In the previous lectures, we explored some examples of the earliest number systems

which were used primarily for counting objects.

These counting numbers are called “natural numbers”.

The natural numbers start at one and can count to arbitrarily large quantities.

As we have seen, Roman numerals are one of many possible ways to represent natural numbers.

The Roman system was eventually replaced with the modern decimal number system

which uses “positional notation” and only ten numeric symbols.

The decimal number system was found to be superior to the ancient Roman system

because of the simple rules it uses to create numbers.

In the decimal system there are ten numeric symbols, 0 through 9, called “digits”.

Depending on the column they occupy

these digits represent the quantity of ones

tens

hundreds

thousands

and so on

which make up the number.

In positional notation, the column occupied by a digit

determines the “multiplier” for that digit.

For example, in the decimal system

the value of the right-most digit is multiplied by 1.

The digit in the next column to the left is multiplied by 10.

The next digit is multiplied by 100 and so on.

The value of a number is the sum of all its digits times their multipliers.

For example, the value of the decimal number 1879

is 1 times 1000

plus 8 times 100

plus 7 times 10

plus 9 times 1.

In any positional notation, each column's multiplier differs from the adjacent column

by a constant multiple called the “base” of the number system.

In the decimal system, each column multiplier is ten times the previous column.

Therefore the decimal system is called a “base-10” number system.

There are an infinite number of columns in the decimal number system

with each column multiplier being ten times bigger than the column to the right.

However, when writing a number, the zeros in front are normally not written.

We can count up to 9

using only the ones column.

Once we reach 9

the ones column starts over at 0

and the tens column increments.

As we continue counting

the tens column counts the number of times

that the ones column has passed from 9 to 0.

In other words, the tens column registers the number of tens which we have counted.

This continues until we reach 99.

At that point the ones and tens columns start over at 0

and the hundreds column increments.

The positional notation system is simple.

Every time a column passes from 9 to 0

the next column to the left increments.

How is it that we ended up with a number system based on multiples of ten?

There is not any good reason for choosing ten over some other number

other than the fact that people have ten fingers

and probably originally communicated quantities using their fingers.

But what if we were cartoon characters with four digits on each hand?

Is it possible that in cartoon land

everyone uses a number system based on multiples of eight instead of ten?

How would a base-8 or “octal” number system work?

In octal there are only eight numeric symbols instead of ten as in decimal.

Instead of 0 through 9 the symbols 0 through 7 are used.

The symbols 8 and 9 are not needed.

Counting in octal is very similar to counting in decimal.

Since there are no symbols for 8 or 9

the highest quantity which can be represented in the ones column is 7.

Counting an eighth item causes the ones column to start over at 0

and the next column to increment.

So the second column counts the number of eights.

Therefore in octal the number following 7 is 10

which looks just like the decimal number ten.

After octal "10" comes octal "11", "12", and so on

until we get to octal "17".

At that point, we go to octal "20".

The second column has now counted two “eights” or sixteen.

We continue like this until we get to the highest number we can represent with two digits

octal "77".

At that point, the ones and eights columns start over at 0 and the third column increments.

The 1 in the third column represents eight “eights” or sixty-four.

Each column multiplier is eight times the previous one.

Every number which can be written in decimal can also be written in octal

although after counting to 7

the way the quantities are represented is completely different.

It is easy to convert an octal number to decimal

when you consider how positional notation works.

Let's take for example, the octal number "1750".

As in decimal, the value of the octal number is the sum of all its digits times their multipliers.

So the number "1750" represents

1 times 512

plus 7 times 64

plus 5 times 8

plus 0 ones

which adds up to the quantity which in decimal is called one-thousand.

You may sometimes see a small subscript 8 or 10 after an octal or decimal number

in case there may be some confusion about which base is being used.

Digital computers use electronic circuits called “flip-flops” to represent numbers.

Each flip-flop can store a single bit which can represent either a 0 or a 1.

Multiple bits can be combined to represent a base-2 or “binary” number.

In the binary number system 0 and 1 are the only two numeric symbols.

Since binary is base-2

each column multiplier is two times the multiplier of the previous digit.

And just like decimal or octal numbers

the value of a binary number is sum of all its digits times their multipliers.

Since the digits are either 1 or 0 the calculation is simple.

We just add the multipliers of all the columns which contain ones.

For example, the binary number 11010

represents 1 sixteen

plus 1 eight

plus 1 two

which is equal to twenty-six.

Even though digital computers store numbers in binary

it can be quite tedious to write down or remember large binary numbers.

For instance, the number one-million in binary is

one one one one

zero one zero one

one zero zero one

one zero one one

zero zero zero zero.

Early in the history of digital computers

engineers found that it was easier to use octal notation

than to deal with long strings of ones and zeros.

Three binary digits can be represented by a single octal symbol.

It is easy to memorize the eight possible combinations of three binary bits.

To convert a multiple-digit octal number to binary

each octal digit in the number is converted to a 3-bit binary equivalent

and the binary digits are all combined into a single binary number.

Any leading zeros can be removed.

To convert a binary number to octal we do the same thing in reverse.

To convert this binary number back to octal

we split it into 3-bit groups starting from the right

and each 3-bit group is then converted to its equivalent octal symbol.

So the octal equivalent to this binary number is "3654660"

a lot easier to remember than all those ones and zeros.

Today, computer storage is normally organized into 8-bit groups called "bytes".

Because of this, many computer engineers prefer to use base-16

otherwise known as “hexadecimal” or “hex” instead of octal.

With hexadecimal, every group of four bits converts to a single hex symbol.

Two hex symbols represent exactly one byte.

Even fewer digits than octal are required to represent a given number

and it's just as easy to convert back and forth to binary.

Hexadecimal numbers use sixteen numeric symbols.

The symbols 0 through 9 are used just as in decimal

but six more symbols are needed.

Instead of making up new symbols, the letters A through F are used

to represent what we call ten through fifteen in decimal.

Counting in hexadecimal works the same way as in decimal or octal

except that hex uses sixteen symbols per digit.

Because each column multiplier is sixteen times larger than the previous column

hexadecimal can represent large numbers

with fewer digits than octal or decimal.

When counting in hexadecimal

after getting to F which is decimal 15

we go to "10" which is decimal 16

then "11", "12", and so on.

Once we reach 1F

we go to "20" which is decimal 32.

When we get to the largest number which we can represent with two hex digits, FF

we go to "100" which is decimal 256

and so on.

As we mentioned

using hex notation, four binary bits can be represented by a single hex symbol.

Each of the sixteen possible combinations of four bits

is equivalent to a single hex digit.

Let's convert the same binary number as before to hex.

Starting from the right, we group the digits into groups of four.

Each group of binary digits is then converted to its equivalent hex symbol.

So we have seen how the same natural number

can be represented in base-2 using two numeric symbols

base-8 using eight symbols

base-10 using ten symbols

and base-16 using sixteen symbols.

But no matter how we choose to write this natural number

it still represents the same quantity.

As you have seen

we use the same basic rules for counting in binary

octal

decimal

and hexadecimal.

The only difference is that each base has a different number of numeric symbols.

So using positional notation

we can create a number system using any natural base we like.

Try creating one of your own.

Who knows, it might catch on!