B=71
A2B=1475
BBBB=20735


Convenience relist...

Base 12, Duodecimal Orders of Magnitude
1 · 12 · 144 · 1728 · 20736 · 248832
Positional
248832 · 20736 · 1728 · 144 · 12 · 1

The Dozenal Society of America has all sorts of information regarding the mathematical and societal aspects of the base twelve number, counting system.

Some of the Comments from the Previous Hosting Platforms

  • Commenter, Will Apse, said: I'm a bit weird about the number 12. As a kid I used pounds, shillings, and pence for money with 12 pennies in the shilling and twenty shillings in the pound (decimalized when I was 12, lol). This might be why I often think about the oddities of 12's. Money and time are rather important.
  • Commenter, DreamerMeg, said: I was brought up in the UK pre decimal money, same as Will Apse, but it's still difficult for me to get my head round the idea of base 12.
  • Commenter, BradMastersOCcal, said: Binary to base 12 is not as clean as base 8 (3 bits) or 16 (4 bits); this allows binary to just overflow into the next number. Base 12 has to use 4 bits but stop at 1011 (B). It is more like BCD. I had no idea this was a popular common base. Thanks, it is a very interesting article.
  • Commenter, DreamerMeg , said: As a child, we had to learn to count in 12's for the UK's monetary system. Counting in 10's is a lot easier. We had a class of 7-year-olds chanting 12 pence is 1 shilling, 18 pence is 1 and sixpence, 24 pence are two shillings, 30 pence is 2 and sixpence, 36 pence is three shillings! The UK went decimal in 1971 but I can still calculate between new money and old money and between decimal measurement systems (SI units) and the old pounds and ounces. Keeps the brain active, but I don't know that it's useful. Good article.

Side Note. Selecting the relevant Label Menu option below provides a much-expanded list of resources. You can then select one of the listed page titles to make it standalone.

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I first published this article at another website on 11/17/2013. However, to keep the information current, relocating to websitewithnoname.com was best. This copyrighted tutorial has served people well for years.

How to Convert Base 4 and Base 8 to Decimal Base 10 System Lessons - Quaternary and Octal Numbering Systems

Latest update: November 28, 2023
Page URL indicates original publication date; meanwhile times change and the updates continue.

Quaternary Base 4, Octal Base 8; Includes Tables

Numbering System to Decimal Base 10 Lessons

HAL says hi.

Introduction and Start of Base Number Systems Tutorials

These four base numbering system lessons use the exact same teaching methodology. As such, when you have learned one, you will have learned them all. There is also some repetitiveness, purpose being to reduce unnecessary scrolling. Comparisons of different base number systems can also prove useful.

If one understands the everyday, base 10 decimal number system; then you already understand the base 2, base 4, base 8, and base 16 numbering systems. You just don’t know that you know yet.

As you know, we use the decimal, base 10 numbering system in our day-to-day lives. The decimal base 10 system has ten numbers (0-9) and orders of magnitude that are times ten. The lowest-order number represents itself times one. The next-order number represents itself times ten. The next order number represents itself times 10 x 10 or itself times 100. The next order number represents itself times 10 x 10 x 10 or itself times 1000. And so on.

An example of the decimal base 10 system would be the number 7824. This number means there are:
  • Four 1’s,
  • two 10’s,
  • eight 100’s,
  • and seven 1000's.
Which represents 4 + 20 + 800 + 7000; for a total of 7824.

Tutorial continues below for the base numbering system lesson of your choice. This is a large file, your selection may take a few seconds to display the correct section.

Table of Contents

There are also separate tutorials for Base 3 Ternary and Base 5 Quinary and Base 12 Duodecimal.

Lessons and examples follow or select from Table of Contents.




Binary Base 2 to Decimal Base 10 – How to Do and Convert Base 2 to/from Base 10 – Number System Conversions – Includes Examples

0's and 1's




This is an ad note. Due to popular demand, the binary base 2 lesson has been moved to MathSchool.etsy.com.



Quaternary Base 4 to Decimal Base 10 – How to Do and Convert Base 4 to/from Base 10 – Number System Conversions – Includes Examples

0 1 2 3
How to Do Quaternary, Base 4 Number System Conversions.
Includes Examples.

Base 4, also known as the quaternary number system, is predominantly used in DNA genotyping and some electronics applications, etc. [A year 2019 update. Scientists have added four more letters to the DNA alphabet, so base 8 may also be relevant.]

This lesson gives you everything you need to know for converting from quaternary aka base 4 to decimal and for converting from decimal aka base 10 to quaternary. If you understand the decimal number system (or the binary base 2 numbering system for that matter), then you already understand the quaternary (base 4) number system.

Per the introduction, base 10 has ten numbers (0-9) and orders of magnitude that are times ten. The orders of magnitude are l, 10, 100 (10x10) , 1000 (10x10x10), etc.

An example would be the number 7112. This number means that there are:
  • two 1’s,
  • one 10’s,
  • one 100’s
  • and seven 1000’s.
Which represents 2 + 100 + 100 + 7000; for a total of 7112.

How to Do the Quaternary Base 4 Numbering System


Base 4 uses the same base 10 structure, the only difference being the orders of magnitude. Base 4 has four numbers (0-3) and orders of magnitude that are times four . The lowest-order number represents itself times one. The next-order number represents itself times four. The next order number represents itself times 4x4 or itself times 16. The next order number represents itself times 4x4x4 or itself times 64. The next order number represents itself times 4x4x4x4 or itself times 256. And so on.

Orders of Magnitude in Base 4

1 · 4 · 16 · 64 · 256 · 1024· 4096 · 16384

Positional

16384 · 4096 · 1024 · 256 · 64 · 16 · 4 · 1

A basic, first example of a quaternary number would be the base 4 number 11111. This would mean there is:
  • one 1,
  • one 4,
  • one 16,
  • one 64,
  • and one 256.
Which represents 1 + 4 + 16 + 64 + 256; for a total of 341 in Base 10 decimal.

Another base 4 example would be the quaternary number 321. This number means that there are:
  • one 1’s,
  • two 4’s,
  • and three 16’s.
Which represents 1 + 8 + 48; for a total of 57 in decimal.

Another base 4 example would be the quaternary number 3023. This number means that there are:
  • three 1’s,
  • two 4’s,
  • no 16’s,
  • and three 64’s.
Which represents 3 + 8 + 0 + 192; for a total of 203 in decimal.

Orders of Magnitude in Base 4

1 · 4 · 16 · 64 · 256 · 1024· 4096 · 16384

Positional

16384 · 4096 · 1024 · 256 · 64 · 16 · 4 · 1

Table: Quaternary (Base 4) to Decimal (Base 10) Conversion Examples

Column headings in the following table are simply a convenience relist of the relevant positional orders of magnitude as applies to each column.

4 · 1
16 · 4 · 1
64 · 16 · 4 · 1
0=0
21=9
200=32
1=1
22=10
222=42
2=2
23=11
223=43
3=3
30=12
333=63
10=4
33=15
1000=64
11=5
100=16
1100=80
12=6
102=18
2000=128
13=7
120=24
2030=140
20=8
122=26
3122=218



Octal Base 8 to Decimal Base 10 – How to Do and Convert Base 8 to/from Base 10 – Number System Conversions – Includes Examples

0 1 2 3 4 5 6 7
How to Do Octal, Base 8 Number System Conversions.
Includes Examples.

Base 8, also known as the octal number system, is mostly used in electronics and some DNA applications, etc.

Here is everything you need to know on how to convert from octal aka base 8 to decimal. And for converting from decimal aka base 10 to octal.

As previously stated: if you understand the decimal (base 10) number system you use every day, then you already understand the octal (base 8) numbering system.

Per the introduction, base 10 has ten numbers (0-9) and orders of magnitude that are times ten. The orders of magnitude are l, 10, 100 (10x10) , 1000 (10x10x10), etc.

An example would be the number 2375. This number means that there are:
  • five 1’s,
  • seven 10’s,
  • three 100’s
  • and two 1000’s.
Which represents 5 + 70 + 300 + 2000; for a total of 2375.

How to Do the Octal Base 8 Numbering System


Electonics and Base 8 Octal

Base 8 uses the same base 10 structure, the only difference being the orders of magnitude. Base 8 has eight numbers (0-7) and orders of magnitude that are times eight. The lowest-order number represents itself times one. The next-order number represents itself times eight. The next order number represents itself times 8x8 or itself times 64. The next order number represents itself times 8x8x8 or itself times 512. And so on.

Orders of Magnitude in Base 8

1 · 8 · 64 · 512 · 4096 · 32768 · 262144

Positional

262144 · 32768 · 4096 · 512 · 64 · 8 · 1

A basic, first example of an octal number would be the base 8 number 11111. This would mean there is:
  • one 1,
  • one 8,
  • one 64,
  • one 512,
  • and one 4096.
Which represents 1 + 8 + 64 + 512 + 4096; for a total of 4681 in Base 10 decimal.

Another base 8 example would be the octal number 321. This number means that there are:
  • one 1’s,
  • two 8’s,
  • and three 64’s.
Which represents 1 + 16 + 192; for a total of 209 in decimal.

Another base 8 example would be the octal number 4075. This number means that there are:
  • five 1’s,
  • seven 8’s,
  • no 64’s,
  • and four 512’s.
Which represents 5 + 56 + 0 + 2048; for a total of 2109 in decimal.

Orders of Magnitude in Base 8

1 · 8 · 64 · 512 · 4096 · 32768 · 262144

Positional

262144 · 32768 · 4096 · 512 · 64 · 8 · 1

Table: Octal (Base 8) to Decimal (Base 10) Conversion Examples

Column headings in the following table are simply a convenience relist of the relevant positional orders of magnitude as applies to each column.

8 · 1
8 · 1
512 · 64 · 8 · 1
0=0
15=13
100=64
1=1
16=14
165=117
2=2
17=15
200=128
7=7
20=16
534=348
10=8
25=21
1000=512
11=9
34=28
1100=576
12=10
50=40
2000=1024
13=11
55=45
2006=1030
14=12
77=63
2011=1033



Hexadecimal Base 16 to Decimal Base 10 – How to Do and Convert Base 16 to/from Base 10 – Number System Conversions – Includes Examples

Hex: 0-9, A a, B b, C c, D d, E e, F f

Hexadecimal (base 16) is the primary base numbering system used by computer programmers. Hex code is used in everything from core dumps to color codes and everything in-between.

How to Do the Hexadecimal Base 16 Numbering System



This is an ad note. Due to popular demand, the hexadecimal base 16 lesson has been moved to MathSchool.etsy.com.


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Base 5 Quinary Numbering System to Base 10 Decimal - Lesson

Latest update: August 8, 2022
Page URL indicates original publication date; meanwhile, times change and the updates continue.

How to learn and do quinary conversion; the base 5 numbering system; 0 1 2 3 4; complete lesson and lots of examples. Includes short table of most-searched-for quinary, base 5 to decimal base 10 questions and answers.

There is also access to other base numbering system lessons. If ever might be needed, well worth noting for future reference.

Base 5 - Quinary

If you understand the decimal (base 10) number system we use in our everyday lives, then you already understand the quinary (base 5) number system. You just don’t know that you know yet.

We use the decimal (base 10) number system in our day-to-day living. Base 10 has ten numbers (0-9) and orders of magnitude are times 10. The lowest-order number represents itself times one. The next-order number represents itself times 10. The next order number represents itself times 10x10 or itself times 100. The next order number represents itself times 10x10x10 or itself times 1000. And so on.

A decimal base 10 example would be the number 2417. This number means there are:
  • seven 1’s,
  • one 10,
  • four 100’s,
  • and two 1000’s.
Which represents 7 + 10 + 400 + 2000; for a total of 2417.

The Quinary, Base 5 Numbering System Uses the Same Structure...

...the only difference being the order of magnitude. Base 5 has five numbers (0 1 2 3 4) and orders of magnitude are times five. The lowest-order number represents itself times one. The next-order number represents itself times 5. The next order number represents itself times 5x5 or itself times 25. The next order of magnitude would be 5x5x5 or 125. And so on.

A base 5 to decimal base 10 example would be the number 3142. This number means there are...
  • two 1’s,
  • four 5’s,
  • one 25,
  • and three 125's.
Which represents 2 + 20 + 25 + 375 for a total of 422 in base 10.

Another base 5 to decimal base 10 example would be the number 2011. This number means there are...
  • One 1,
  • one 5,
  • no 25's,
  • and two 125's.
Which represents 1 + 5 + 0 + 250; for a total of 256 in base 10.

Some of the Most Searched for Base 5 Questions and Answers

  • What is 5 (base 10) in base 5? Answer is 10 in base 5.
  • What is 10 (base 10) in base 5? Answer is 20 in base 5.
  • What is 23 (base 10) in base 5? Answer is 43 in base 5.
  • What is 25 (base 10) in base 5? Answer is 100 in base 5.
  • What is 27 (base 10) in base 5? Answer is 102 in base 5.


Quinary / Base 5 Orders of Magnitude

1 - 5 - 25 - 125 - 625 - 3125 - 15625 - 78125 - 390625 - 1953125

Positional

1953125 - 390625 - 78125 - 15625 - 3125 - 625 - 125 - 25 - 5 - 1

List of Base 5 to Decimal Base 10 Numbering System Conversion Examples

  • 1 = 1
  • 2 = 2
  • 3 = 3
  • 4 = 4
  • 10 = 5
  • 11 = 6
  • 12 = 7
  • 13 = 8
  • 14 = 9
  • 20 = 10
  • 21 = 11
  • 22 = 12
  • 23 = 13
  • 24 = 14
  • 30 = 15
  • 31 = 16
  • 32 = 17
  • 33 = 18
  • 34 = 19
  • 40 = 20
  • 41 = 21
  • 42 = 22
  • 43 = 23
  • 44 = 24
  • 100 = 25
  • 101 = 26
  • 102 = 27
  • 103 = 28
  • 104 = 29
  • 110 = 30
  • 111 = 31

Quinary / Base 5 Orders of Magnitude (convenience relist)

1 - 5 - 25 - 125 - 625 - 3125 - 15625 - 78125 - 390625 - 1953125

Positional

1953125 - 390625 - 78125 - 15625 - 3125 - 625 - 125 - 25 - 5 - 1

List of More Base 5 to Decimal Base 10 Examples
  • 200 = 50
  • 220 = 60
  • 222 = 62
  • 300 = 75
  • 303 = 78
  • 333 = 93
  • 404 = 104
  • 420 = 110
  • 1000 = 125
  • 1001 = 126
  • 1010 = 130
  • 1100 = 150
  • 1234 = 194
  • 2020 = 260
  • 2030 = 265
  • 3020 = 385
  • 3411 = 481
  • 4000 = 500
  • 4040 = 520
  • 4242 = 572
  • 4321 = 586
  • 4333 = 593
  • 4444 = 624

Quinary / Base 5 Orders of Magnitude (convenience relist)

1 - 5 - 25 - 125 - 625 - 3125 - 15625 - 78125 - 390625 - 1953125

Positional

1953125 - 390625 - 78125 - 15625 - 3125 - 625 - 125 - 25 - 5 - 1

If interested, here is access to the tutorials for Base 2, 4, 8, 16 (Binary, Quaternary, Octal, Hexadecimal); all on one page. If not needed now, maybe useful for future use.

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