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

Latest update: March 6, 2024
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Quaternary Base 4, Octal Base 8; Includes Tables

Numbering System to Decimal Base 10 Lessons

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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

• Base 2 – Binary (removed)
  
• Base 4 – Quaternary
• Base 8 – Octal
• Base 16 – Hexadecimal (removed)

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

Lessons and examples follow

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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

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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

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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

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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

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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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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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How to Do Ternary or Trinary, Base 3 Numbering System Conversions Lesson / Tutorial Examples

Latest update: February 3, 2024

If you understand the everyday decimal (base 10) number system, then you already understand the ternary, base 3 counting and numbering system. You just don’t know you know yet. The complete lesson immediately follows the short semantics note about "ternary" versus "trinary".

Base 3 Conversion - Base 3 to Base 10 and Back - 0 1 2

How to Learn the Ternary Base 3 Numbering System

A complete lesson and examples.

Semantics Note

Ternary is the primary descriptor used to identify base 3 (using the digits 0 1 2) in mathematics as relates to numbering systems. Trinary is the primary descriptor used to identify base three as relates to logic (using the digits -1 0 +1); but the term has also been used in place of ternary. This page does not address the logic definition of trinary. This page is about and explains the base 3 number system; usually called ternary, but sometimes referred to as trinary.

A Quick Review of 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 example would be the number 3528. This number means that there are:

• Eight 1’s,
• two 10’s,
• five 100’s,
• and three 1000's.

Which represents 8 + 20 + 500 + 3000 for a total of 3528.

The Ternary or Base 3 Numbering System...

...uses the same structure, the only difference being the orders of magnitude. Base 3 or ternary has three numbers: 0, 1, and 2.

The orders of magnitude are times three.

• The lowest order number represents itself times one.
• The next order number represents itself times 3.
• The next order number represents itself times 3 x 3, or itself times 9.
• The next order of magnitude would be 3 x 3 x 3, or itself times 27.
• The next order of magnitude would be 3 x 3 x 3 x 3, or itself times 81.

And so on.

Orders of Magnitude in Base 3

• 1 · 3 · 9 · 27 · 81 · 243 · 729 · 2,187 · 6,561

Positional

• 6,561 · 2,187 · 729 · 243 · 81 · 27 · 9 · 3 · 1

A basic, first example of a ternary number would be the base 3 number 11111. This would mean there are:

• one 1,
• one 3,
• one 9,
• one 27,
• and one 81.

Which represents 1 + 3 + 9 + 27 + 81 for a total of 121 in Base 10 decimal.

Another base 3 example would be the number 1120. This number means that there are:

• No 1’s,
• two 3’s,
• one 9,
• and one 27.

Which represents 0 + 6 + 9 + 27 for a total of 42 in base 10 decimal.

Another base 3 example would be the number 2101. This number means there are:

• One 1,
• No 3's,
• One 9,
• And two 27’s.

Which represents 1 + 0 + 9 + 54 for a total of 64 in base 10 decimal.

More Ternary (Base 3) to Base 10 Conversion Examples

9 · 3 · 1	9 · 3 · 1	27 · 9 · 3 · 1
0=0	110=12	220=24
1=1	111=13	221=25
2=2	112=14	222=26
10=3	120=15	1000=27
11=4	121=16	1001=28
12=5	122=17	1002=29
20=6	200=18	1010=30
21=7	201=19	1011=31
22=8	202=20	1012=32
100=9	210=21	1020=33
101=10	211=22	1021=34
102=11	212=23	1022=35

(Convenience relist)

Orders of Magnitude in Base 3

• 1 · 3 · 9 · 27 · 81 · 243 · 729 · 2,187 · 6,561

Positional

• 6,561 · 2,187 · 729 · 243 · 81 · 27 · 9 · 3 · 1

Other base numbering systems:  Try c. 2024: Search results for base (websitewithnoname.com)

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

Latest update: June 9, 2024

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

How to Do Hexadecimal, Base 16 Number System Conversions.
Includes Examples.

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.

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 5681. This number means there are:

• one 1’s,
• eight 10’s,
• six 100’s,
• and five 1000’s.

Which represents 1 + 80 + 600 + 5000; for a total of 5681.

Base 16 uses the same base 10 structure, the only difference being the orders of magnitude.

How to Do the Hexadecimal Base 16 Numbering System

Beware Miscalculations

The orders of magnitude are times sixteen. The lowest-order number represents itself times one. The next-order number represents itself times sixteen. The next order number represents itself times 16x16 or itself times 256. The next order number represents itself times 16x16x16 or itself times 4096. And so on.

Hexadecimal Orders of Magnitude:

1 · 16 · 256 · 4096 · 65536 · 1048576

Positional:

1048576 · 65536 · 4096 · 256 · 16 · 1

Base 16 aka hex has sixteen numbers (0-F). The first ten numbers are the usual 0 thru 9. The next six numbers are A=10, B=11, C=12, D=13, E=14, F=15.

Altogether we have:
0=0, 1=1, 2=2, 3=3, 4=4, 5=5, 6=6, 7=7, 8=8, 9=9,
A=10, B=11, C=12, D=13, E=14, F=15.

A basic, first example of a hexadecimal number would be the base 16 number 11111. This would mean there is:

• one 1,
• one 16,
• one 256,
• one 4096,
• and one 65536.

Which represents 1 + 16 + 256 + 4096 + 65536; for a total of 69905 in Base 10 decimal.

Another base 16 example would be the hex number 5C7F. This number means there are:

• fifteen 1’s,
• seven 16’s,
• twelve 256’s,
• and five 4096’s.

Which represents 15 +112 +3072 + 20480; for a total of 23679 in decimal.

Another base 16 example would be the hex number D24A. This number means there are:

• ten 1’s,
• four 16’s,
• two 256’s,
• and thirteen 4096’s.

Which represents 10 +64 +512 + 53248; for a total of 53834 in decimal.

Hexadecimal Orders of Magnitude

1 · 16 · 256 · 4096 · 65536 · 1048576

Positional

1048576 · 65536 · 4096 · 256 · 16 · 1

Table: Hexadecimal (Base 16) 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. There is no significance attached as to where one column ends and the next one begins.

A=10, B=11, C=12, D=13, E=14, F=15

16 · 1	256 · 16 · 1	65536 · 4096 · 256 · 16 · 1
0=0	16=22	101=257
1=1	17=23	111=273
2=2	1A=26	200=512
9=9	1C=28	3E4=996
A=10	1F=31	3E8=1000
B=11	20=32	BAD=2989
F=15	21=33	FFF=4095
10=16	27=39	1000=4096
11=17	2A=42	1004=4100
12=18	77=119	2BAD=11181
13=19	BD=189	DEAD=57005
14=20	FF=255	10000=65536
15=21	100=256	10100=65792

Simply a Sequential List of Hexadecimal Numbers

Table created using the Microsoft Excel formula: “=DEC2HEX(cell address here)”.

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An Existential Definition of the Binary Encoding of the Universe

Latest update: July 25, 2022. Page URL indicates original publication date; meanwhile, times change and the updates continue.

Theory of a Physical Reality Philosophy -
A Math - Physics - Metaphysics Approach

Alternate titles

• How to Interpret 0 and 1
• The Numbers 0 and 1 Defines Nonexistence versus Existence
• What Is the Mathematical and Metaphysical Concept and Significance of the Numbers 0 and 1
• 1 - The First Non Zero Number - What Is the Meaning of Zero and One
• Binary Reality Fact of Existence
• It's All Ones and Zeros
• Does Negative Zero Exist

One Represented by Many

The Basic Interpretation of 0 and 1

This is a somewhat metaphysical approach as to the existence, meaning, concept, and significance of the numbers 0 and 1. What are the meanings of 0 and 1? What does 0 and 1 really signify?

Zero and One are the most important numbers of all the numbers in the universe. Zero and One defines the difference between existence (1) and nonexistence (0). All other numbers signifying existence can only exist when 1 exists. One is the opposite of none.

1 and 0 are the demarcation defining what is and what is not.

All numbers other than 1 are more of 1 or are none.

0 and 1 are the only numbers times themselves that are themselves. All other numbers become other numbers.

A successful division of 1 by a number other than 1 can only occur when the 1 is not a true 1. The 1 was, in fact, the sum of smaller 1's.

1 is not the first prime number. Contrary to popular belief, 1 is not a prime number at all.

0 exists when 1 or 1's do not exist. -1 cannot exist unless 1 exists. -0 does not exist. or does it?

Does -0 Exist? Otherwise known as Does the Concept of Negative Zero Exist?

If -0 equals 1, then -1 would equal 0; but it does not. So in math, -0 does not exist.

Logic and metaphysical logic, however could be different. Negative can mean not or can mean opposite. A Not 1 means it is equal to anything other than one. An opposite to one would be equal to 0. So not 1 and opposite to 1 have different meanings.

Applying the same logic to zero, however, gives us different results. A not zero means it is equal to anything other than zero; an opposite to zero means it is equal to anything other than zero. So not zero and opposite to zero are equal, whereas not one and opposite to one are not.

It therefore follows that stating -0 is also stating that the concept of not is the same as the concept of opposite. In other words, not equals opposite. That is not true. The meaning of not does not mean the same as the meaning of opposite. So -0 must be interpreted as a false statement, thus -0 does indeed not exist.

The Diverse or Abstract Interpretation of 1 and the Universe

The concept of the number 1 is greatly dependent on the undefined premise of "one what?". Every time science thinks it has found a true one, it invariably turns out to not be so. As an example, the atom was once thought to be the one true basic building block of the universe. But then it turned out the atoms were composed of the smaller 1's of electrons, protons, and neutrons. And now we have quarks, leptons, strangelets, etc. to contend with.

Who knows? Maybe there is no true one. Everything will always be discovered to be composed of something smaller. However, in its most basic form (if we can ever find it), all else derives from one. One is the definition, reality, concept, and source of all existence.

The Binary Base 2 Numbering System

Nothing versus something is the root of the binary system of numbers.

We use Base 10 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 ten. The next order number represents itself times 10x10 or itself times 100. And so on.

An example would be the number 742. This number means that there are:

• two 1’s,
• four 10’s,
• and seven 100’s.

Which represents 2 + 40 + 700; for a total of 742.

The base 2 binary number system uses the same structure, the only difference being the order of magnitude. Base 2 has two numbers (0 and 1) and orders of magnitude that are times two. The lowest-order number represents itself times one. The next-order number represents itself times 2. The next order number represents itself times 2x2 or itself times 4. And so on.

An example would be the number 110. This number means that there are:

• No 1’s,
• one 2,
• and one 4.

Which represents 0 + 2 + 4; for a total of 6.

Other binary examples are:

• 0=0, 1=1
• 10=2, 11=3
• 100=4, 101=5, 110=6, 111=7
• 1000=8, 1001=9, 1010=10, 1011=11, 1100=12, 1101=13, 1110=14, 1111=15
• 10000=16

Binary is the mathematical representation of how the universe encodes itself.

Here are the quick lessons on this and the other: Base Number Counting Systems. All on one page.

.

1 and the Base 10 Pyramid

And what page about the base 10 number 1 would be complete without the usual acknowledgement of the series of 1's times themselves (squared)...

• 1² = 1
• 11² = 121
• 111² = 12321
• 1111² = 1234321
• 11111² = 123454321
• 111111² = 12345654321
• 1111111² = 1234567654321
• 11111111² = 123456787654321
• 111111111² = 12345678987654321

The binary paradigm is the foundation of the universe, including fractals (recursive geometric shapes) and chaos theory (the butterfly effect).

NASA deep space photo demonstrating fractals/chaos theory

There is no reason to believe the concept of "many" excludes the concept of multiple realities.

A visual representation of the theory of parallel and/or alternate universes.

Humankind relative to the universe

There are no extra pieces in the universe (-Deepak Chopra). You are here for a reason.

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