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History of the Computer – It’s a Binary World – How Computers Count
Many have cursed their computer for taking things too literally! It’s easy to blame the computer when something goes wrong.
You get to the market checkout and the item you bought “on special” comes out at full price. The manager has to be called to fix it, and what does he say? “We’ve had problems with the computer, it has the wrong price on some things.”
Enter a list of addresses into your word processor and print the party invitations for next week. Then you will find that today’s date has been inserted into the signature block – by the computer!
Maybe you’ve heard the expression “Garbage in garbage out”? Someone, at some point, told the computer to do what it did, it didn’t deliberately decide to mess with you. Computers can only do what they are told, they are more. logical that Spock and take everything literally.
Let’s see why they are so pedantic!
The world around us has many aspects that work in the same way as a computer. There are many examples of opposites, for example up and down, left and right, forward and backward. A light can be on or off, maybe it’s night or day. Yes or no? You can think of many others. This system of two possible states is called a binary system. If it’s not one, it must be the other.
A computer uses the binary system to perform all its functions, the basic unit, originally a vacuum tube, then a transistor, then a chip, is used thousands of times to make the total unit. The on/off light we mentioned above is controlled by a switch. In the computer, this switch is a transistor, which is either on or off.
Now let’s get to the math! Don’t worry, it’s very simple math! In fact, it’s so simple that we just count up to 1. That’s right, we can only have two states, so we count from 0 to 1. (This is another thing that computers are pedantic about, they insist on starting at zero).
The binary system is a number system. You are familiar with the decimal system which has 10 numbers from 0 to 9 (think like a computer 0 is first). You can create all kinds of number systems for whatever purpose you want. You probably know about a dozen (12) and have heard of half a dozen as well. If you’ve used a computer a lot, you may have come across the hexadecimal system. This one has 16 “numerals” 0-9 and AF. Another numbering system used by computer scientists is the octal system which has 8 numbers, from 0 to 7.
Okay, so how do we count with just 0’s and 1’s. Simple, exactly the same way you count in decimal. The first ten numbers are fine, 0 to 9, but what? Let’s start again but add a 1 making 10 or “one, zero”. This takes us to “one, nine” and we go to “two, zero”, and so on to “nine, nine” and then add a 1 again to make 100: “one, zero, zero”.
DECIMAL 0-9, 10-19, 20-…..-99, 100.
If you’ve followed me so far, you’re ready for the binary sequence, it’s much simpler. Starting at zero we have 0.1, and that’s it. We follow the same rule and add a 1, making “one,zero”. Next comes “one, one”; then “one, zero, zero”; -“one, zero, one”; etc. They are equivalent to decimals 0,1,2,3,4,5. How does this relate to computers? This is the following.
BINARY – 0 1 10 11 100 101
DECIMAL– 0, 1, 2, 3, 4, 5
In our computer we have transistor switches as described above. For the math example we just saw, we need 3 switches. All of them represent a binary digit or bit. To represent a decimal 1, these switches would be OFF,OFF,ON or 001. For a decimal 5 we would have ON,OFF,ON or 101. By extension you can see that with 4 switches we could go to 1111 or 15. Decimal.
TRANSISTORS [OFF OFF ON] [ON OFF ON] [ON ON ON ON]
BINARY…… 001 101 1111
DECIMAL….. 1, 5, 15
Another point to keep in mind is that each binary digit, or bit, has a value. Just as in decimal we have units, tens, hundreds, etc. in binary the values are 1,2,4,8,16,32,64,128, etc. etc. The binary code 1111 mentioned above is therefore 1+2+4+ 8=15. what would BINARY 1010 be in decimal?
BIT VALUE 8 4 2 1
BINARY…. 1 0 1 0
If you wanted to find out which binary 100101100 was in decimal, you could add the individual values. In fact, people who work on basic machines need to know “machine code”! For them 1010 would be A in hexadecimal or 12 in octal.
One of the reasons for using octal or hexadecimal code is to allow humans to interpret machine codes. Some mainframe computers use “words” made up of 24, 32, 36, or 72 bits. They are displayed or printed in groups of three for octal or four for hexadecimal. For example, the 24-bit binary word on a computer can be interpreted as shown here.
BINARY 100 111 000 011 010 000 011 100
OCTAL.. 4 7 0 3 2 0 3 4
BINARY 1001 1100 0011 0100 0001 1100
HEX… 9 D 3 4 1 D
This probably seems like a very long way to work out numbers, until you remember that these “switches” can operate at a speed of nanoseconds, on the order of 1,000,000,000 times per second, making large calculations possible.
That’s probably enough to digest in one sitting. Next we will see how a computer adds and multiplies.
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