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©2009 by L. Lagerstrom
Number Systems
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The decimal (base 10) system
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Computers and 0s and 1s
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The binary (base 2) system
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Bits and bytes
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Representing negative and large numbers
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The IEEE double precision system
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Computational limitations
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Accuracy, precision (sig. digits), and errors
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Representing text
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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.