Introduction to Variables:The Numeric Systems

Introduction


When a computer boots, it “loads” the operating system. If you want to use a program, you must find it either on the Start menu or from its directory and take the necessary action to open it. Such a program uses numbers, characters, meaningful words, pictures, graphics, etc, that are part of the program. As these things are numerous, so is the size of the program, and so is the length of time needed to come up. Your job as a programmer is to create such programs and make them available to the computer, then to people who want to interact with the machine.

To write your programs, you will be using alphabetic letters that are a, b, c, d, e, f, g, h, I, j, k, l, m, n, o, p, q, r, s, t, v, w, x, y, z, A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z. You will also use numeric symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Additionally, you will use characters that are not easily readable but are part of the common language; they are ` ~ ! @ # $ % ^ & * ( ) _ + - = : “ < > ; ‘ , . /. Some of these symbols are used in the C# language while some others are not. When creating your programs, you will be combining letters and/or symbols to create English words or language instructions.

Some of the instructions you will give to the computer could consist of counting the number of oranges, converting water to soup, or making sure that a date occurs after January 15. After typing an instruction, the compiler would translate it to machine language. The computer represents any of your instructions as a group of numbers. Even if you ask the computer to use an orange, it would translate it into a set of numbers. As you give more instructions or create more words, the computer stores them in its memory using a certain amount of space for each instruction or each item you use.


There are three numeric systems that will be involved in your programs, with or without your intervention.

The Binary System



When dealing with assignments, the computer considers a piece of information to be true or to be false. To evaluate such a piece, it uses two symbols: 0 and 1. When a piece of information is true, the computer gives it a value of 1; otherwise, its value is 0. Therefore, the system that the computer recognizes and uses is made of two symbols: 0 and 1. As the information in your computer is greater than a simple piece, the computer combines 0s and 1s to produce all sorts of numbers. Examples of such numbers are 1, 100, 1011, or 1101111011. Therefore, because this technique uses only two symbols, it is called the binary system.

When reading a binary number such as 1101, you should not pronounce "One Thousand One Hundred And 1", because such a reading is not accurate. Instead, you should pronounce 1 as One and 0 as zero or o. 1101 should be pronounced One One Zero One, or One One o One.

The sequence of the symbols of the binary system depends on the number that needs to be represented.

The Decimal System



The numeric system that we are familiar with uses ten symbols that are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Each of these symbols is called a digit. Using a combination of these digits, you can display numeric values of any kind, such as 240, 3826 or 234523. This system of representing numeric values is called the decimal system because it is based on 10 digits.

When a number starts with 0, a calculator or a computer ignores the 0. Consequently, 0248 is the same as 248; 030426 is the same as 30426. From now on, we will represent a numeric value in the decimal system without starting with 0: this will reduce, if not eliminate, any confusion.

Decimal Values: 3849, 279, 917293, 39473
Non- Decimal Values: 0237, 0276382, k2783, R3273

The decimal system is said to use a base 10. This allows you to recognize and be able to read any number. The system works in increments of 0, 10, 100, 1000, 10000, and up. In the decimal system, 0 is 0*100 (= 0*1, which is 0); 1 is 1*100 (=1*1, which is 1); 2 is 2*100 (=2*1, which is 2), and 9 is 9*100 (= 9*1, which is 9). Between 10 and 99, a number is represented by left-digit * 101 + right-digit * 100. For example, 32 = 3*101 + 2*100 = 3*10 + 2*1 = 30 + 2 = 32. In the same way, 85 = 8*101 + 5*100 = 8*10 + 5*1 = 80 + 5 = 85. Using the same logic, you can get any number in the decimal system. Examples are:

2751 = 2*103 + 7*102 + 5*101 + 1*100 = 2*1000 + 7*100 + 5*10 + 1 = 2000 + 700 + 50 + 1 = 2751

67048 = 6*104 + 7*103 + 0*102 + 4*101 + 8*100 = 6*10000 + 7*1000+0*100+4*10+8*1 = 67048

Another way you can represent this is by using the following table:

etc Add 0 to the preceding value 1000000 100000 10000 1000 100 10 0

When these numbers get large, they become difficult to read; an example is 279174394327. To make this easier to read, you can separate each thousand fraction with a comma. Our number would become 279,174,394,327. You can do this only on paper, never in a program: the compiler would not understand the comma(s).

The Hexadecimal System



While the decimal system uses 10 digits (they are all numeric), the hexadecimal system uses sixteen (16) symbols to represent a number. Since the family of Latin languages consists of only 10 digits, we cannot make up new ones. To compensate for this, the hexadecimal system uses alphabetic characters. After counting from 0 to 9, the system uses letters until it gets 16 different values. The letters used are a, b, c, d, e, and f, or their uppercase equivalents A, B, C, D, E, and F. The hexadecimal system counts as follows: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, and f; or 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. To produce a hexadecimal number, you use a combination of these sixteen symbols.

Examples of hexadecimal numbers are 293, 0, df, a37, c23b34, or ffed54. At first glance, the decimal representation of 8024 and the hexadecimal representation of 8024 are the same. Also, when you see fed, is it a name of a federal agency or a hexadecimal number? Does CAB represent a taxi, a social organization, or a hexadecimal number?

From now on, to express the difference between a decimal number and a hexadecimal one, each hexadecimal number will start with 0x or 0X. The number will be followed by a valid hexadecimal combination. The letter can be in uppercase or lowercase.
Legal Hexadecimals: 0x273, 0xfeaa, 0Xfe3, 0x35FD, 0x32F4e
Non-Hex Numbers: 0686, ffekj, 87fe6y, 312


There is also the octal system but we will not use it anywhere in our applications.