The number system is a notational scheme that deals with using a symbol to represent a quantity.

A point to ponder: We know that a quantity can be both finite and infinite. In the real world, many things around us are quantifiable and can be accounted for. Trees in a garden are finite. The population of a country is finite. In contrast, sand particles are seemingly infinite (by being intractable and ubiquitous). Star systems are seemingly infinite (by observation). Prime number sequences are infinite (by conjecture). It is also understood that tangible and intangible things exist in nature in both finite and infinite states. A finite entity can be made infinite just by endless replication. An infinite and intangible entity can be harnessed as a finite unit by giving it a name and identity. Can you think of some examples in this universe (for example, is this one of many universes or is it the only one and infinitely expanding)?

In my experience, there is a lot of confusion regarding number systems, even with some experienced IT folk. Quantities and the representation of these quantities such as symbols/notations are primarily separate entities. A notation system and what it represents are completely different things, although because of ubiquity and visibility, the meanings are exchanged and we take it for granted that they are both one and the same, and that creates the confusion. We normally count using our fingers because it seems natural to us. We have five digits per hand and they can be utilized to count up to 10 units. So, we developed a decimal counting system. Note that the numbers 0 to 9 constitute the whole symbol set for whole numbers. While defining a symbol set, although we use the symbols that are designed through the centuries that have passed and have their place, it is not mandatory to define numbers only in that particular set. Nothing prevents us from developing our own symbol set to notate quantities.

An example symbol set = {NULL, ALPHA, TWIN, TRIPOD, TABLE}, and we can substitute pictures in the above set, which directly map to {0, 1, 2, 3, 4}. Can you think of your own symbol set?

The largest number in the symbol set (9) is one less than that of the base (10). Also, zero was a relatively late addition, with many cultures not realizing that null or nothing can also be symbolized. Using zero in a number system is the crux to developing a position-based encoding scheme. You can only occupy something where there is a void that acts as a container, so to speak. So, you can think of 0 as a container for symbols and as a placeholder for new positions. In order to count 10 objects, we reuse the first two symbols from the symbol set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. This is key to understanding number systems in vogue. What happens is that each number position/column is taken as an accumulator. You start at your right and move towards the left. The value in the first column changes in place as long as the symbol set has a symbol for it.

When the maximum value is achieved, a new column toward the left is taken and the counting continues with the initial symbols in the symbol set. Thus, think of each column as a bucket that is a larger bucket than the one to the right of it. Further, each new column represents the largest quantity of the last column. Here, the number of columns used or the number of symbol places denotes the range of quantities that can be represented. We can only use the symbols in the symbol set. Thus, if we had a set of infinite symbols for each quantity, we would not have to reuse the symbols to represent larger quantities, but that would be very unwieldy as we humans don't have a very good memory span.

To reiterate, think of the columns as containers. Once you are out of symbols for that particular column, you reuse the first symbol greater than zero. Thereafter, you reset the previous column to zero, and start increasing the symbol count till it reaches the maximum in the set. You then repeat the process for the specific quantity to be represented. Study the following image to gain more understanding visually:

You can also look at the number system notation as a histogram-based notation that uses symbols instead of rectangles, wherein the quantity is represented by the total count in a compact representation. The histogram is essentially a statistical tool to find the percentage of an entity or a group of entities and other control points such as features of the entities in a population that contains the entities.

Think of it as a frequency count of a particular entity. Here, each entity refers to the base power group that each digit towards the left represents.

So, taken as a summation of weights, each position that can be seen as representing a total Frequency Count of how many of that position's relative quantity. Instead of drawing 15 lines to denote 15 objects, we use the symbols 1 and 5 to denote 5 objects and 10 more, with 5 joining the 1 and then taking the place of 0, which acts as a container or placeholder to give the combined notation of 15.

For a larger number such as 476, you can see this notation as a count of how many 100s, 10s, and the core symbol set values. So, 400 = 4 * 100 or there are 4 hundreds, and 7 * 10 or that there are 7 tens and 6 values more. The reason you add these up is because they each represent a part of the total.

Can you repeat the process with a new base? Why don't you try base 3? The solution will be given later in this chapter, but you must try it yourself first.

Have you ever wondered why the base 8 (octal) does not have the numbers 8 and above in its notation? Use the same rules that you have just read to reason why this notation system works the way it does. It follows the same number symbol-based position-relative notation. You can also think of this as weights being attached to the positions as they are positioned towards the left. Finally, as each row signifies a count of the next quantity, you essentially sum up all the position values according to their weights.

We are accustomed to using the above formula as an automated response for counting numbers without ever giving much thought to the reasoning behind this notational system. It's taken for granted that you never question it.

The hexadecimal base notation also works in the same way. The reasoning behind using 16 as a quantity belies the fact that a permutation of 2 symbols {0, 1} to a length of 4 gives us 16 different patterns. Since 4 bits used as a basic block works for grouping bit sequences as per engineering conventions, the nibble is the smallest block unit in computing. The bit is the smallest individual unit present. The minimum value is 0000 and the largest value is 1111. These unique patterns are represented using symbols from the alphabet and the numbers 0 to 9.

You can replace the alphabet symbols A to F with any shape, picture, pattern, Greek letter, or visual glyph. It's just that the alphabets are already a part of our communication framework, so it makes sense to reuse them. So, the convention of grouping 4 bits to form a pattern using a notation that expresses the same thing provides a much more convenient way to look at binary information. Since our visual acuity is much sharper when we form groups and patterns, this system works for engineers and analysts, who need to work with binary data (in a domain-agnostic manner) on a regular basis. Note that as per convention, hexadecimal values are prefixed with 0x or post-fixed with H to denote hexadecimal notation.

The hexadecimal symbol set = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}

Permutations are also the foundational mathematics behind data type representation. So, we have taken 4 bits to form a nibble and 8 bits to form a byte. What is a byte? Taken simply, it is a series of symbols from the set {0, 1} to a length of 8, which represents the permutated value of that particular sequence as a quantity. It could also be further used to represent a packed data type where each bit position denotes a toggle of a value as on or off, similar to an array of switches. Since we work with programming languages, the binary data types are of a primary interest as they work like an index into a table where the entire range from the minimum to the maximum is already worked out as part of a finite series. Thus, the bit pattern 00001111 gives the value of 15 out of a total of 2^8 values. Why 2^8? This is because when you need to compute unique values out of a symbol set to a specific length, you take the total number of symbols to the power of the length to get the maximum value. However, you also have to take into account the primary conditions for permutations and its difference from combinations all relating to symbol usage and being ordered or not. As a rule, to reuse the symbols in a specific order, you can take powers, as in the case of permutations. However, if using a symbol removes it from the participation of the next position's symbol set, you need to take factorials, as in the case of combinations. They all fall into a branch of mathematics called Combinatorics. Likewise, do you see the logic behind primitive data types such as int, short, char, and float? When using custom data types, such as structs and classes, you are effectively setting up a binary data structure to denote a specific data type that could be a combination of primitive data types or user-defined ones. Since the symbol set is the same for both primitive/primary and data types, it is the length of the data structure assigned per data type that gives meaning to the structure.

For a simple exercise, find the unique ways in which you can arrange the letters {A, B, C}, where each symbol can be reused to a length of 3, that is, each position can use any symbol from the set above. Thereafter, find the unique ways in which you can combine the symbols, without repeating any previous pattern but in any sequence. You will find that you get 27 patterns from the first exercise and 6 patterns from the second. Now, build a formula or try to model this pattern. You get (base^(pattern length)) and factorial (base). This is how binary notation is used to encode quantities, which are being denoted by symbols (which can also be mapped to a scheme), which in turn are based on the principles of human language, and therefore, all information can be encoded in this manner.

Computers do not even understand the symbol 1 (ASCII 0x31) and the symbol 0 (ASCII 0x30). They only work with voltage levels and logic gates as well as combined and sequential circuits such as D flip-flops for memory. This complex dance is orchestrated by a clock that sets things in motion (a regular pulse of n cycles/s aids in encoding schemes, in much the same way as in music, the rhythm brings predictability and stability that greatly simplifies encoding/decoding and transmission); much like a conductor, in tandem with the microprocessor, which provides built-in instructions that can be used to implement the algorithm given to it. The primary purpose of using various notation systems is that doing so makes it more manageable for us to work with circuit-based logic and provides an interface that looks familiar so that humans can communicate with the machine. It's all just various layers of abstraction.

The following table shows how the base 3 notation scheme can be worked out:

Can you write a program to display the base notation of bases from 2 to 16? A maximum of base 36 is possible by using alphabets and numbers as the symbol set after which new symbols or combinations of existing symbols have to be used to map a new symbol set. I think that this would be a great exercise in programming fundamentals.