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Number SystemsDecoded Binary, Decimal, and Hexadecimal Expert Reference Series of White Papers... These are the decimal, binary, and hexa-decimal number systems.. These alternative numbe

Number Systems

Decoded

Binary, Decimal, and Hexadecimal

Expert Reference Series of White Papers

In many IT-related fields of study, it is extremely valuable to have a good understanding of the number sys-tems that are often encountered For many people, not appreciating things such as binary and hexadecimal is

a stumbling block that keeps them from advancing their knowledge If you cringe a little bit at the mention of these topics, then this white paper is definitely for you

Of course, you may have learned these things many years ago, and you would like a refresher course on num-ber systems Or, perhaps you are tired of not understanding the gobbledygook that appears in file dumps, in routing tables, and in your protocol analysis tools

Whatever your particular reasons for reading this white paper are, I welcome your attention

What You Will Learn

We all have an instinctive feel for what numbers are They are used on a regular basis to identify a particular house on a street, a channel on the television, or how much to pay for an item

Numbers are used in a variety of ways; they are most often used to convey a quantity to someone—twelve chairs, for instance Numbers are also used in verbal codes, such as 10-4 (ten-four) Sometimes they are part of

an address (such as zip codes)

Despite the intended use, numbers are constructed in the same way and follow the same basic set of rules Understanding those rules is the key to advancing your appreciation of numbers

Several systems of numbers are commonly used in computing circles These are the decimal, binary, and hexa-decimal number systems

These alternative number systems have three things in common: a base number (sometimes called a radix); a set of digits or numerals to work with; and a positional notation in which the position of a given digit

affects how it is interpreted

Each number system employs:

• ABase or radix value around which the number system is organized

• A set of Digits or symbols used in forming numbers

• APositional Notation meaning that the position in which a digit appears conveys information about the significance or weight of that digit

George Mays, CISSP, CCNA, A+, Security+, Network+

Number Systems Decoded

Binary, Decimal, and Hexadecimal

Figure A Number System Names, Bases, and Digits

Each number system has a name: Decimal, Binary, and Hexadecimal Since “hexadecimal” does not flow trip-pingly off our tongues, we often simply say “Hex” instead

The decimal number system is encountered most commonly This is the number system that everyone learns as

a child As a result, people are most comfortable with decimal

Everything in a digital world reduces to binary form – everything! The information stored on your disk drive is stored in a binary form The information transported across your network is transmitted in a binary form Binary is the frank language of computers And while people may not be as comfortable with binary, comput-ers love it

Hexadecimal was invented to intimidate those who are new to information technologies It is the number sys-tem that is used to confuse students

Just kidding, hex is used as a kind of shorthand for binary Binary numbers tend to be rather long and cumber-some to type, write down, or say So hex is used to convey the same information in a compact fashion It turns out to be very simple to convert from hex to binary – you will learn how

The “base” value for a number system is simply the number around which the system is organized Everything orbits around the value 10 for base 10 (decimal) numbers For example, there are ten digits to work with Notice that base 2 (binary) numbers have only two digits that can be used

You do see the pattern, don’t you? If base 10 has ten digits to work with, and base 2 has two digits to work with, then base 16 must have – how many digits? You guessed it, sixteen They are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A,

B, C, D, E, and F

The sixteen hex digits sometimes leave people scratching their head because we use the symbols A, B, C, D, E, and F as digits These symbols simply represent quantities, as

follows:

These symbols are letters of the alphabet You know that; I

know that However, in hexadecimal numbers, they are digits,

perfectly good digits

Someone might point out to you that there are 3 windows in

the classroom The digit 3 conjures up something in your brain –

the notion of the quantity 3 In like manner, I could note that

there are B desks in the classroom That should make you think

of the quantity 11, hence 11 desks are present

Hexidecimal Digit Decimal Value

Figure B Hex Digit Values in Decimal

Base 10

Let’s consider a simple base 10 number, 1585 The number is structured in a very orderly way To begin with, note that the digits all come from the decimal symbol set (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) and that the symbol 5 appears twice in the number

We us the term “positional notation” to reflect the fact

that the 5 appearing in the rightmost position is

interpret-ed differently than the one appearing two positions to the

left That is to say that the position in which a digit

appears has everything to do with the significance or

weight that we attach to it In this case the rightmost 5

implies 5 ones and that to the left implies 5 hundreds

Take a look at Figure C It is obvious to most people how decimal values are organized The rightmost position

is the “ones place.” To the left is the “tens place.” To the left again is the “hundreds place,” and so on Notice that each “place” or position carries a weight 10 times the one to its right (Just what you’d expect? This is the base 10 number system, after all.)

A careful analysis of our example value reveals that the number 1585 is intended to convey to us, as readers

of the number, a quantity that is comprised of 1 thousand, 5 hundreds, 8 tens, and 5 ones

That is pretty simple, right? There are no surprises here This is stuff from the third grade in elementary school But what about the other number systems? They must be different somehow

Base 2

Let us now examine a binary example, 1001 Start by noting that the digits come from the smaller binary sym-bol set, just 0s and 1s And, in a manner similar to the previous example, the symsym-bol 1 appears twice

The rightmost position in a binary number is the ones

place, as it was in the decimal example (See Figure D.)

But as we progress to the left, each position carries a

weight that is 2 times the one on its right Thus we

observe the ones place, twos place, fours place, eights

place, and so on

So in binary values you have only 2 digits to work with, zeroes and ones, and the “places” in the numbers are each weighted 2 times the position to their right That’s why it is called the base 2 numbering system

Analyzing our example value (1001), we discover that this number is intended to convey to the reader a quan-tity comprised of 1 eight and 1 one, or 9 (in decimal)

Base 16

Now consider a hexadecimal example, 12A Just as with the other number systems, the digits come from the allowable symbol set, in this case 0 through 9 and A through F

Figure D Binary Notation Figure C Positional Notation

A few observations are in order before proceeding First,

some people seem to think that there is some difference

between a lower case “a” and an upper case “A.” There

is not You will see both cases regularly and the

differ-ence does not matter Second, as noted previously, some

people cringe seeing letters of the alphabet in numbers

Put your confusion aside Just as the

regular-old-every-day digit 5 makes you think of the number of fingers on

your hand, the hex digit C should make you think of the number of doughnuts that come in a package (Of course, a baker’s dozen would be a hex D.)

Consider Figure E The rightmost position is again the 1s place What is different is that as you move to the left, each position carries a weight that is 16 (get it, base 16) times the position on the right So you end up with a progression like 1, 16, 256, 4096, and so on

Our example value then, 12A, denotes 1 two-hundred-fifty-six, 2 sixteen’s, and A ones With a little quick men-tal gymnastics one realizes the A ones means 10 (decimal) ones to mere mormen-tals So this hex value conveys the notion of 256 plus 32 plus 10, or the quantity 298 decimal

Intermediate Summary

You have learned that three number systems are used commonly in IT: decimal; binary; and hexadecimal Also, these number systems are structured in a similar manner They each have a “base” value of 10, 2, and

16, respectively And, each has a limited symbol set (digits) that can be used in constructing numbers For deci-mal the set consists of the digits 0…9, for binary there are only 0s and 1s, and for hex the digits 0…9 and A…F are used

Each of the number systems employs a positional notation in which a given digit’s position within a number dictates the weight that we associate with it The rightmost position is always the 1s place As you move to the left, the positions carry a weight that is the base value times its right-hand neighbor The progressions 1, 10,

100, and so on appear for decimal numbers With binary numbers we see a progression like 1, 2, 4, 8, 16, 32, etc And for hex we observe the progression 1, 16, 256, 4096, and so on

Recognizing the Number System

How do you know whether the value 101 is decimal, binary, or hex? You don’t, unless there is additional clari-fication provided to you

The number system being used is very often conveyed by the context in which you observe the number By default, we tend to assume that a number is decimal unless otherwise indicated But if you were reading a paper on binary masking (whatever that is) you might reasonably assume that the value is binary You get the idea

Mathematicians use subscripts to indicate the number system So if you see 10110then you are dealing with

101 base 10, decimal The value 1012means that this is a binary (base 2) value And, as you would expect,

10116implies a hexadecimal (base 16) value

Figure E Hex Notation

Programmers have various notations that you will encounter as well It is extremely common to encounter a number like 0x101 The 0x prefix is C-programmer-speak for a hex constant So this pops up a lot

Assembly language programmers have their own notations For example, in such a program you might

encounter 101h, meaning a hex value Similarly, 101b implies a binary value

Decimal to Binary Conversion

Converting a decimal value to binary is like making change at a cash register Except, instead of using 1, 10, and 100 dollar bills you use “Bin-o-Bucks”* The denominations are 1, 2, 4, 8, 16, and so on

Make yourself a cheat-sheet like this:

How would you “pay” somebody 18 in bin-o-bucks? You would give them 1 sixteen and 1 two Like this:

And since you are not paying out any eights, fours, or ones, just put a zero in those positions to indicate that fact You end up with:

The binary value 10010 represents the decimal value 18 Easy Try it again by converting the value 3710to bin-o-bucks (Here is your cheat-sheet.)

Now check your result To pay out 37 you would use 1 thirty-two, 1 four, and 1 one Get it? 32 + 4 + 1 = 37 You answer should look like:

Do not make this more complicated than it is Conversion from decimal to binary is, in fact, this simple If you practice just a little bit, then you can easily become quite good at this

Binary to Decimal Conversion

Simply extend the change-making analogy that was just described For example, if someone just handed you

an 8 dollar bill and a 4 dollar bill then they just gave you:

And 8 + 4 is 12 So a binary 1100 is the representation for a decimal 12

Try this again for yourself The customer just handed you a 64, an eight and a two Here is your cheat sheet:

Your result should look like this:

64 + 8 + 2 = 74 So you just received 74 bucks

Decimal to Hexadecimal

Stick with the change-making analogy Only replace the bin-o-bucks with “Hex-A-Bucks”* The denominations

of the bills are 1s, 16s, 256s, and 4096s Your cheat-sheet looks like this:

4096 256 16 1

If you had to pay out 3510dollars, that would call for the use of two 16s and three 1s

4096 256 16 1

So the value 23 is the hex representation of the decimal value 35

Try your hand at this You need to pay out 2510in hex-a-bucks What do you get? Here is your cheat-sheet:

4096 256 16 1

Check your answer You should have gotten this:

4096 256 16 1

The answer is one 16 plus nine 1s 16 + 9 = 25

Hexadecimal to Decimal

Extend the hex-a-bucks analogy from the previous exercises to convert in the opposite direction, from hex to decimal this time

Start with you hex cheat-sheet:

4096 256 16 1

Now, let’s say that you are asked to convert 5416to base 10 Plug 54 into your cheat-sheet You get:

4096 256 16 1

That is five 16s and four 1s 5 x 16 = 80, add the 4 x 1, you get 84 decimal

Can you do this one: 1E16= ?10 Here is your cheat sheet:

4096 256 16 1

That would be 1 sixteen and E ones Recall that the symbol E depicts the quantity 14 in decimal So that is one

16 and 14101s; 16 + 14 is 30 decimal

Hexadecimal to Binary

Hex to binary is very easy Conversion is done using a simple substitution technique Each hex digit converts to four binary digits Here is what the hex digits look like in binary:

Hex Binary Hex Binary Hex Binary Hex Binary

Do you understand where these numbers come from? Let’s take a couple of them and analyze them to be sure that you appreciate what is going on here, starting with the hex 6 What does 616look like in binary?

Notice the 4 and the 2 bit (binary digit) positions are set to 1 The other positions are 0 One 4 plus one 2 equals 6

Or how about the 916table entry? In binary it looks like:

That’s one 8 plus one 1, or 9

Now that you know how this substitution table was constructed, let’s convert a sample hex value, how about B8, to binary Simply write binary 1011 in place of the hex B followed by binary 1000 in place of the 8 You get

1011 1000 binary (I put a space in there for clarity – binary numbers don’t actually have spaces in them.)

Hex value to be converted: B8

Substitute 1011 for hex B: 1011 8

Substitute 1000 for hex 8: 1011 1000

Try another example 4F116= ?2 In place of the 4 write 0100, in place of the F write 1111, and in place of the

1 write 0001 You get 0100 1111 0001

Hex value to be converted: 4F1

Substitute 0100 for hex 4: 0100 F 1

Substitute 1111 for hex F: 0100 1111 1

Substitute 0001 for hex 1: 0100 1111 0001

Binary to Hexadecimal

Converting a binary value to hex is similarly easy Just reverse the substitution process described above Working from right to left in the binary value, substitute one hex digit for each group of four binary digits An example is the best way to appreciate this

Binary value to be converted: 10110110

Identify groups of four bits: 1011 0110

Substitute hex 6 for 0110: 6

Substitute hex B for 1011: B 6

If the number of binary digits (bits) is not a multiple of four, then just pad it on the left with additional zeroes For example:

Binary value to be converted: 1010110110

Identify groups of four bits: 10 1011 0110

Pad the left with zero bits: 0010 1011 0110

Substitute hex B for 1011: 2 B 6

Substitute hex 2 for 0010:

Helpful Tools for Working with Number Systems

There is no shortage of tools to help you work with the

number systems Though hardly the best tool, the Windows

Calculator can be useful

On a Windows system, begin at the Start button, choose All

Programs, then Accessories Finally, choose Calculator The

familiar Windows Calculator will pop up You are probably

used to the wimpy balance-your-checkbook mode that

appears above But if you pick the View pull-down menu

and select Scientific, the fancier calculator features are

revealed

Figure F Calculator

Toward the upper left of the calculator you will see a

group box that contains four radio buttons: Hex, Dec,

Oct, and Bin The Bin button has been clicked in the

figure above and a binary value entered into the

calculator: 1010110110

Clicking on the Hex button, the calculator converts the binary value to hex: 2B6 You can use this technique to convert between any of the supported number sys-tems If you are curious, Oct stands for the octal, or base 8, number system

Another tool that you might like is my Number Converter You can obtain this from my website: HYPERLINK

"http://www.gwmays.com" www.gwmays.com Just follow the link to Number Converter Version 2 It can be downloaded as a zip file or, more conveniently, as a self-extracting zip, the exe

This tool has only one input field at

the upper left The program

con-verts any number that you enter

here to each of the number

sys-tems that we learned about A

note: the Number Converter

“guesses” what kind of value you

entered; you may have to click the

Type of Input Value button to

cor-rect the Number Converter if it

guesses wrong

Another tried and true tool is a conversion chart The conversion chart on the next page covers all values between 0 and 255 decimal

Figure G Hex Calculator

Figure H Hex Conversion

Figure I Number Converter 2.0