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Introduction to Computer Science

Lecture 2.1.

Symbols

Number Systems

Our Number system

• From the time our species began counting, we’ve used our fingers to help.

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• Most civilizations have consequently based their number systems around ten.

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• There is nothing inherently special about our number system apart from its relationship to the physiology of the human hand

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• Had our species developed with eight or twelve fingers, our ways of counting would be different

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• It is no coincidence that the word digit can refer to fingers or toes as well as to numbers

Babylonian Number System

• The only significant exceptions are a few number systems built around five, or 20 or 60, which are all closely related to 10

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• The Babylonian number system based on 60 persists in our reckoning of the time in seconds and minutes

Indo-Arabic Number System

• The number system we used today is known as the Indo-Arabic

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• It is of Indian Origin but was brought to Europe by Arab mathematicians

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• Of particular renown is the Persian mathematician Muhammad ibn Musa al-Khwarzimi

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• The word Algorithm comes from his name

Indo-Arabic Number System

The Indo-Arabic number system differs from previous numbers in three ways:

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1. The Indo-Arabic number system is positional

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• A particular digit represents a different quantity depending on its position

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• The position is more significant than the digits themselves

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• 100 and 99 are very close to each other even though they don’t share a number

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• 100 and 10 are much further apart although they differ from each other by just one digit

Indo-Arabic Number System

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2.

Virtually all earlier number systems had something that the Indian-Arabic system does not

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• Special symbol for ten

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3.

Virtually all earlier number systems lacked something that the Indo-Arabic system has: zero

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• Lacking a zero made mathematical operations very difficult

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• Particularly multiplication and division

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Indo-Arabic Number System

• The whole structure of the Indo-Arabic number system is revealed in the way we pronounce them

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• Take 4825, for instance, we say

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“four thousand, eight hundred and twenty five”

Indo-Arabic Number System

• Since each position corresponds to a power of ten, a special symbol for ten is not required

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• Because ten is represented by setting the 1 in a different position and using 0 as a placeholder

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Elegance of Positional Systems

• What’s also really nice is that fractional quantities shown as digits to the right of a decimal point follow this same pattern.

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• The number 42705.684 is:

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4 x 10,000 +

2 x 1000 +

7 x 100 +

0 x 10 +

5 x 1 +

6 ÷ 10 +

8 ÷ 100 +

4 ÷ 1000 +

• Notice that the last three lines use division rather than multiplication.

Elegance of Positional Systems

• This number 42705.684 can also be written as shown below on the left

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• Using powers of 10, the number can be written as shown below on the right

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• Note how the exponents go down to zero then become negative

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Addition in a Positional System

• When you add decimal numbers of any length, you follow a procedure that breaks down the problem into steps.

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• Each step involves nothing more complicated than adding pairs of single-digit numbers

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• Addition table for each digit in decimal system is given on the right

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• Find the two digits you wish to add in the top row and top column and follow down and across to get the sum

Alternative Number Systems

• What if human beings had only four fingers on each hand?

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• Instead of basing our number system on ten, we would have probably had a number system based on eight

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• Such a number system is called octal or base eight

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• Just like in our decimal system (aka base 10) number system, we don’t have a symbol for ten, we would probably not have a symbol for eight or nine

Multiplication in a Positional System

• Similarly, when you need to multiply two decimal numbers, you follow a somewhat more complicated procedure but still one that breaks down the problem so that you need do nothing more complex than adding or multiplying single-digit decimal numbers

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• Multiplication table for digits in decimal points is given on the right