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Introduction to Binary Numbers

How Computers Store Numbers

Computer systems are constructed of digital electronics. That means that their electronic circuits can exist in only one of two states: on or off. Most computer electronics use voltage levels to indicate their present state. For example, a transistor with five volts would be considered "on", while a transistor with no voltage would be considered "off." Not all computer hardware uses voltage, however. CD-ROM's, for example, use microscopic dark spots on the surface of the disk to indicate "off," while the ordinary shiny surface is considered "on." Hard disks use magnetism, while computer memory uses electric charges stored in ...view middle of the document...

.. 997, 998, 999, 1000, 1001, 1002, .... You should get the picture at this point.

Another way to make this clear is to write decimal numbers in expanded notation. 365, for example, is equal to 3×100 + 6×10 + 5×1. 1032 is equal to 1×1000 + 0×100 + 3×10 + 2×1. By writing numbers in this form, the value of each column becomes clear.

The binary number system works in the exact same way as the decimal system, except that it contains only two digits, 0 and 1. Start counting in binary: 0, 1, Oops! There are no more binary digits. In order to keep counting, we need to add a second column worth twice the value of the column before. We continue counting again: 10, 11, Oops! It is time to add another column again. Counting further: 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111.... Watch the pattern of 1's and 0's. You will see that binary works the same way decimal does, but with fewer digits.

Binary uses two digits, so each column is worth twice the one before. This fact, coupled with expanded notation, can be used convert between from binary to decimal. In the binary system, the columns are worth 1, 2, 4, 8, 16, 32, 64, 128, 256, etc. To convert a number from binary to decimal, simply write it in expanded notation. For example, the binary number 101101 can be rewritten in expanded notation as 1×32 + 0×16 + 1×8 + 1×4 + 0×2 + 1×1. By simplifying this expression, you can see that the binary number 101101 is equal to the decimal number 45.

An easy way to convert back and forth from binary to decimal is to use Microsoft Windows Calculator. You can find this program in the Accessories menu of your Start Menu. To perform the conversion, you must first place the calculator in scientific mode by clicking on the View menu and selecting Scientific mode. Then, enter the decimal number you want to convert and click on the "Bin" check box to convert it into binary. To convert numbers from binary to decimal, click on the "Bin" check box to put the calculator in binary mode, enter the number, and click the "Dec" check box to put the calculator back in decimal mode.

How Hexadecimal Works

Binary is an effective number system for computers because it is easy to implement with digital electronics. It is inefficient for humans to use binary, however, because it requires so many digits to represent a number. The number 76, for example, takes only two digits to write in decimal, yet takes seven digits to write in binary (1001100). To overcome this limitation, the hexadecimal number system was developed. Hexadecimal is more compact than binary but is still based on the digital nature of computers.

Hexadecimal works in...

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