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

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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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