The Numeric Systems
 

Introduction

When you launch a program, there is an initial setup that the program has to go through to be ready. A program has numbers, characters, words, pictures, graphics, etc, that are part of the application. 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 write instructions that can make such programs available to the computer and usable to people who want to interact with the machine.

To write your programs, you will be using alphabetic characters 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 symbols that are not easily readable but are part of the common language; they are ` ~ ! @ # $ % ^ & * ( ) _ + - = : “ < > ; ‘ , . /. Some of these symbols are used in the Object Pascal 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 would consist of counting the number of oranges, converting water to soup, or making sure that a date occurs after January 15. After you have typed an instruction, the compiler would translate it to machine language. This is because the computer does not understand the language you and I speak or write. 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 the computer memory using a certain amount of space for each instruction or each item you use.

There are three numbering systems that will be involved in your programs, with or without your intervention. The decimal system provides the counting techniques that you use everyday. The hexadecimal system is an intermediary system that allows you to know how the computer deals with numbers. The binary system is the actual system that the computer uses to find out (almost) everything in your program.

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.

Why learn the binary system? You need to know what the binary system looks like because you will be writing instructions to the computer, and the computer does not understand English.

 

 

The Decimal System

The numeric system that we have always used uses a set of 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).

Why use the decimal system? Because, to represent numbers, that is the system that you and I are familiar with.

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 Latin language 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, and F. Once again, to produce a 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 a $ symbol. The number will be followed by a valid hexadecimal combination. Legal Hexadecimals:$273, $feaa, $fe3, $35FD, $32F4eNon-Hex Numbers: 0686, ffekj, 87fe6y, 312

Why learn or use the hexadecimal system? Because the computer does not understand the decimal system, which is our everyday base of counting items, and because we do not understand or are not familiar with the binary system, the hexadecimal system provides an intermediary system. Also, the hexadecimal system use shorter forms of representing the same numbers in the decimal system.

Signed and unsigned

The numbers we have used so far were counting from 0, then 1, then 2, and up to any number desired, in incrementing values. Such a number that increments from 0, 1, 2, and up is qualified as positive. By convention, you do not need to let the computer or someone else know that such a number is positive: by just displaying or saying it, the number is considered positive.

In real life, there are numbers counted in decrement values. Such numbers start at –1 and move down to -2, -3, -4 etc. These numbers are qualified as negative.

When you write a number “normally”, such as 42, 502, or 1250, that is, if there is no sign on the left side of the number, it is positive. Such a number, because it displays no sign, is referred to as unsigned. If you want to express the number as negative, you use the – sign on the left side of the number. When displaying such a sign, the number is referred to as signed. Therefore, if the number does not have the – sign, the compiler considers such a number as unsigned and it is positive. On the other hand, a number qualifies as signed when it can be positive or negative, depending on how it is used.

 

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