iSubnet

Thanks for taking the time to help me out. The podcast is linked from the sidebar. Be sure to click on the "album art" window in the lower left corner in iTunes to get the full effect of the graphics. The text is broken up into sections as blog postings. The Next==> link at the bottom of each one will take you to the next section. The title is a link to the previous one. In this case, it will take you to the evaluation questions. Please give your feedback as comment postings. Please feel free to post on relevant pages in addition to the evaluation page. Comments on the process (including the evaluation process) are especially welcome.

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A mathematician would correct me to say that the topic is correctly termed "positional numeral systems" w/ differing bases. However, in the vernacular, it's referred to as "number systems", so that's what I'm calling it too. Mathematicians also call this "place value notation", which is probably the most useful thing to describe how we're actually going to look at these ideas.

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So let's start with that numeral system that you're the most familiar with, that you've been working with since before kindergarten - decimal. "Decimal" comes from the latin "decimus" which means a tenth. Which brings us to the actual crux of the matter; that decimal is a base 10 number system. It's called "base 10" because are ten different values, zero through nine, that any of the numeral places can hold. We could create a 'value table' to describe or dissect the values of a numeral in a given place in the number, even though you don't think of it that way.


You know this; of course you do. There's the ones position, there's the tens, the hundreds, the thousands and so on. If we look at a number like 1536. One thousand five hundred and thirty six. We tend to evaluate it working from right to left, from the least significant value to the greater values. We have six in the ones position, three in the tens, five in the hundreds, and one in thousands. And this all seems like second nature to us right now, because it is. Because this is our native number system. Most people speculate that this is because we have ten digits on our hands, ten fingers (counting our thumbs).

But it's important for us to hang onto these ideas that we don't even think about in decimal, these ideas of

• position (place) value
• base value (range of valid numbers for each position)
• place significance (least significant & most significant values of placement)

because it's all going to get pretty kooky when we start to do some mathematics on other base systems. Actually, it's going to start to get a little kooky right now if we look a little deeper into the way we understand base 10, the way we construct the value table we used a moment ago.


What we think of as the ones place might better be called the zero-eth position. Huh? Stay with me. In terms of number sets, you may recall the "natural numbers", the counting numbers. Negative numbers are not part of the set of natural numbers. Historically there has been some issue about whether this set includes zero or not, because zero was a rather advanced abstraction. Many ancient societies, including the Romans, didn't have a numeral for zero. It isn't naturally where we begin to count. We tend to think that if there aren't any, we can't count them. But if we think of our value table as an inventory sheet, then we can begin to appreciate that zero is, indeed, a natural number. It's a number that is less than one and so it is the first counting number.


Back to the the least significant digit. We call it the ones place because 1 is what we multiply by that positional numeral in order to get the number value. But why exactly is that? If we think about it, why is the first position in a base 10 system not ten? Well, it's because the place value is the result of ten (the base) to the zero-eth power. 100 = 1. We'll be working with exponents but this is a part that you may not remember and will have to memorize. Anything to the power of zero equals one. It just does. Therefore, the least significant decimal digit is ones. Six times one is six. The next place significance value is the tens because ten to the power of one equals ten. Three time ten equals thirty. Ten to the power of two (ten times ten) is a hundred and five times a hundred is five hundred. Ten to the power of three (ten times ten times ten) is a thousand and so on.

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The other base system that concerns us most is base 2. It is also called "binary" because it is composed of two parts. This is the native number system of computers. Computers only have two fingers, if you will; on or off. That's it. Back in the old days there were relays that were either open or closed. (In fact, Grace Hopper found that her program wasn't working right because there was a moth caught in a relay, preventing it from signaling as closed. That was the first computer bug. True story.) Nowdays we have little pulses of plus or minus 3.3 volts or whatever signaling level the chip expects. But computers still only have those two fingers, on or off. Because geeks are cool, we have shortened the term "binary digit", to get the word "bit".


Constructing a binary value table looks different but he principle is the same. The rightmost least significant bit is also the ones place but the full range of values is zero or one. Because of, this, the place values are dictated by powers of two. Two is the base number (because there are only two possible values for a binary digit.) This is the fundamental property of the base 2 number system. We will be doing lots of work with of powers of base two. The sooner you get comfortable with "2 to the power of" values, the happier you're going to be. Extra resources will be available to help with that.

Let's start with 2 to the power of zero, the least significant bit. 2 to the power of zero is? What? It's one. Why? Just because anything to the power of zero is one. Working our way leftwise, to the next most significant bit, this position (or place) value is 2 to the power of one, which is? It's two. Because? Anything to the power of one is the number itself. It's like we're back learning multiplication tables. The next most significant digit value is 2 to the power of 2. The exponent, of course, indicates how many instances of the base is multiplied by itself; 2 times 2. That's 4. And the most significant of these four digits, the leftmost position multiplier has the value of 2 to the power of 3. 2 times 2 times 2, is 3 2's multiplied together, or 8.

Let's take a look at a binary number such as 0101. Working from the least significant digit, there is a 1 in the 1's place, 0 in the 2's, 1 in the 4's, and 0 in the 8's. We add that up as 1 + 4 and get 5. 0101 in binary equals 5 in decimal. Okay with that? How about 1010? 0 1's, 1 2's, 0 4's, and 1 8's. Right? So 2 + 8 = 10. 1010 in binary equals 10 in decimal. If we have four places then the highest number possible is 1111, which 1 + 2 + 4 + 8, which equals a decimal 15. For the same reason that the highest 4 digit decimal value is 9,999.

You've heard of bits and bytes. We just said that a bit is a binary digit. A byte is a grouping of 8 bits interpreted together. How many possible values are there in a byte? If there are 8 bits, then we're looking at 27 possible values, or 256. There's a rather arcane term, called the "nibble". That's 4 bits - or half of a byte. Cute. Nobody ever talks about a nibble, but it's going to be a handy unit for some of our purposes.

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We're going to be working with binary a lot more, a whole lot more. But for the sake of comparison, now is a good time to look a another "differently-based number system" that will crop up in networking, base 16. Base 16 is also called "hexadecimal", which means something like "six plus ten". It's called "hex" for short, but it can feel like something of a curse, too.

We're getting familiar with the types of things that we look for in a number system. Since it is called base 16, we can infer that there is a range of 16 values for a given digit. Obviously, our decimal range of 0 thru 9 is insufficient. Hex digits follow the sequence 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. Pretty tricky, huh? (The letters are usually in caps, by the way.) To build a value table, we start with the rightmost, least significant position. That is worth 16 to the zero-eth power, which is 1. (Because anything to the power of zero is one.) In the next-most significant position is the 16's place, because 16 to the power of 1 is 16. You can see that it gets mathier in a hurry. The third digit from the rightmost has the value multiplier of 162. That's 16 times 16 or 256. The fourth position is 163, 16 times 16 times 16. (It's also 256 times 16, of course.) That's 4096!

If we want to translate a number like FEED (which is a very cool number), it's going to get very big. There's a bit of notation that you sometimes see that indicates that the number is hex and that is a zero and an x. You call it hex, if it's pronounced at all. So 0xD times 1 = 13, if you can wrap your brain around that. Then in the 16's place, E (which is 14) times 16 = 224. But E times 256 = 3584. And an F in the 4th digit is F (or 15) times 4096 = 61,440. So FEED = 13 + 224 + 3584 + 61,440; which equals 65,261. If you need to pause this and put your head down between your knees for a few deep breaths, go ahead.

Why do you suppose that base 16 is going to be useful if computers only have 2 fingers, if their native number system is binary? Let's harken back to that binary value table that we built. Do we remember the greatest value that we said we could get in our 4 bit nibble? It was 15; the range of values for 4 bits is 0 thru 15. 16 different values. Aha! That's pretty sneaky - we can put 4 binary digits, a whole nibble, into 1 hex digit. Yes, computers may only have 2 fingers, but they can count on them really, really quickly. If we think that 65,261 was an unwieldy representation, what will 0xFEED look like in binary? for starters, how many binary digits do we expect? If it's 4 hex digits and each hex digit represents 4 bits, it's going to be a 16 digit binary number, right? While this is just the kind of translation fun that we're going to get deep into shortly, for now you'll just have to trust me that 0xFEED = 1111111011101101.

Actually, you don't have to trust me. If you're using iTunes, bring up your computer's calculator. Set it to programmer or scientific mode, whatever it is on your OS. You can translate from decimal to binary to hex to your heart's content. It's a great way to check your work as you practice translating by hand. Make no mistake about it, you have got to be able to do all of these calculations by hand and you absolutely must thoroughly understand the principles. The same principles that apply to subnetting apply to more "advanced" concepts like supernetting. Once you really get it well enough that the subnetting questions on your CCNA test are gimmees and you're working in the field, then you can use a subnet calculator. But you really can't rely on them if you don't completely understand the principles of the math. Until then, use them only to check your calculations.

Okay, where were we? We were comparing these number systems and their application to working with our 2-fingered computers. You've probably seen some hex data if you've done any html or other basic computer color manipulation. The standard is that a color is specified by red, green, and blue levels of 0-255 represented in hex. For instance, you can get a nice sea green with 0x009966, complimented by lovely shade of cantaloupe, or 0xFF9933. Notice that they both have the same highish levels of green (the 99 byte) and a slight variance in the blue. It's the complete presence or absence of red that changes the color. What a difference a byte makes.

Of course, one bit can be the difference between a match or no match, but the byte is the primary grouping of bits that we use. In the calculators I just mentioned, you might notice mention of the octal number system. Because base 8 represents the byte (8 bits of data), it has been used in other computer related functions. We won't need octal, but the byte will come up all the time. You'll see hardware addresses of network adapters represented by 6 hex bytes. The address 00:16:cb:bc:5a:37 is much easier to read than it's 48-bit binary equivalent. By the way, you will also see the MAC address notation in these other formats (00-16-cb-bc-5a-37 and 0016.cbbc.5a37).

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