Slide 1
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The Role of Mathematics
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Slide 2
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Roger Penrose
‘The Road to Reality’
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Slide 3
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History and Background
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A case can be made that mathematics pre-dates human
employment and articulation of mathematics.
- animals exhibit “number sense”.
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Mathematics appears to develop from a strictly pragmatic
point of view.
- tally sticks for counting
- geometry for construction
- calculating taxes, dates, navigation
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Slide 4
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Number Systems
In general ‘modern’ number systems are fairly easy to
define. The base or radix of a number system defines
the number of digits which can be used to construct
numbers.
Consider the decimal (Radix = 10, digits 0  9) system
as an example. All numbers are constructed from
the digits 0  9 and successive powers of 10.
eg. 364 = 3 x 102 + 6 x 101 + 4 x 100 = 300 + 60 + 4.
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Slide 5
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To write the same number in the binary system
(Radix = 2) we may only employ 2 digits (0 and 1)
and must use powers of 2 rather than powers of 10,
therefore 364dec = 101101100bin .
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101101100 = 1 x 28 + 0 x 27 + 1 x 26 + 1 x 25
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+0x
24
+1x
23
+1x
22
+0x
21
+0x
20
= 256 + 0 + 64 + 32 + 0 + 8 + 4 + 0 + 0 = 364
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Slide 6
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For comparison consider 364 in the octal (Radix = 8)
system.
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364dec= 554oct = 5 x 82 + 5 x 81 + 4 x 80
= 320 + 40 + 4 = 364dec
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Example: Convert 210dec to hexadecimal (Radix 16).
(note: digits in hex  0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F)
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Slide 7
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For comparison consider 364 in the octal (Radix = 8)
system.
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364dec= 554oct = 5 x 82 + 5 x 81 + 4 x 80
= 320 + 40 + 4 = 364dec
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Example: Convert 210dec to hexadecimal (Radix 16).
(note: digits in hex  0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F)
210dec = 13 x 161 + 2 x 160
= 208 + 2 = 210 = D2hex
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Slide 8
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Babylonian Base 60 Number System
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Slide 9
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Why choose a base 60 system?
No exact answer but here are some thoughts:
60 has divisors of 1,2,3,4,5
System of measures allowing for 1/3’s
Equilateral triangle (60o angles)
360 days in the year
Counting to 60 on your fingers
Mixing of cultures (10,6 / 5,12)
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Slide 10
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Problems for the Babylonians:
No zero
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Ambiguity in displaying numbers
Counting Numbers
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Slide 11
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The Role of Zero
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The Greeks – “How can nothing be
something?”
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Slide 12
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Kinds of Numbers
Whole Numbers–{ 1,2,3,4,5….}
Integers–{…-4,-3,-2,-1,0,1,2,3,4,...}
Rational Numbers–{ratio of 2 integers}
Irrational Numbers{non-terminating, non-repeating decimal representation}
Transcendental Numbers{Irrational but non-algebraic}
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Slide 13
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Real Numbers - {Rational, Irrational}
- Numbers that can be defined by an
infinite decimal representation.
Do the real numbers have physical significance?
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Slide 14
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Complex Numbers (Imaginary Numbers)
z = a +ib where i = √-1
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The Complex Plane
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i
Real Number Line
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