The Binary Number System

The number system that you are familiar with, that you use every day, is the decimal number system, alsocommonly referred to as the base-10 system. When you perform computations such as 3 + 2 = 5, or 21 –7 = 14, 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. Of course itis not just you: It is the way that everyone thinks—and has always thought—about numbers and arithmetic. Evidence exists that Egyptians were using a decimal number system five thousand years ago.

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 eversince civilization started counting. In dealing with the inner workings of a computer, though, you are going to have to learn to think in adifferent number system, the binary number system, also referred to as the base-2 system.Before considering why we might want to use a different number system, let’s first consider: Why do weuse base-10? The simple answer: We have 10 fingers. Before the days of calculators and computers, wecounted on our hands (many of us still do!).

Consider a child counting a pile of pennies. He would begin: “One, two, three, …, eight, nine.” Upon

reaching nine, the next penny counted makes the total one single group of ten pennies. He then keeps

counting: “One group of ten pennies… two groups of ten pennies… three groups of ten pennies … eight

groups of ten pennies … nine groups of ten pennies…” Upon reaching nine groups of ten pennies plus nine additional pennies, the next penny counted makes the total thus far: one single group of one hundred pennies. Upon completing the task, the child might find that he has three groups of one hundred pennies,five groups of ten pennies, and two pennies left over: 352 pennies.

More formally, the base-10 system is a positional system, where the rightmost digit is the ones position (the number of ones), the next digit to the left is the tens position (the number of groups of 10), the nextdigit 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). In decimal notation, we write a number as a string of symbols, where each symbol is one of these ten digits, and to interpret a decimal number, we

multiply each digit by the power of 10 associated with that digit’s position.

There is nothing essentially “easier” about using the base-10 system. It just seems more intuitive only
because 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.
So, long ago, humans looked at their hands, saw ten fingers, and decided to use a base-10 system. But
how many fingers does a computer have?
Consider: Computers are built from transistors, and an individual transistor can only be ON or OFF (two
options). Similarly, data storage devices can be optical or magnetic. Optical storage devices store data in
a specific location by controlling whether light is reflected off that location or is not reflected off that
location (two options). Likewise, magnetic storage devices store data in a specific location by
magnetizing the particles in that location with a specific orientation. We can have the north magnetic pole
pointing in one direction, or the opposite direction (two options).
Computers can most readily use two symbols, and therefore a base-2 system, or binary number system, is
most appropriate. The base-10 number system has 10 distinct symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. The
base-2 system has exactly two symbols: 0 and 1. The base-10 symbols are termed digits. The base-2
symbols are termed binary digits, or bits for short. All base-10 numbers are built as strings of digits (such
as 6349). All binary numbers are built as strings of bits (such as 1101). Just as we would say that the
decimal number 12890 has five digits, we would say that the binary number 11001 is a five-bit number.
The point: All data in a computer is represented in binary. The pictures of your last vacation stored on
your hard drive—it’s all bits. The YouTube video of the cat falling off the chair that you saw this
morning—bits. Your Facebook page—bits. The tweet you sent—bits. The email from your professor
telling you to spend less time on vacation, browsing YouTube, updating your Facebook page and sending
tweets—that’s bits too. Everything is bits.
To understand how computers work, you have to speak the language. And the language of computers is
the binary number system.