# BINARY NUMBER SYSTEM

Digital logic and memory devices are based on two electrical states (on and off), it is natural to use a number system, called the binary number system, which contains only two symbols, namely 0 and 1.

Why we use only binary number?

The number system that you are familiar with, that you use every day, is the decimal number system, also commonly referred to as the base-10 system. When you perform computations such as 5 + 2 = 7, or 25 – 9 = 16, you are using the decimal number system. This system, which you likely learned in first or second grade, is ingrained into your subconscious; it’s the natural way that you think about numbers. It is the way that everyone thinks and has always thought about numbers arithmetic.

The Roman numeral system, predominant for hundreds of years, was also a decimal number system (though organized differently from the Arabic base-10 number system that we are most familiar with). Indeed, base-10 systems, in one form or another, have been the most widely used number systems ever since civilization started counting

workings of a compute are also based on the binary number system, also referred to as the base-2 system.

Why do we use base-10 ?

Base-10 system is a positional system, where the rightmost digit is the position of the one (the number of ones), the next digit to the left is the tens position (the number of groups of 10), the next digit to the left is the hundreds position (the number of groups of 100), and so forth. The base-10 number system has 10 distinct symbols, or digits (0, 1, 2, 3,…8, 9)

For example, consider the decimal number: 6349

It is the only system that you have used extensively, and, again, the fact that it is used extensively is due to the fact that humans have 10 fingers. If humans had six fingers, we would all be using a base-6 system, and we would all find that system to be the most intuitive and natural.

All data in a computer is represented in binary. The pictures of your last vacation stored on your hard drive are also in all bits. The YouTube video of the cat falling off the chair that you saw this morning is also in bits. Your Facebook page is also in bits. The tweet you sent is also in bits. The email from your professor telling you to spend less time on vacation, browsing YouTube, updating your Facebook page, and sending tweets that are bits too. Everything is bits.

## Binary number system

Since you have been using the 10 different digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 all your life, you may wonder how it is possible to count and do arithmetic without using all 10. Actually, there is no advantage in using 10 counting digits rather than, say, 8, 12, or 16. The 10-digit system (called the decimal system, since the word “decimal” means “based on 10”) probably came into universal use because man first started to count by using his fingers, and there happen to be 10 of them.

To see how to count by using other than 10 digits, notice how we count in the ordinary decimal system. We represent a number higher than 9, the highest digit, by a combination of two or more digits. The number next after 9 is 10, and then 11, etc. After we reach 99, the highest number that can be written with two digits, we start using three digits. The number next after 99 is 100, and then comes 101, etc

we write a binary number as a string of symbols, but now each symbol is a 0 or a 1. To explain a binary number, we multiply each digit by the power of 2 associated with that digit’s position.

For example, consider the binary number 1101

Since binary numbers can only contain the two symbols 0 and 1, numbers such as 25 and 14520 cannot be binary numbers

We say that all data in a computer is stored in binary as 1’s and 0’s. It is important to keep in mind that values of 0 and 1 are logical values, not the values of a physical quantity, such as a voltage. The actual physical binary values used to store data internally within a computer might be, for instance, 5 volts and 0 volts, or perhaps 3.3 volts and 0.3 volts or used to reflect and no reflection. The two values that are used to physically store data can differ within different portions of the same computer. All that really matters is that there are two different symbols, so we will always refer to them as 0 and 1.

A string of eight bits (such as 11000110) is termed a byte. A collection of four bits (such as 1011) is smaller than a byte and is hence termed a nibble

Given a fixed number of n bits, known as a word, which the arithmetic unit of a computer is designed to handle, then there are 2n separate binary numbers that can be accommodated. For example, in 8 bits, one can accommodate the binary numbers corresponding to decimal 0 to 255 (256 different numbers).

Current computers have word lengths of 32 or 64 bits

### Binary Number Registers

In a computer, a number is represented physically in a register, which has a fixed length. For example, a register having 16 binary digits (“bits”) limits the amount of information that can be represented

• Note also that each bit must have a value of either 0 or 1
• Note also that there is no provision for a binary point in a physical register. The position of the binary point must be inferred based on convention or other information.