Difference between revisions of "Computer Number Systems"

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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.  
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.
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 are 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, groups of 4 bits form a single hex digit.


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 or floating point numbers are represented in binary.
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 or floating point numbers are represented in binary.
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== 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.  
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:[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  [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.


== Basics ==  
== Format of ACSL Problems ==


===  Evaluation of binary, octal, and hex numbers ===
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.


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.
To be successful in this category, you must know the following facts cold:
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 $$ {\bf 4} \cdot 8^3+ {\bf 1} \cdot 8^2 + {\bf 7} \cdot 8^1 + {\bf 0} \cdot 8^0 = 4\cdot 512 + 1\cdot 64 + 7\cdot 8 + 0\cdot 1 = 2048+64+56+0=2168$$
# The decimal value of each hex digit A, B, C, D, E,
# The binary value of each hex digit A, B, C, D, E, F
# Powers of 2up  to 4096
# Powers  of 8, up to 4096
# Powers  of  16, up to 65,536


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 Problems ==


*Step 1: Write down the hexadecimal number: $ 3AF _{16}$
=== Sample Problem 1 ===
*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 ===
Solve for $x$ where $x_{16}=3676_8$.


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.
'''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 $x$.


'''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 $a \cdot 512 + b \cdot 64 + c \cdot 8 + d$.
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:
$$\begin{align}
3676_8 &= 011 ~  110 ~ 111 ~ 110_2  & \text{convert each octal digit into base 2}\hfill\cr
&= 0111 ~ 1011 ~ 1110_2  & \text{group by 4 bits, from right-to-left}\hfill\cr
&= 7 ~ \text{B} ~ \text{E}_{16}  & \text{convert each group of 4 bits into a hex digit}\cr
\end{align}$$


:Step 1: How many times does the largest power of 8 go into the number: 1375 / 512 = 2, remainder of 351.
=== Sample Problem 2 ===
: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)
Solve for $x$ in the following hexadecimal equation: $ x= \text{F5AD}_{16} - \text{69EB}_{16}$


'''Method 2: Repeated Division. '''
'''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. 


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


::1375/8 = 171, remainder is 7
Combining these results of each column, we get a final answer of $8BC2_{16}$.
::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.
=== Sample Problem 3 ===
 
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:


 
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?


since D=13 and 13+3=16 so leave the 0 and carry the 16 as a 1Then 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.
'''Solution:''' There are 13 numbers that have ascending digits in both bases from 100
to 200They are (in base 10):
123 (7B), 124, 125, 126, 127 (7F), 137 (89), 138, 139 (8B), 156 (9C), 157, 158, 159 (9F), 189 (BD)


Subtracting in base 8 is as follows:   
== Video Resources ==


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.
There are many YouTube videos about computer number systems. Here are a handful that cover the topic nicely, without too many ads:


== 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.
|-
 
| <youtube width="300" height="180">https://youtu.be/aW3qCcH6Dao</youtube>
To be successful in this category, you must know the following fact cold:
| [https://youtu.be/aW3qCcH6Dao ''Number Systems - Converting Decimal, Binary and Hexadecimal'' ('''Joe James''')]


# The decimal values  of  hex  digits  A,  B,  C,  D,  E,  F 
An introduction to number systems, and how to convert between decimal, binary and hexadecimal numbers.
# The binary value of hex digits A, B, C, D, E, F
# Powers  of  2,  up  to 4096 
# Powers  of  8, up  to 4096 
# Powers  of  16, up  to  65,536


== Sample Problems ==
|-
| <youtube width="300" height="180">https://youtu.be/m1JtWKuTLR0</youtube>
| [https://youtu.be/m1JtWKuTLR0 ''Lesson 2.3 : Hexadecimal Tutorial'' ('''Carl Herold''')]


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


== Other Resources ==
|-
| <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''')]


There are lots and lots and lots of YouTube videos about computer number systems. Here are a handful that cover the topics, without too many ads:
A fun introduction to hexadecimal numbers, focusing a bit
on using hex numbers for specifying RGB colors.  


TBD
|-
| <youtube width="300" height="180">https://youtu.be/jvx-NrILgpE</youtube>
| [https://youtu.be/jvx-NrILgpE ''Collins Lab: Binary & Hex'' ('''Adafruit Industries''')]


TBD
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. 


TBD
|}

Revision as of 22:02, 20 August 2018

All digital computers – from supercomputers to your smartphone – 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, 3 bits together can take on 8 different values.

Computer scientists use the 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 are represented in a computer. Since binary numbers representing moderate values quickly become rather lengthy, bases eight (octal) and sixteen (hexadecimal) are frequently used as shorthand. In octal, groups of 3 bits form a single octal digit; in hexadecimal, groups of 4 bits form a single hex digit.

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 or floating point numbers are represented in binary.

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.

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 decimal value of each hex digit A, B, C, D, E, F
  2. The binary value of each hex digit A, B, C, D, E, F
  3. Powers of 2, up to 4096
  4. Powers of 8, up to 4096
  5. 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

There are many YouTube videos about computer number systems. Here are a handful that cover the topic nicely, without too many 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.