9p. Number systems _printable_

9p. Number systems _printable_ - Number Systems The...

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1 ©2009 by L. Lagerstrom Number Systems The decimal (base 10) system Computers and 0s and 1s The binary (base 2) system Bits and bytes Representing negative and large numbers The IEEE double precision system Computational limitations Accuracy, precision (sig. digits), and errors Representing text Note: As a supplement to this presentation, see the associated video clips (Number systems 1 and Number systems 2). ©2009 by L. Lagerstrom The Decimal (Base-10) System We are so used to thinking about and using numbers like 5,283 that we often forget exactly what it means. Sometime way back in elementary school we learned that each digit has a place value. The number 5,283 literally means "5 thousands, 2 hundreds, 8 tens, and 3 ones." In other words, moving from right to left among the digits, each place value represents an increasing power of 10. We can represent this as follows: ____ ____ ____ ____ ____ ____ ____ 10 6 10 5 10 4 10 3 10 2 10 1 10 0 And so on for increasing powers of 10 to the left. (We can also have negative powers of 10 going to the right, i.e., 10 -1 (tenths), 10 -2 (hundredths), etc.) Since we are using powers of 10, we have 10 numerals to work with in filling the places (i.e., 0 through 9). This system is called the decimal or base-10 number system. ©2009 by L. Lagerstrom Computers and 0s and 1s When modern computers were first being developed in the 1930s, 40s, and 50s, the decision was made to design them around a binary, or base-2, number system. Whereas the decimal system has 10 numerals, the binary system only has 2, the numerals 0 and 1. This might seem like an impossible restriction--how can we have a computer do regular math operations with only 0s and 1s? We'll see in a minute how this is overcome, but first why in the world make this restriction in the first place? The reason is that the greatest engineering system in existence in the 1930s and 40s was the phone system. And the phone system was based on electromagnetic switches that could be turned on and off to make or break connections and route calls through the system. Scientists and engineers realized that the easiest way to build a computing machine was to base it on this well-understood technology. ©2009 by L. Lagerstrom Computers and 0s and 1s, cont. Since a simple switch only has two states, off or on (or open or closed), these states can represent the numbers 0 and 1. A binary number such as 101110, therefore, can be represented by a series of six switches that are on, off, on, on, on, off, respectively (i.e., 1 for on, 0 for off). It turns out that once we can represent binary numbers using switches, we can also design circuits to manipulate the numbers, do arithmetic, etc. Early experimental computers used electromagnetic switches, but these mechanical switches were soon replaced by vacuum tubes and then transistors, which could be made much smaller and faster.
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