Understanding the Binary System
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Most of the information we receive from the world around us comes in quantities that vary continuously. For example, the changing frequencies making the music we hear, or the graduations of colour in an image. If values vary smoothly like this, we refer to them as analog data. These are impossible to represent using digital data, which describe only ‘fixed jumps’ in values.
In the past engineers used analog slides rules as computer devices. Modern computers are digital and need to convert or digitise analog data to process it.
Once we have digitised data, how does a computer process it?
Digital computers are built using switches that are either “on” or “off” and work very fast and accurately. Originally, computer switches were mechanical, the next generation used vacuum tubes, the next used transistors and today computers use integrated circuits each containing billions of transistors acting as switches.
Modern computers use low voltages or high voltages to represent a 0 or 1. We call these values binary digits or bits for short. Real-world analogue quantities need to be approximated using just these two symbols. Before we learn how this is done we need to make sure we understand the binary number system.
At first it seems quite difficult to work with binary numbers, but with a little practice it can be fun. You may have already learnt how binary numbers work. Many people think digital means only binary (base 2). This is not true. When we count using the 10 symbols 0-9 we are still counting digitally, although using base 10. It is possible to count in any base. Other important ones are octal (using eight symbols) and hexadecimal (16 symbols).
To count in binary (also called base 2) we follow exactly the same idea, but we use only two symbols. We could use any symbols, but traditionally we pick 0 and 1, which is appropriate because it reminds us of switches being on or off. Counting in base 2 works the same way as counting in base 10.
Using two columns we can write four different numbers: 00, 01, 10, 11
Task 7: Counting in binary
Column values in the decimal system are powers of 10. In the same way, column values in the binary system columns are powers of 2.
Complete the decimal table ( Table 2.2) for 1-15 by counting in 1s down the rows using the digits 0-9.
Complete the binary table (Table 2.3) for 1-15 by counting in 1s down the rows using only 0 or 1.
Important: In either table you can find a number’s value by adding the column values in the headings. So in binary: 1 0 1 means 1 x 4 + 0 x 2 = 1 x 1 (see table 2.3)
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