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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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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 · · 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 · · 3 · 1

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



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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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Scope and Meaning of Metaphysics - Examples - Thinking Outside the Box

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

What Is Metaphysics and the Metaphysical –
Science / Philosophy / Spiritual

An artist's rendition of metaphysics - Source: NASA

The Undefined Science

Metaphysics – not even Stanford and Wikipedia can agree on neither a succinct nor an encompassing meaning or definition for metaphysics. This is because:
  • The definitions and meanings of metaphysics and the metaphysical keep changing.
  • The study and science of metaphysics includes an almost unlimited number of subcategories.
  • The most respected minds in academia, the sciences, and even in the general population are continually unable to reach an agreement as to the perfect meaning or definition.
  • There are disputes as to what does and does not belong in the study or category of metaphysics and the metaphysical.

One definition or meaning for metaphysics could be the sciences and non-sciences that cannot be put in another category are put into the category of metaphysics. However, this meaning or definition would not be complete; because there are sciences that are studied and categorized in other categories that are also included as a part of metaphysics and the metaphysical.

For certain disciplines, the discipline is a subset of metaphysics; and metaphysics is a subset of the discipline. In other words, each is a subset of the other depending on the context and/or the parameters used.

An example:
A. There is the philosophical part of metaphysics.
B. There is the metaphysical part of philosophy.

What is Metaphysics and the Metaphysical -
Thinking Outside the Box

Problems can have more than one solution.

The scope of subjects such as cosmology, ontology, physics, philosophy, existence, time and space, astrobiology, perception, etc are generally undisputed as being both part of metaphysics and outside of metaphysics. It all depends on the subcategory of the main subject being addressed.

Other examples and subjects continually being associated with the scope of metaphysics include paranormal, astrology, New Age, spiritual, meditation, wisdom, karma, determinism, fate, shaman, unity, reiki, psychic, mind and matter, meaning of life, feng shui, etc.; the metaphysical and quasi-metaphysical list can be a long one.

Metaphysics is the perfect place to “think outside the box”. One is allowed to research any theory or hypothesis you wish. You are allowed to follow any chain of logic to wherever it goes. Combining what first appears to be unrelated subjects to derive unusual metaphysical conclusions is perfectly acceptable. In other words, the scope of metaphysics allows one to explore and experiment without constraint.

Philosophy vs Metaphysics -
Perception / Perspective / Reality



What Is Spiritual Metaphysics

Spiritual metaphysics versus religion. What is spiritual metaphysics? It is similar, but not the same as religion.
  • What is religion? Religion is where one makes up whatever they want.
  • Spiritual metaphysics on the other hand, attempts to base its conclusions on logic. As humankind's logic and thinking abilities evolve, so does spiritual metaphysics.

Metaphysical and Philosophical Thoughts, Quotes, Ramblings, Examples... And possibly a little scattered humor.

  • In the beginning there was nothing, which exploded.
  • With all that goes on in life and reality, the only possible conclusion is we are all lab rats.
  • Most everything means nothing.
  • Time can sometimes cure a problem all by itself. Though can be an excellent excuse for procrastination; it is more often than not, not a wise choice.
  • Free will exists. But it is limited. And it is more limited with some than with others.
  • Sometimes one can actually choose whether a problem exists or not.
  • Faith is the death of intelligence.
  • The fatal flaw of logic is its presumption of being aware of all relevant premises. However, awareness of all relevant factors is seldom the true situation. The best one can hope for is a probable prediction.
  • No one belongs here more than you.
  • No one belongs here less than you.
  • One of the most important lessons a person can learn in life is that other people are as real behind their eyes as you are behind yours.
  • Miscommunication can cause more problems than all other factors combined. Always examine a perceived negative statement twice; it may not be negative at all. Give others the benefit of the doubt.
  • Time and Space. Can't live with it. Can't live without it.
  • Life is an iterative process.
More quotes of all kinds: Best Quotes.

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Negative Customer Review re: MasterCool Evaporative / Swamp Cooler - And Problems

Latest update: April 23, 2023. Page URL indicates original publication date; meanwhile, times change and the updates continue.

Another summer is on the way. Here is an unfortunately negative product reliability customer review of MasterCool evaporative aka swamp coolers, plus water cooler pump installation and replacement instructions.

It should be noted that water cooler pumps are the least of MasterCool's poor quality control and reliability problems.

The primary purpose of this page is a review of the MasterCool and same-company other brands of evaporative aka swamp coolers. The detailed How-to-Replace-Water-Cooler-Pump segment was included simply because it is such a frequent problem with any and all evaporative swamp coolers. The pump failure is a side issue, it is the other poor quality standards and reliability issues of MasterCool evaporative aka swamp coolers that resulted in the publishing of this negative review. These issues are addressed immediately following the water cooler pump replacement segment.

MasterCool window evaporative cooler.

My personal experience and what this review specifically is referring to is the MasterCool MCP44 Evaporative Cooler. However, the same company that makes MasterCool also makes the Champion and Essick brands of evaporative swamp coolers. If a company makes a poorly manufactured one brand or model, it's a pretty good bet their other brands are of equally poor quality. That's a decision only you can make. At minimum, you definitely want to at least avoid MasterCool.

The first problem was the failed water pump, it lasted a little more than a year. If you are only here for the review, you will find the first paragraph immediately below the cooler pump picture informative; then might as well skip past the numbered list to the next section where things really get interesting. There are also pictures after the review showing what the inside of the MasterCool window evaporative swamp cooler looks like.


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About and How to Replace or Install the MCP44 Series MasterCool Evaporative Water Cooler Pump (instructions will probably work equally well with many other brands and models)


Model ESK5500 Cooler Pump.
There is additional information and pictures of the MasterCool evaporative swamp cooler with the back panel removed following the review.

I acquired the new MasterCool MCP series evaporative swamp cooler unit a few years ago. The water pump lasted a little longer than a year. If you think getting the back panel off the evaporative cooler unit is a major project, just wait until it's time to correctly put the thing back on. There sat the water pump; "Made in China" it duly informed me. Water pumps are the known weak link when it comes to evaporative swamp coolers, the manufacturers know this and so try to make the component fairly easy to replace. Here is the procedure, don't forget to check out the additional pictures and warnings following the main review.

Read the entire list and check out the additional pictures and information a few times before beginning the actual step-by-step pump replacement process. You will then have a pretty good idea of the overall procedure and there will be fewer surprises. In other words, now that you know the hazard points, things will go a lot quicker and easier. In fact, once that back panel is off, the whole procedure will pretty much be intuitive. All the information and pictures makes it sound a lot more complicated than it really is. "Check list" might be a better description than "step-by-step".
  1. Check the electrical info on the replacement water pump and plug into a wall socket for a couple seconds to be sure the new pump works. Note if the new pump already has a protective screen wrapped around the bottom of it.
  2. Unplug the entire unit from the wall socket and turn off the water feed. 
  3. Read the manual. Among other things, it tells you how to remove the back panel and about the water hose you need to disconnect at the top and inside of the unit before you can completely remove the back panel. The additional pictures further down this page have more info.
  4. Drain/siphon water from tray.
  5. Very carefully retrieve the water-protected water pump electrical power cord from the enclosure. Definitely peruse the additional pictures and information further down the page before attempting this, otherwise you might accidentally unplug the cord while it is still in the protected enclosure. If that happens, retrieval of the inside cord could be a major problem. Once the plugin part is reached and extracted, do NOT unplug yet.
  6. Unbolt and/or unscrew the clamps/brackets/etc. that are holding the pump in place.
  7. Pull off the water hose from the pump. There may or may not be a clamp you have to undo first.
  8. Remove the still plugged in old pump and set aside elsewhere on the tray. If no new screen was provided with the new pump, retrieve and clean the filter screen wrapped around the bottom of the old pump.
  9. More than likely, the entire bottom of the unit is filled with peeled paint and other debris. Now is a good time to clean up and get rid of all that. You'll have to move the old pump around while doing this, maybe place it on top of the bracket. Do not unplug it.
  10. Bring out the new pump. It will also be made in China, apparently no other options are  available for this unit. If needed, wrap old screen around new pump as it was on the old pump.
  11. Place the new pump where the old pump originally was.
  12. Make sure the water hose, electrical cord, and bracket(s) are all completely untangled from each other. Review pictures.
  13. Unplug the old pump. Do not let go of the cord coming from inside of the housing, otherwise it might slip back inside; you do not want that to happen. Plug in the new pump. Reinsert electrical cord back into the water-protected location. Don't reattach plastic cap yet.
  14. Reconnect the water hose.
  15. Position everything as you want it to be and attach pump to all the previous bracket and other connection points.
  16. Check tray. Remove all tools, parts, rags, the old pump if it is still laying there, and everything else that doesn't belong.
  17. Turn the water feed back on and confirm the water level rises to the level you wish it to be, give it at least 15 minutes. The higher, the better; but not above the overflow drain height. Adjust float if necessary.
  18. Time for the test. Unhook/Pull away the water hose from the plastic holders on evaporative swamp cooler and make sure the hose is pointed at the ground and well away from the pump and tray. The absolute last thing you want to do is spray either of those pumps with water. Plug evaporative swamp cooler back in the wall socket; turn the pump on for several seconds to check that everything works. Give the fan a couple seconds as well. Turn everything off and re-unplug the cooler.
  19. Push the water hose back into the plastic holders on the swamp cooler. Review pictures and accompanying information. Reattach plastic cap. Give everything one last, good look over.
  20. Time to reinstall the back panel. Brute strength and ingenuity will be required to get that thing setting back on top of the tray. Reconnect the hose at the top. You should be able to get your hand in there; grab the rubber hose; and force it back to the top of the tube. Twist and turn the hose as needed to remove any kinks.
  21. Time for more brute strength, ingenuity, and persistence. You will need to lift the panel about an inch or two above the tray; line it up with the sliders on the sides of the cooler; then shove panel flat against the cooler and pull down, hooking the panel back onto the sliders. Multiple attempts will probably be required. When there are no gaps on the sides and between the tray and panel, it probably means you succeeded. I didn't bother putting the two screws back in, that panel wasn't going anywhere. Recheck the water feed valve is still on.
  22. Plug the unit back in the wall socket; ponder that switch panel for a moment before reaching for it... Check the back panel that all the pads are getting wet. If they are, then it looks like you were successful in reinstalling the back panel correctly. Good luck.
If the water cooler pump had been the only incident, I would not have written this negative review. It is what happened next that pretty much made this review mandatory.

The Day the MasterCool Died...

Everything worked fine for another year or two. Then came that fateful morning...

It was going to be a hot one, temps in the 90's were on the way. I turned the water pump on to soak the pads as usual for five minutes before turning on the fan. The little, green light glowed benignly and the water pump whirred happily.

I then went back and turned on the fan. The fan started up. Then the whole unit suddenly shut down. The fan. The pump. The switch lights. Everything.

I tried again. Everything shut down again. I tried different combinations of turning on the various switches. Self-shutdown every time. I tried using the remote instead. Same results.

I somehow sensed and knew I was already doomed. But I went through the motions and checked the house circuitry and fuse/switch box six ways from Sunday. There were no problems; that cooler was getting uninterrupted power.

I tried messing with the switches again, same results. Then the symptoms changed. At first, the pump and the fan worked fine when each was turned on alone. But whenever I turned the second one on, that's when the whole system would shut down.

Suddenly though, now with each attempt, the switch lights would flicker on and off at random for a few seconds before shutdown occurred. In other words, the lights would start doing a strange, little dance; water pump and blower fan sometimes automatically responding accordingly, sometimes not.

As for checking for loose wire connections, chip creep, or anything else a non-expert might be able to fix; forget it. The switch panel circuit board location was completely inaccessible.

I'm standing there looking at the thing after it had done its little dance and shutting itself off for the 30th time or whatever. Then MasterCool decided it was time to perform the coup de grĂ¢ce to any remaining hopes of repair. It turned itself on. That's right. The unit started itself, all the lights happily flickering back and forth for several seconds with the fan and pump sometimes joining in, and then once again shutting itself down . That control circuit board was not only toast, it was unsafe.

"And that truly is indeed that," thought I. I unplugged the unit for the last time.

I'm sitting at my desk, pondering my next move, when I happened to glance down and notice the back page of the owner's manual (printed in China by the way). It proudly informed me the Essick, Champion, MasterCool family of evaporative swamp coolers are designed, assembled, and serviced in the USA. What they don't mention is all the components were made in China or elsewhere.

As far as I'm concerned, lying by omission is still lying. I put the odds at 99.9% that circuit board switch panel was made in China. Even if it wasn't; it was still a low quality, poorly manufactured component no matter where it was made. And you can bet Champion, Essick, MasterCool brands all use the same supplier(s). That's when I decided to write this review.

Back page of MasterCool manual.

This page is just my opinion. However, evaporative swamp coolers are an expensive proposition. Please do the homework and research before making that final decision. Suddenly having your cooling system malfunction during a hot summer day is not a pleasant experience.

Side note. Here's a video review (opens in new tab or window) from a new buyer of the MasterCool MCP44 series brand. He has both positive and negative things to say about it. He also mentions one very serious flaw. Basically, every insect in the neighborhood will end up inside your home; he'll tell you all about that. I'd wondered where all those moths and gnats were coming from, I even had an indoor mosquito. Now I know.

I would avoid MasterCool evaporative swamp coolers at all costs. And since Essick and Champion are made by the same company with the same poor quality standards, I would seriously try to avoid those as well. Unfortunately, there seems to be some sort of monopoly situation in play. Other brands are hard to find. If anyone can recommend another brand, there are probably a whole lot of people who would very seriously appreciate hearing about it in the comments section.

An Update. A contractor recently told me how to get the front panel off to access the circuit board(s). Basically, you scrape off the plastic at certain points (four, circular, indented, perimeter) and the screws are underneath.

[End of Review. Pictures follow.]

Here Are Some Pictures of What You Will Find When You Remove the Back Panel of the MCP44 Series Evaporative Swamp Cooler


There are two more screw holes at the base. Both were covered with silicon dry gel.
When I removed the gel, no screws were present. Your results may vary.

Once the screws are removed, slide the back panel up off the side slots (remember about those side slots, they'll be your nemesis when reinstalling the panel) and pull panel away from the cooler just a little bit. Do not try to remove the panel yet, the water hose is still attached. Using a flat edge screwdriver will help pry/pull the thing off. Won't be that difficult.

Success. The panel can now be removed. Brute strength and ingenuity will be required.

Welcome to the inside of the MasterCool evaporative swamp cooler.
Note the water hose running along the length of the left side, it is easily detachable and re-attachable to the three plastic holders you see. You'll be doing that when you do your quick test at the end.
The two pumps you see in the tray are the water pump and purge pump. Note the two electrical wires running from the pumps to the square, black hole at the bottom-left of the fan housing.
At lower-right, on the outside, is the water feed.

Where the pump wires go. That black, square aperture in the previous picture was originally covered with the white, square, plastic cover you see. You will need to remove the two screws and plastic cover. These three items are prime candidates for getting lost, so I put them in my empty, front pants pocket.

Now comes the risky business part. You need to fish those wires out of that hole. Do it slowly, carefully, and gently jiggle the wires whenever feeling any resistance. Pulling the wires too hard will unplug them while they are still inside the housing. You do not want that to happen.

You will eventually end up with the two plugs on the outside. Keeping in mind the warning in the first set of instructions, pull out the old cooler pump plug and plug in the new one at the appropriate time and as described.
Reinsert the wires back into the housing; and adjust everything the way you want it to be. Then finally put the plastic cover and two screws back on the electrical access aperture.


Everything reattached and ready to go. Time to do the test as described in the first segment.


After the test, refasten the hose back into the plastic holders. Check entire length for kinks.

A side note picture of the floater, controlling the water level in the tray. Slightly bending the floater rod up or down will change the water level accordingly.


A side note, example picture close up of how the water feed and valve might be connected to the water supply. Configurations vary.

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