2. Number Bases
2.1. Decimal (Base Ten)
The decimal number system is used in everyday life
The decimal number system is base ten; it consists of ten different symbols/numerals
\(0, 1, 2, 3, 4, 5, 6, 7, 8, 9\)
These symbols have an ordering to them
\(0 \rightarrow 1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 5 \rightarrow 6 \rightarrow 7 \rightarrow 8 \rightarrow 9\)
These symbols also have a meaning/value associated with them
The symbol \(5\) means five
It conveys a magnitude
The magnitude of these values correspond to the ordering
\(0 < 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8 < 9\)
2.1.1. Counting in Base Ten
Counting in base ten consists of moving to the next symbol
Counting with Ten Symbols \(0\)
Zero
\(1\)
One
\(2\)
Two
\(3\)
Three
\(4\)
Four
\(5\)
Five
\(6\)
Six
\(7\)
Seven
\(8\)
Eight
\(9\)
Nine
\(????\)
Ten?
However, there are only ten unique symbols
This means it’s not possible to count past nine with the symbols alone
This is where symbol position comes in
2.1.1.1. Position
First, consider that each number has an infinite number of zeros in front of it
Why this is will be discussed shortly
Consider the below table where only seven of the leading zeros are included
Thus, these numbers will each have a total of eight symbols
Each of these symbols/numerals, in this context, is called a digit
Digit comes from latin (digitus), meaning finger or toe, of which, humans typically have ten of each
Values Expressed with Eight Digits \(00000000\)
Zero
\(00000001\)
One
\(00000002\)
Two
\(00000003\)
Three
\(00000004\)
Four
\(00000005\)
Five
\(00000006\)
Six
\(00000007\)
Seven
\(00000008\)
Eight
\(00000009\)
Nine
To count past the value nine, a new rule is introduced
If the no more symbols are left
Reset the symbol to zero in the current position
Change to the subsequent symbol in the next position
Counting with Digits …
…
\(00000008\)
Eight
\(00000009\)
Nine
\(00000010\)
Ten
\(00000011\)
Eleven
…
…
\(00000018\)
Eighteen
\(00000019\)
Nineteen
\(00000020\)
Twenty
…
…
\(00000099\)
Ninety Nine
\(00000100\)
One Hundred
\(00000101\)
One Hundred and One
…
…
Consider the number \(101\)
Although this number contains two ones, they have a different meaning because of their position
2.1.1.2. Names
Each of these positions has a name
From right to left, they are
Digit Position Names First Digit
Ones
\(10^{0}\)
\(1\)
Second Digit
Tens
\(10^{1}\)
\(10\)
Third Digit
Hundreds
\(10^{2}\)
\(100\)
Fourth Digit
Thousands
\(10^{3}\)
\(1000\)
Fifth Digit
Ten Thousands
\(10^{4}\)
\(10000\)
Sixth Digit
Hundred Thousands
\(10^{5}\)
\(100000\)
Seventh Digit
Millions
\(10^{6}\)
\(1000000\)
Eighth Digit
Ten Millions
\(10^{7}\)
\(10000000\)
…
…
…
…
As a consequence of the counting pattern, each position corresponds to a value that is the base raised to some power
Consider the number \(123\)
The symbol \(1\) in the hundreds position is \(1 \times 10^{2}\)
There is one hundred
The symbol \(2\) in the tens position is \(2 \times 10^{1}\)
There are two tens
The symbol \(3\) in the ones position is \(3 \times 10^{0}\)
Three ones
Thus, the number is \(1 \times 10^{2} + 2 \times 10^{1} + 3 \times 10^{0}\)
It may feel strange to think about the number \(123\) like this
But this is what the decimal encoding of the number is conveying
The value \(123\) represented with Canadian Dollars. There is one hundred, two tens, and three ones.
2.1.1.3. Consequence of Finite Digits
Although with integers, there are an infinite number of leading zeros and positions
Only eight digits were used to encode the numbers in the above examples
Ths means there is a limit to the number of unique positive integer values that can be expressed
The largest number is \(99999999\)
Ninety nine million, nine hundred and ninety nine thousand, nine hundred and ninety nine
Obviously there are values larger than this, but they are not representable with only 8 digits
2.2. Binary (Base Two)
Decimal is the typical way numbers are encoded in every day life
However, base ten (decimal) is only an encoding
It’s not a number, it’s a way to represent a number
Other bases could just as easily be used
For example, binary (base two)
Instead of ten symbols, only two are used — \(0, 1\)
\(0\) means zero, like base ten
\(1\) means one, like base ten
The symbols have an ordering to the magnitude of values, like base ten
\(0 \rightarrow 1\)
\(0 < 1\)
Position matters, like base ten
The value of a \(1\) depends on where it is
There are an infinite number of leading zeros
\(1 == 00000001 == 0000000000000001\)
In decimal, the symbols are called digits
In binary, they are called bits
Binary information digit — bit
The counting rules are the same
Increment to the next symbol
If there are no more symbols
Reset to \(0\)
Increment the symbol in the next position
Counting in Binary \(00000000\)
Zero
\(00000001\)
One
\(00000010\)
Two
\(00000011\)
Three
\(00000100\)
Four
\(00000101\)
Five
\(00000110\)
Six
\(00000111\)
Seven
\(00001000\)
Eight
…
…
With eight bits, the largest positive integer that could be represented is \(255\)
Can represent the numbers 0 through to 255
Like base ten, the specific bit position carries different values
These values are always the base to some power
Although these bit positions don’t really go by specific names, they can be named like the digits
Bit Position Names First Digit
Ones
\(2^{0}\)
\(1\)
Second Digit
Twos
\(2^{1}\)
\(2\)
Third Digit
Fours
\(2^{2}\)
\(4\)
Fourth Digit
Eights
\(2^{3}\)
\(8\)
Fifth Digit
Sixteens
\(2^{4}\)
\(16\)
Sixth Digit
Thirty-twos
\(2^{5}\)
\(32\)
Seventh Digit
Sixty-fours
\(2^{6}\)
\(62\)
Eighth Digit
One hundred and twenty eights
\(2^{7}\)
\(128\)
…
…
…
…
Like base ten, the binary number can be broken down to the sum of its positional values
Consider the number \(1111011\)
One Sixty-four
One thirty-two
One sixteen
One eight
Zero fours
One two
One one
Thus, the number is
\(1 \times 2^{6} + 1 \times 2^{5} + 1 \times 2^{4} + 1 \times 2^{3} + 0 \times 2^{2} + 1 \times 2^{1} + 1 \times 2^{0}\)
\(1 \times 64 + 1 \times 32 + 1 \times 16 + 1 \times 8 + 0 \times 4 + 1 \times 2 + 1 \times 1\)
\(64 + 32 + 16 + 8 + 0 + 2 + 1\)
\(123\)
Note
Position really matters — have you ever counted to \(31\) on one hand?
Typically, when counting with fingers, the position of the finger is ignored. This means that the biggest number one could count to on one hand is five. This is effectively base 1.
However, it is possible to make use of the position of each finger to get more out of your hand. Try it yourself.
The number 19 in binary represented with fingers. The thumb is the least significant bit (ones) and the pinky finger is the most significant bit (sixteens).
2.3. Hexadecimal (Base 16)
Using a base larger than ten is fine
However, there are only ten conventional numerals
Because of this, additional symbols are needed to represent values greater than 9
Consider hexadecimal — base 16
Often called hex
Although any symbol could be used to represent the values 10 through 15, the letters \(A\) through \(F\) are used
Like counting in decimal and binary, the same rules apply to hexadecimal
The symbols have an ordering to their magnitudes
\(0 < 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8 < 9 < A < B < C < D < E < F\)
Counting works the same way
Leading zeros also exist, but are excluded below for brevity
Counting in Hexadecimal \(0\)
Zero
\(1\)
One
…
…
\(9\)
Nine
\(A\)
Ten
\(B\)
Eleven
\(C\)
Twelve
\(D\)
Thirteen
\(E\)
Fourteen
\(F\)
Fifteen
\(10\)
Sixteen
\(11\)
Seventeen
…
…
\(FF\)
Two Hundred and Fifty Five
\(100\)
Two Hundred and Fifty Six
\(101\)
Two Hundred and Fifty Seven
…
…
Like binary, the hexadecimal digits don’t really have names, but they can be named to get a sense of their meaning
Hexadecimal Digit Position Names First Digit
Ones
\(16^{0}\)
\(1\)
Second Digit
Sixteens
\(16^{1}\)
\(16\)
Third Digit
Two Hundred and Fifty Sixes
\(16^{2}\)
\(255\)
Fourth Digit
Four thousand and Ninty Sixes
\(16^{3}\)
\(4096\)
…
…
…
…
Like decimal and binary, a hexadecimal number can be broken down into the sum of its positional values
Consider the number \(7B\)
Seven Sixteens
Eleven Ones
Thus, the number can be calculated as follows
\(7 \times 16^{1} + 11 \times 16^{0}\)
\(7 \times 16 + 11 \times 1\)
\(112 + 11\)
\(123\)
2.4. Converting Numbers Between Bases
Although only three bases were discussed so far, but there is nothing stopping one from using base 5, or 123, or …
A subscript is added to the number to specify the base when there is a potential for ambiguity
For example, consider the number \(10\)
What number is that?
Is it a base ten number?
Base two?
Hexadecimal?
Ambiguity is eliminated by including the subscript
\(10_{10}\) means ten
\(10_{2}\) means two
\(10_{16}\) means sixteen
In a computing context, binary and hexadecimal are often written as
0b10and0x10respectively
2.4.1. Converting to Decimal
It is often useful to convert numbers from other bases to decimal since that is what is used in everyday life
Saying \(123\) is a lot easier for us to understand than saying \(1111011_{2}\) or \(7B_{16}\)
However, an encoding is only an encoding — these all mean the same thing
For both the binary and hexadecimal bases discussed, numbers were already converted to decimal
\(1111011_{2}\)
\((1 \times 2^{6} + 1 \times 2^{5} + 1 \times 2^{4} + 1 \times 2^{3} + 0 \times 2^{2} + 1 \times 2^{1} + 1 \times 2^{0})_{10}\)
\((1 \times 64 + 1 \times 32 + 1 \times 16 + 1 \times 8 + 0 \times 4 + 1 \times 2 + 1 \times 1)_{10}\)
\((64 + 32 + 16 + 8 + 0 + 2 + 1)_{10}\)
\(123_{10}\)
\(7B_{16}\)
\((7 \times 16^{1} + 11 \times 16^{0})_{10}\)
\((7 \times 16 + 11 \times 1)_{10}\)
\((112 + 11)_{10}\)
\(123_{10}\)
In general, multiply the value of the digit in base ten by the value of the base in base ten to the power of its corresponding position
For example, given some number of \(n\) digits \(d\) in base \(b\)
\((d_{n-1}d_{n-2}...d_{2}d_{1}d_{0})_{b}\)
The value in base ten would be calculated as follows
\((d_{n-1} \times b^{n-1} + d_{n-2} \times b^{n-2} + ... + d_{2} \times b^{2} + d_{1} \times b^{1} + d_{0} \times b^{0})_{10}\)
Consider the number \(J204_{20}\) — note that \(J_{20} == 19_{10}\) in this example
\((19 \times 20^{3} + 2 \times 20^{2} + 0 \times 20^{1} + 4 \times 20^{0})_{10}\)
\((19 \times 8000 + 2 \times 400 + 0 \times 20 + 4 \times 1)_{10}\)
\((152000 + 800 + 0 + 4)_{10}\)
\(152804_{10}\)
2.4.2. Converting from Decimal
To convert a number from decimal to some arbitrary base \(b\)
Divide the number by \(b\) and keep the quotient \(q_{0}\) and remainder \(r_{0}\)
The value of \(r_{0}\) is the digit for the least significant position
Divide \(q_{0}\) by the base \(b\) to get a new quotient \(q_{1}\) and remainder \(r_{1}\)
The value of \(r_{1}\) is the digit for the next position
Repeat until the quotient is zero
Consider converting the number \(123_{10}\) to binary
Converting \(123_{10}\) to Binary Digit/Bit Position
Division
\(q\)
\(r\)
\(0\)
\(123/2\)
\(61\)
\(1\)
\(1\)
\(61/2\)
\(30\)
\(1\)
\(2\)
\(30/2\)
\(15\)
\(0\)
\(3\)
\(15/2\)
\(7\)
\(1\)
\(4\)
\(7/2\)
\(3\)
\(1\)
\(5\)
\(3/2\)
\(1\)
\(1\)
\(6\)
\(1/2\)
\(0\)
\(1\)
Therefore, the number \(123_{10}\) is \(1111011_{2}\)
Consider converting the number \(123_{10}\) to hexadecimal
Converting \(123_{10}\) to Hexadecimal Hex Digit Position
Division
\(q\)
\(r\)
\(0\)
\(123/16\)
\(7\)
\(11_{10}\) or \(B_{16}\)
\(1\)
\(7/16\)
\(0\)
\(7\)
Therefore, the number \(123_{10}\) is \(7B_{2}\)
2.4.3. Converting Between Arbitrary Bases
Converting between arbitrary bases will not be covered in detail
If one really wants to convert between base, they can do it in two steps with decimal
For example, converting from base 5 to base 7
Convert from base five to decimal
Convert from decimal to base 7
However, there is an interesting trick when converting between bases when one is an even power of the other
Consider binary and hexadecimal
\(2^{4} == 16\)
Each group of four bits corresponds to a single hexadecimal digit
This is perhaps best illustrated with an example
\(1111011_{2}\)
Take the least significant batch of four bits and convert to hexadecimal
\(1011_{2} == 11_{10} == B_{16}\)
Take the next batch of four bits, add leading zeros if necessary, and convert to hexadecimal
\(0111_{2} == 7_{10} == 7_{16}\)
One can then combine the hexadecimal digits to create the hexadecimal number
\(1111011_{2} == 7B_{16}\)
This works because, with any grouping of four bits, sixteen unique values can be represented
\(0\) through \(15\)
With hexadecimal, sixteen unique values can be represented with a single hexadecimal digit
\(0\) through \(F\)
This works in general when one base is an even power of another
In the below table, observe that
Base four can be obtained by combining two bits from base two
Base eight can be obtained by combining three bits from base two
Base sixteen can be obtained by combining four bits from base two
Base sixteen can also be obtained by combing two base four digits
Numbers in Base two, Four, Eight, and Sixteen Base Two
Base Four
Base Eight
Base Sixteen
\(0000\)
\(00\)
\(00\)
\(0\)
\(0001\)
\(01\)
\(01\)
\(1\)
\(0010\)
\(02\)
\(02\)
\(2\)
\(0011\)
\(03\)
\(03\)
\(3\)
\(0100\)
\(10\)
\(04\)
\(4\)
\(0101\)
\(11\)
\(05\)
\(5\)
\(0110\)
\(12\)
\(06\)
\(6\)
\(0111\)
\(13\)
\(07\)
\(7\)
\(1000\)
\(20\)
\(10\)
\(8\)
\(1001\)
\(21\)
\(11\)
\(9\)
\(1010\)
\(22\)
\(12\)
\(A\)
\(1011\)
\(23\)
\(13\)
\(B\)
\(1100\)
\(30\)
\(14\)
\(C\)
\(1101\)
\(31\)
\(15\)
\(D\)
\(1110\)
\(32\)
\(16\)
\(E\)
\(1111\)
\(33\)
\(17\)
\(F\)
2.5. For Next Time
Read Chapter 2 of your text
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