Difference between revisions of "Computer Number Systems"

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All digital computers – from supercomputers to your smartphone – ultimately can do one thing: detect whether an electrical signal is on or off.  
All digital computers, from supercomputers to your smartphone, are electronic devices and ultimately can do one thing: detect whether an electrical signal is on or off. That basic information, called a ''bit'' ('''bi'''nary digi'''t'''), has two values: a 1 (or ''true'') when the signal is on, and a 0 (of ''false'') when the signal is off.  
That basic information, called a ''bit'' ('''bi'''nary digi'''t'''), has two values: a 1 (or ''true'') when the signal is on, and a 0 (of ''false'') when the signal is off. Larger values can be stored by a group of bits. For example, 3 bits together can take on 8 different values.  


Computer scientists use the [https://en.wikipedia.org/wiki/Binary_number binary number system] (that is, base 2) to represent the values of bits. Proficiency in the binary number system is essential to understanding how numbers and information is represented in a computer. Since binary numbers representing moderate values quickly become rather lengthy, bases eight ([https://en.wikipedia.org/wiki/Octal octal]) and sixteen ([https://en.wikipedia.org/wiki/Hexadecimal hexadecimal]) are frequently used as shorthand. In octal, groups of 3 bits form a single octal digit; in hexadecimal, group of 4 bits form a single hex digit.
Larger values can be stored by a group of bits. For example, there are 4 different values stored by 2 bits (00, 01, 10, and 11), 8 values for 3 bits (000, 001, 010, 011, 100, 101, 110, and 111), 16 values for 4 bits (0000, 0001, ..., 1111), and so on. However, large numbers, using 0s and 1s only, are quite unwieldy for humans. For example, a computer would need 19 bits to store the numbers up to 500,000! We use of different number systems to work with large binary strings that represent numbers within computers.


In this category, we will focus on conversion between binary, octal, decimal, and hexadecimal numbers. There may be some arithmetic in these bases, and occasionally, a number with a fractional part. We will not cover how negative numbers are represented in binary.
== Different Number Systems ==
 
A ''number system'' is the way we name and represent numbers. The number system we use in our daily life is known as '''decimal number system''' or '''base 10''' because it is based on 10 different digits: 0, 1, 2, ..., 8, and 9. The base of number is written as subscript to a number. For example, the decimal number "twelve thousand three hundred and forty-five" is written as $12345_{10}$. Without a base or explicit context, it's assumed that a number is in base 10.
 
In computer science, apart from the decimal system, three additional number systems are commonly used: '''binary''' (base-2), '''octal''' (base-8), and '''hexadecimal''' or just '''hex''' (base-16). Binary numbers are important because that is how numbers are stored in the computer. Octal and hexadecimal are used to represent binary numbers in a user-friendly way. Each octal symbol represents 3 binary bits and each hex digit represents 4 binary bits. For example, the decimal number 53201, stored as in the computer as the binary number 10000001111000101100, is represented by the octal number 2017054, and the hex number 81E2C. The table below compares the number systems:
:{| class="wikitable"
!Number System
!Base
!Digits Used
!Examples
|-
|Binary
|2
|0,1
|$10110_{2}$, $10110011_{2}$
|-
|Octal
|8
|0,1,2,3,4,5,6,7
|$75021_{8}$, $231_{8}$, $60012_{8}$
|-
|Decimal
|10
|0,1,2,3,4,5,6,7,8,9
|$97425_{10}$ or simply 97425
|-
|Hexadecimal
|16
|0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
|$54A2DD0F_{16}$
|}
 
 
In general, $N$ bits have $2^N$ different values. The computers aboard the Apollo spacecraft had 8-bit words of memory. Each word of memory could store 256 different values, and the contents were displayed using 2 hex characters.
 
The following table shows the first 20 numbers in decimal, binary, octal, and hexadecimal:
:{| class="wikitable" style="text-align:center"
!Decimal
!Binary
!Octal
!Hexadecimal
|-
|0
|0
|0
|0
|-
|1
|1
|1
|1
|-
|2
|10
|2
|2
|-
|3
|11
|3
|3
|-
|4
|100
|4
|4
|-
|5
|101
|5
|5
|-
|6
|110
|6
|6
|-
|7
|111
|7
|7
|-
|8
|1000
|10
|8
|-
|9
|1001
|11
|9
|-
|10
|1010
|12
|A
|-
|11
|1011
|13
|B
|-
|12
|1100
|14
|C
|-
|13
|1101
|15
|D
|-
|14
|1110
|16
|E
|-
|15
|1111
|17
|F
|-
|16
|10000
|20
|10
|-
|17
|10001
|21
|11
|-
|18
|10010
|22
|12
|-
|19
|10011
|23
|13
|-
|20
|10100
|24
|14
|}
 
== Converting to Decimal ==
 
As we learned in the first or second grade, the decimal value of a decimal number is simply sum of each digit multiplied by its place value:
 
:$12345 = 1×10^{4}+ 2×10^{3}+ 3×10^{2}+ 4×10^{1}+ 5×10^{0} =  10000 + 2000+ 300 +40+5 = 12345$
 
:$3079 = 3×10^{3}+ 0×10^{2}+ 7×10^{1}+ 9×10^{0} = 3000 + 70 + 9 = 3079$
 
Analogously, the decimal value of a number in an arbitrary base is the sum of each digit multiplied by its place value. Here are some examples of converting from one base into base 10:
 
:$1101_{2}  = 1 × 2^3  + 1 × 2^2  + 0 × 2^1  + 1 × 2^0 = 8 + 4  + 0  + 1  = 13_{10}$
 
:$175_{8}  = 1 × 8^2  + 7 × 8^1  + 5 × 8^0 = 1 × 64 + 7 × 8 + 5 × 1 = 64 + 56 + 5 = 125_{10}$
 
:$A5E_{16}  = 10 × 16^2 + 5 × 16^1 + 14 × 16^0 = 10 × 256 + 5 × 16 + 14 × 1 = 2560 + 80 + 14 = 2654_{10}$
 
== Converting from Decimal ==
 
The algorithm to convert a number from an arbitrary base requires finding how many times successive powers of the base go into the number, starting with the largest power of the base less than the starting number. For example, converting 3306 from base 10 to octal proceeds as follows:
 
1) The largest power of 8 that is less than or equal to 3306 is 512 ($8^3$). Divide 3306 by 512:
:$3306 = 6*8^3 + 234$ 
 
2) The next power of 8 is 64 ($8^2$). Divide 234 by 64:
:$234 = 3 *8^2 + 42$
 
3) The next smaller power of 8 is 8 ($8^1$). Divide 42 by 8:
:$42 = 5*8^1 + 2$
 
4) Finally, the next smaller power of 8 is 1 ($8^0$). Divide 2 by 1:
:$2 = 2 * 8^0$
 
The answer is $6352_8$.
 
== Converting between Binary, Octal, and Hexadecimal ==
 
Converting from octal to binary is simple: replace each octal digit by its corresponding 3 binary bits. For example:
 
:$375_{8} = \ \ 011\ \ \ 111\ \ \ 101_{2} = 11111101_2$
 
Converting from hex to binary is also simple: replace each hex digit by its corresponding 4 binary bits. For example:
 
:$FD_{16} = \ \ 1111\ \ \ 1101_{2} = 11111101_2$
 
Converting from binary to either octal or hex is pretty simple as well: group the bits by 3s or 4s (starting at the right), and convert each group:
 
:$10000001111000101100 = 10\ 000\ 001\ 111\ 000\ 101 \ 100 = 2017054_8$
 
:$10000001111000101100 = 1000\ 0001\ 1110\ 0010\ 1100 = 81E2C_{16}$
 
Converting between base 8 and 16 is easy by expressing the number in base 2 (easy to do!) and then converting that number from base 2 (another easy operation)! This is shown below in Sample Problem #1.
 
== Using Hexadecimal Numbers to Represent Colors ==
 
Computers use hexadecimal numbers to represent various colors in computer graphics because all computer screens use combinations of red, green, and blue light or RGB to represent thousands of different colors. Two digits are used for each so the hexadecimal number “#FF0000” represents the color red, “#00FF00” represents green, and “#0000FF” represents blue.  The color black is “#000000” and white is “#FFFFFF”.
The hash tag or number sign is used to denote a hexadecimal number.  $FF_{16} = F (15) × 16 + F (15) × 1 = 240 + 15 = 255_{10}$ so there are 0 to 255 or 256 different shades of each color or $256^{3} = 16,777,216$ different colors.
The following web site has nearly every color name, along with its hex code and decimal values: 
:https://www.rapidtables.com/web/color/RGB_Color.html
For example “salmon” is “#FA8072” which represents the decimal numbers 250 (hex FA), 128 (hex 80), and 114 (hex 72).


== Resources ==
== Resources ==


[[https://ryanstutorials.net/ Ryan's Tutorials]] covers this topic in beautifully. Rather than trying to duplicate that work, we'll point you to the different sections:
[https://ryanstutorials.net/ Ryan's Tutorials] covers this topic beautifully. Rather than trying to duplicate that work, we'll point you to the different sections:


:[https://ryanstutorials.net/binary-tutorial/ 1. Number Systems] - An introduction to what numbers systems are all about, with emphasis on decimal, binary, octal, and hexadecimal. ACSL will typically identify the base of a number using a subscript. For example, $123_8$ is an octal number, whereas $123_{16}$ is a hexadecimal number.  
:[https://ryanstutorials.net/binary-tutorial/ 1. Number Systems] - An introduction to what numbers systems are all about, with emphasis on decimal, binary, octal, and hexadecimal. ACSL will typically identify the base of a number using a subscript. For example, $123_8$ is an octal number, whereas $123_{16}$ is a hexadecimal number.  
Line 18: Line 227:
:[https://ryanstutorials.net/binary-tutorial/binary-negative-numbers.php 4. Negative Numbers] - ACSL problems will not cover how negative numbers are represented in binary.  
:[https://ryanstutorials.net/binary-tutorial/binary-negative-numbers.php 4. Negative Numbers] - ACSL problems will not cover how negative numbers are represented in binary.  


:[https://ryanstutorials.net/binary-tutorial/binary-floating-point.php 5. Binary Fractions and Floating Point] - The first part of this section is relevant to ACSL: fractions in other bases. ACSL will not cover floating point numbers in other basis. So, focus on the section [https://ryanstutorials.net/binary-tutorial/binary-floating-point.php#convertfraction Converting to a Binary Fraction], but keep in mind that ACSL problems may also cover octal and hexadicmal fractions.
:[https://ryanstutorials.net/binary-tutorial/binary-floating-point.php 5. Binary Fractions and Floating Point] - The first part of this section is relevant to ACSL: fractions in other bases. ACSL will not cover floating point numbers in other basis. So, focus on the section [https://ryanstutorials.net/binary-tutorial/binary-floating-point.php#convertfraction Converting to a Binary Fraction], but keep in mind that ACSL problems may also cover octal and hexadecimal fractions.
 
The  [https://coolconversion.com/math/binary-octal-hexa-decimal/ CoolConversion.com online calculator] is another online app for practicing conversion from/to decimal, hexadecimal, octal and binary; this tool shows the steps that one goes through in the conversion.
 
The AskNumbers.com site has a nice description of the conversions between [https://www.asknumbers.com/hex-to-octal.aspx binary, octal, decimal and hexadecimal] numbers.
 
== Format of ACSL Problems ==
 
The problems in this category will focus on converting between binary, octal, decimal, and hexadecimal, basic arithmetic of numbers in those bases, and, occasionally, fractions in those bases.
 
To be successful in this category, you must know the following facts cold:
 
# The binary value of each octal digit 0, 1, ..., 7
# The binary value of each hex digit 0, 1, ..., 9, A, B, C, D, E, F
# The decimal value of each hex digit 0, 1, ..., F 
# Powers  of  2,  up  to 4096
# Powers  of  8,  up  to 4096
# Powers  of  16,  up  to  65,536


The  [https://coolconversion.com/math/binary-octal-hexa-decimal/ CoolConversion.com online calculator] is another online app for practicing conversion from/to decimal, hexadecimal, octal and binary; this tool shows the steps that one goes through in the conversion.
== Sample Problems ==


== Basics ==  
=== Sample Problem 1 ===


===  Evaluation of binary, octal, and hex numbers ===
Solve for $x$ where $x_{16}=3676_8$.


The binary number system consists of two symbols, 0 and 1. The value of a binary number is found by looking at the bit in each position and multiplying it by the value of that position.
'''Solution:''' One method of solution is to convert $3676_8$ into base 10, and then convert that number into base 16 to yield the value of $x$.
For example, the binary number '''101011''' has a decimal value of $$ {\bf 1} \cdot 2^5 + {\bf 0}  \cdot  2^4 + {\bf 1}  \cdot 2^3 + {\bf 0}  \cdot  2^2 +{\bf 1} \cdot 2^1 + {\bf 1}  \cdot 2^0 = 32+ 0+8 + 0 +2 + 1 = 43$$


The base 8 number system, octal, uses the symbols 0, 1, 2, 3, 4, 5, 6 and 7.  To convert an octal number into decimal, simply multiply each digit by the value of that position. For example, the octal number '''4170''' has decimal value of $$ {\bf 4} \cdot 8^3+ {\bf 1} \cdot  8^2 + {\bf 7} \cdot 8^1 + {\bf 0}  \cdot  8^0 = 4\cdot 512 + 1\cdot 64 + 7\cdot 8 + 0\cdot 1 = 2048+64+56+0=2168$$
An easier solution, less prone to arithmetic mistakes, is to convert from octal (base 8) to hexadecimal (base 16) through the binary (base 2) representation of the number:
   
$$\begin{align}
3676_8 &= 011 ~  110 ~ 111 ~ 110_2  & \text{convert each octal digit into base 2}\hfill\cr
&= 0111 ~ 1011 ~ 1110_2 & \text{group by 4 bits, from right-to-left}\hfill\cr
&= 7 ~ \text{B} ~ \text{E}_{16}  & \text{convert each group of 4 bits into a hex digit}\cr
\end{align}$$


The base 16 number system, hexadecimal, uses 16 symbols: the symbol 0 through 9, A, B, C, D, E, and F. The decimal value of a hex number follows the same pattern as we've just seen for determining the decimal value of a binary or octal number.
=== Sample Problem 2 ===


*Step 1: Write down the hexadecimal number: $ 3AF _{16}$
Solve for $x$ in the following hexadecimal equation: $ x= \text{F5AD}_{16} - \text{69EB}_{16}$
*Step 2: Write each hexit as an increasing power of 16: $ 3 \cdot 16^2 + A \cdot 16^1 + F \cdot 16^0 $
*Step 3: Convert each hexit to decimal: $ 3\cdot 256 + 10\cdot 16 + 15\cdot 1 $
*Step 4: Perform the math: $ 768 + 160 + 15 =  943$


===  Converting from decimal to binary, octal, and hex ===
'''Solution:''' One could convert the hex numbers into base 10, perform the subtraction, and then convert the answer back to base 16. However,
working directly in base 16 isn't too hard.
As in conventional decimal arithmetic, one works from right-to-left, from the least significant digits to the most. 


The are fundamentally two ways to convert a base 10 into another base: ''finding powers'' and ''repeated division''. Here is the conversion of 1375 into octal using these two methods.
:The rightmost digit becomes 2, because D-B=2. 
:The next column is A-E. We need to ''borrow'' a one from the 5 column, and 1A-E=C
:In the next column, 4-9=B, again, borrowing a 1 from the next column.
:Finally, the leftmost column,  E-6=8


'''Method 1: Finding powers.''' Find how many times each decreasing power of that base can be divided evenly into the number and repeating the process with the remainder. The powers of 8 are: 1, 8, 64, 512, 4096, .... SInce 4096 is larger than 1375, we know that the form of the octal number will be $a \cdot 512 + b \cdot 64 + c \cdot 8 + d$.
Combining these results of each column, we get a final answer of $8BC2_{16}$.


:Step 1: How many times does the largest power of 8 go into the number: 1375 / 512 = 2, remainder of 351.
=== Sample Problem 3 ===
:Step 2: How many times does the next smaller power of 8 go into the remainder: 351 / 64 = 5, remainder of 31.
:Step 2 (again):  How many times does the next smaller power of 8 go into the remainder: 31 / 8 = 3, remainder of 7.
:Step 2 (again):  How many times does the next smaller power of 8 go into the remainder: 7 / 1 = 7, remainder of 0.


At this point, the number can be read off: 2 * 512 + 5 * 64 + 3*8 + 7 = 2537 (base 8)
How many numbers from 100 to 200 in base 10 consist of distinct
ascending digits and also have distinct ascending hex digits when
converted to base 16?


'''Method 2: Repeated Division. '''
'''Solution:''' There are 13 numbers that have ascending digits in both bases from 100
to 200.  They are (in base 10):
123 (7B), 124, 125, 126, 127 (7F), 137 (89), 138, 139 (8B), 156 (9C), 157, 158, 159 (9F), 189 (BD)


:Step 1: Divide (1375)10 successively by 8 until the quotient is 0:
== Video Resources ==


::1375/8 = 171, remainder is 7
===ACSL Videos===
::171/8 = 21, remainder is 3
::21/8 = 2, remainder is 5
::2/8 = 0, remainder is 2


:Step 2: Read from the bottom to top as 2537. This is the octal equivalent of decimal number 1375.
 
Adding in bases other than 10 means that you must carry the value of that base and subtracting in bases other than 10 means that you must borrow the value of that base if necessary.  For example:


 
{|


since D=13 and 13+3=16 so leave the 0 and carry the 16 as a 1. Then E=14 and A=10 so 1+14+10 = 25 so leave the 9 and carry the 16 as a 1.  E=14 so 1+14+9=24 so leave the 8 and carry the 16 as 1.  Finally, F=15 so 1+15=16 which is 10.
|-
| <youtube width="300" height="180">https://youtu.be/2v5hDcKCfbo</youtube>
| [https://youtu.be/2v5hDcKCfbo ''Computer Number Systems - ACSL Short Problems Topic'' ('''Next Generation Academy''')]


Subtracting in base 8 is as follows:   
A great tutorial covering this ACSL category, and shows how
to solve many ACSL problems that have appeared in previous contests.


Borrow 1=8 from the 7 since 2+8-6=4.  Therefore, 6–5=1.  Then, borrow 1=8 from the 4 since 5+8–7=6.  Then the last digit on the left is a 3.


== Format of Problems ==
|}


The  key  here  is  to  convert  between  hexadecimal  (base  16)  and  octal  (base  8);  using  the  binary  representation  is  often  easier  than  going  through  base  10.  Some  problems  require  simple  arithmetic  in  hexadecimal  and  octal,  and  once  in  a  while  you’ll  see  of  “decimal  point”. 
=== Other Videos ===


Facts  you  must  know  cold: 
There are many YouTube videos about computer number systems. Here are a handful that cover the topic nicely Some of the videos contain ads; ACSL is not responsible for the ads and does not receive compensation in any form for those ads.  
#The  values  of hex  digits  A,  B,  C,  D,  E,  F 
# Powers  of  2,  up  to,  say,  4096 
# Powers  of  8,  up  to  say,  4096 
# Powers  of  16,  up  to  say,  65,536
#.Convert  from  base  16  to  base  2,  and  vice  versa 
# Convert  from  base  8  to  base  2,  and  vice  versa
# Convert between octal and hex, via base 2


== External Resources ==
{|


This Binary tutorial is divided into 3 sections. In general I recommend you work through them in order but if you've come here just to learn about a specific topic then who am I to slow you down, just head straight on over.
|-
| <youtube width="300" height="180">https://youtu.be/aW3qCcH6Dao</youtube>
| [https://youtu.be/aW3qCcH6Dao ''Number Systems - Converting Decimal, Binary and Hexadecimal'' ('''Joe James''')]


Number Systems - Read on below to discover the theory behind numbers.
An introduction to number systems, and how to convert between decimal, binary and hexadecimal numbers.  
Conversions - How to convert between binary and decimal, hexadecimal and octal.
Arithmetic - Learn how to perform various arithmetic operations with binary numbers.
Negative Numbers - Learn how to manage negative numbers in binary.
Floating point and fractions - Learn how to convert decimal numbers to and from binary fractions and floating point.


Binary Tutorial - 1. Number Systems
|-
https://ryanstutorials.net/binary-tutorial/
| <youtube width="300" height="180">https://youtu.be/m1JtWKuTLR0</youtube>
| [https://youtu.be/m1JtWKuTLR0 ''Lesson 2.3 : Hexadecimal Tutorial'' ('''Carl Herold''')]


The video focuses on hexadecimal numbers: their relationship to binary numbers and how to convert to decimal numbers.


https://ryanstutorials.net/binary-tutorial/binary-conversions.php
|-
Binary Tutorial - 2. Binary Conversions
| <youtube width="300" height="180">https://youtu.be/WR67syBDzew</youtube>
| [https://youtu.be/WR67syBDzew ''Hexes and the Magic of Base 16 - Vaidehi Joshi - May 2017' ('''DonutJS''')]


Binary Tutorial - 3. Binary Arithmetic
A fun introduction to hexadecimal numbers, focusing a bit
https://ryanstutorials.net/binary-tutorial/binary-arithmetic.php
on using hex numbers for specifying RGB colors.  


Binary Tutorial - 5. Binary Fractions and Floating Point
|-
https://ryanstutorials.net/binary-tutorial/binary-floating-point.php
| <youtube width="300" height="180">https://youtu.be/jvx-NrILgpE</youtube>
| [https://youtu.be/jvx-NrILgpE ''Collins Lab: Binary & Hex'' ('''Adafruit Industries''')]


=== test ===
A professionally produced video that explains the number systems, how and why binary numbers are fundamental to computer science, and why hexadecimal is important to computer programmers. 


fooo
|}

Latest revision as of 13:23, 4 July 2022

All digital computers, from supercomputers to your smartphone, are electronic devices and ultimately can do one thing: detect whether an electrical signal is on or off. That basic information, called a bit (binary digit), has two values: a 1 (or true) when the signal is on, and a 0 (of false) when the signal is off.

Larger values can be stored by a group of bits. For example, there are 4 different values stored by 2 bits (00, 01, 10, and 11), 8 values for 3 bits (000, 001, 010, 011, 100, 101, 110, and 111), 16 values for 4 bits (0000, 0001, ..., 1111), and so on. However, large numbers, using 0s and 1s only, are quite unwieldy for humans. For example, a computer would need 19 bits to store the numbers up to 500,000! We use of different number systems to work with large binary strings that represent numbers within computers.

Different Number Systems

A number system is the way we name and represent numbers. The number system we use in our daily life is known as decimal number system or base 10 because it is based on 10 different digits: 0, 1, 2, ..., 8, and 9. The base of number is written as subscript to a number. For example, the decimal number "twelve thousand three hundred and forty-five" is written as $12345_{10}$. Without a base or explicit context, it's assumed that a number is in base 10.

In computer science, apart from the decimal system, three additional number systems are commonly used: binary (base-2), octal (base-8), and hexadecimal or just hex (base-16). Binary numbers are important because that is how numbers are stored in the computer. Octal and hexadecimal are used to represent binary numbers in a user-friendly way. Each octal symbol represents 3 binary bits and each hex digit represents 4 binary bits. For example, the decimal number 53201, stored as in the computer as the binary number 10000001111000101100, is represented by the octal number 2017054, and the hex number 81E2C. The table below compares the number systems:

Number System Base Digits Used Examples
Binary 2 0,1 $10110_{2}$, $10110011_{2}$
Octal 8 0,1,2,3,4,5,6,7 $75021_{8}$, $231_{8}$, $60012_{8}$
Decimal 10 0,1,2,3,4,5,6,7,8,9 $97425_{10}$ or simply 97425
Hexadecimal 16 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F $54A2DD0F_{16}$


In general, $N$ bits have $2^N$ different values. The computers aboard the Apollo spacecraft had 8-bit words of memory. Each word of memory could store 256 different values, and the contents were displayed using 2 hex characters.

The following table shows the first 20 numbers in decimal, binary, octal, and hexadecimal:

Decimal Binary Octal Hexadecimal
0 0 0 0
1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F
16 10000 20 10
17 10001 21 11
18 10010 22 12
19 10011 23 13
20 10100 24 14

Converting to Decimal

As we learned in the first or second grade, the decimal value of a decimal number is simply sum of each digit multiplied by its place value:

$12345 = 1×10^{4}+ 2×10^{3}+ 3×10^{2}+ 4×10^{1}+ 5×10^{0} = 10000 + 2000+ 300 +40+5 = 12345$
$3079 = 3×10^{3}+ 0×10^{2}+ 7×10^{1}+ 9×10^{0} = 3000 + 70 + 9 = 3079$

Analogously, the decimal value of a number in an arbitrary base is the sum of each digit multiplied by its place value. Here are some examples of converting from one base into base 10:

$1101_{2} = 1 × 2^3 + 1 × 2^2 + 0 × 2^1 + 1 × 2^0 = 8 + 4 + 0 + 1 = 13_{10}$
$175_{8} = 1 × 8^2 + 7 × 8^1 + 5 × 8^0 = 1 × 64 + 7 × 8 + 5 × 1 = 64 + 56 + 5 = 125_{10}$
$A5E_{16} = 10 × 16^2 + 5 × 16^1 + 14 × 16^0 = 10 × 256 + 5 × 16 + 14 × 1 = 2560 + 80 + 14 = 2654_{10}$

Converting from Decimal

The algorithm to convert a number from an arbitrary base requires finding how many times successive powers of the base go into the number, starting with the largest power of the base less than the starting number. For example, converting 3306 from base 10 to octal proceeds as follows:

1) The largest power of 8 that is less than or equal to 3306 is 512 ($8^3$). Divide 3306 by 512:

$3306 = 6*8^3 + 234$

2) The next power of 8 is 64 ($8^2$). Divide 234 by 64:

$234 = 3 *8^2 + 42$

3) The next smaller power of 8 is 8 ($8^1$). Divide 42 by 8:

$42 = 5*8^1 + 2$

4) Finally, the next smaller power of 8 is 1 ($8^0$). Divide 2 by 1:

$2 = 2 * 8^0$

The answer is $6352_8$.

Converting between Binary, Octal, and Hexadecimal

Converting from octal to binary is simple: replace each octal digit by its corresponding 3 binary bits. For example:

$375_{8} = \ \ 011\ \ \ 111\ \ \ 101_{2} = 11111101_2$

Converting from hex to binary is also simple: replace each hex digit by its corresponding 4 binary bits. For example:

$FD_{16} = \ \ 1111\ \ \ 1101_{2} = 11111101_2$

Converting from binary to either octal or hex is pretty simple as well: group the bits by 3s or 4s (starting at the right), and convert each group:

$10000001111000101100 = 10\ 000\ 001\ 111\ 000\ 101 \ 100 = 2017054_8$
$10000001111000101100 = 1000\ 0001\ 1110\ 0010\ 1100 = 81E2C_{16}$

Converting between base 8 and 16 is easy by expressing the number in base 2 (easy to do!) and then converting that number from base 2 (another easy operation)! This is shown below in Sample Problem #1.

Using Hexadecimal Numbers to Represent Colors

Computers use hexadecimal numbers to represent various colors in computer graphics because all computer screens use combinations of red, green, and blue light or RGB to represent thousands of different colors. Two digits are used for each so the hexadecimal number “#FF0000” represents the color red, “#00FF00” represents green, and “#0000FF” represents blue. The color black is “#000000” and white is “#FFFFFF”.

The hash tag or number sign is used to denote a hexadecimal number. $FF_{16} = F (15) × 16 + F (15) × 1 = 240 + 15 = 255_{10}$ so there are 0 to 255 or 256 different shades of each color or $256^{3} = 16,777,216$ different colors.

The following web site has nearly every color name, along with its hex code and decimal values:

https://www.rapidtables.com/web/color/RGB_Color.html

For example “salmon” is “#FA8072” which represents the decimal numbers 250 (hex FA), 128 (hex 80), and 114 (hex 72).

Resources

Ryan's Tutorials covers this topic beautifully. Rather than trying to duplicate that work, we'll point you to the different sections:

1. Number Systems - An introduction to what numbers systems are all about, with emphasis on decimal, binary, octal, and hexadecimal. ACSL will typically identify the base of a number using a subscript. For example, $123_8$ is an octal number, whereas $123_{16}$ is a hexadecimal number.
2. Binary Conversions - This section shows how to convert between binary, decimal, hexadecimal and octal numbers. In the Activities section, you can practice converting numbers.
3. Binary Arithmetic - Describes how to perform various arithmetic operations (addition, subtraction, multiplication, and division) with binary numbers. ACSL problems will also cover basic arithmetic in other bases, such as adding and subtracting together 2 hexadecimal numbers. ACSL problems will not cover division in other bases.
4. Negative Numbers - ACSL problems will not cover how negative numbers are represented in binary.
5. Binary Fractions and Floating Point - The first part of this section is relevant to ACSL: fractions in other bases. ACSL will not cover floating point numbers in other basis. So, focus on the section Converting to a Binary Fraction, but keep in mind that ACSL problems may also cover octal and hexadecimal fractions.

The CoolConversion.com online calculator is another online app for practicing conversion from/to decimal, hexadecimal, octal and binary; this tool shows the steps that one goes through in the conversion.

The AskNumbers.com site has a nice description of the conversions between binary, octal, decimal and hexadecimal numbers.

Format of ACSL Problems

The problems in this category will focus on converting between binary, octal, decimal, and hexadecimal, basic arithmetic of numbers in those bases, and, occasionally, fractions in those bases.

To be successful in this category, you must know the following facts cold:

  1. The binary value of each octal digit 0, 1, ..., 7
  2. The binary value of each hex digit 0, 1, ..., 9, A, B, C, D, E, F
  3. The decimal value of each hex digit 0, 1, ..., F
  4. Powers of 2, up to 4096
  5. Powers of 8, up to 4096
  6. Powers of 16, up to 65,536

Sample Problems

Sample Problem 1

Solve for $x$ where $x_{16}=3676_8$.

Solution: One method of solution is to convert $3676_8$ into base 10, and then convert that number into base 16 to yield the value of $x$.

An easier solution, less prone to arithmetic mistakes, is to convert from octal (base 8) to hexadecimal (base 16) through the binary (base 2) representation of the number:

$$\begin{align} 3676_8 &= 011 ~ 110 ~ 111 ~ 110_2 & \text{convert each octal digit into base 2}\hfill\cr &= 0111 ~ 1011 ~ 1110_2 & \text{group by 4 bits, from right-to-left}\hfill\cr &= 7 ~ \text{B} ~ \text{E}_{16} & \text{convert each group of 4 bits into a hex digit}\cr \end{align}$$

Sample Problem 2

Solve for $x$ in the following hexadecimal equation: $ x= \text{F5AD}_{16} - \text{69EB}_{16}$

Solution: One could convert the hex numbers into base 10, perform the subtraction, and then convert the answer back to base 16. However, working directly in base 16 isn't too hard. As in conventional decimal arithmetic, one works from right-to-left, from the least significant digits to the most.

The rightmost digit becomes 2, because D-B=2.
The next column is A-E. We need to borrow a one from the 5 column, and 1A-E=C
In the next column, 4-9=B, again, borrowing a 1 from the next column.
Finally, the leftmost column, E-6=8

Combining these results of each column, we get a final answer of $8BC2_{16}$.

Sample Problem 3

How many numbers from 100 to 200 in base 10 consist of distinct ascending digits and also have distinct ascending hex digits when converted to base 16?

Solution: There are 13 numbers that have ascending digits in both bases from 100 to 200. They are (in base 10): 123 (7B), 124, 125, 126, 127 (7F), 137 (89), 138, 139 (8B), 156 (9C), 157, 158, 159 (9F), 189 (BD)

Video Resources

ACSL Videos

Computer Number Systems - ACSL Short Problems Topic (Next Generation Academy)

A great tutorial covering this ACSL category, and shows how to solve many ACSL problems that have appeared in previous contests.


Other Videos

There are many YouTube videos about computer number systems. Here are a handful that cover the topic nicely Some of the videos contain ads; ACSL is not responsible for the ads and does not receive compensation in any form for those ads.

Number Systems - Converting Decimal, Binary and Hexadecimal (Joe James)

An introduction to number systems, and how to convert between decimal, binary and hexadecimal numbers.

Lesson 2.3 : Hexadecimal Tutorial (Carl Herold)

The video focuses on hexadecimal numbers: their relationship to binary numbers and how to convert to decimal numbers.

Hexes and the Magic of Base 16 - Vaidehi Joshi - May 2017' (DonutJS)

A fun introduction to hexadecimal numbers, focusing a bit on using hex numbers for specifying RGB colors.

Collins Lab: Binary & Hex (Adafruit Industries)

A professionally produced video that explains the number systems, how and why binary numbers are fundamental to computer science, and why hexadecimal is important to computer programmers.

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