Transcript Numbers: Real, Imaginary, Complex, and beyond

Numbers: Real, Imaginary,
Complex, and Beyond ...
Roger House
Scientific Buzz Café
Coffee Catz
Sebastopol, CA
2010 February 3
Copyright © 2010 Roger House
Number Systems
 - natural numbers
 - integers
 - rational numbers
 - real numbers
 - complex numbers
 - quaternions
 - octonions
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The natural numbers 
 = { 1, 2, 3, ... }
 stands for natural.
• Also called whole numbers and
counting numbers.
• There are infinitely many of them.
• Seemingly not too interesting, but ...
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The natural numbers 
• There are three kinds of natural numbers:
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primes
composites
• 1 is called a unit; it is unique; all other
natural numbers can be generated by
simply adding up enough 1’s.
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Prime and composite
• A natural number is prime if it is not 1 and
can only be divided evenly by itself and 1:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
• A natural number is composite if it has
divisors other than itself and 1:
4 = 2•2, 6 = 2•3, 8 = 2•2•2, 9 = 3•3,
1001 = 7•11•13
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Fundamental Theorem of
Arithmetic
• Theorem. Every natural number other
than 1 can be represented uniquely as a
product of prime numbers.
30 = 2•3•5
819 = 3•3•7•13
1,048,576 = 220
• So the natural numbers can all be
generated from the primes.
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Goldbach’s Conjecture
• Conjecture: Every even integer greater than
2 can be written as the sum of two primes.
4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5,
10 = 3 + 7 = 5 + 5, 194 = 67 + 127
• Conjectured by Christian Goldbach (16901764) in 1742; no proof after 269 years.
• Known to be true for all even numbers < 1018.
 is not so simple as it may seem.
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The integers 
 = { ..., -2, -1, 0, 1, 2, ... }
 stands for Zahl, the German word for
number.
• The integers consist of the natural
numbers, their negatives, and zero.
• We find negative numbers useful for things
like the temperature and bank balances,
but how did they originate?
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x2 + x = 6
• We seek a number which when squared
and added to itself results in 6.
• Try x = 2:
22 + 2 = 4 + 2 = 6
• Try x = -3: (-3)2 + (-3) = 9 - 3 = 6
• So, in order to find all solutions (roots,
zeros) to the equation, we need a bigger
number system, a system which includes
negative numbers.
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Negative numbers
• Negative numbers were resisted as
something strange and peculiar, not really
numbers.
• The great philosopher and mathematician
René Descartes (1596-1650) referred to
solutions like x = -3 as “false or less than
anything”.
• But they were much too useful to resist for
long; now they are every bit as “true” and
“real” as positive numbers.
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The rational numbers 
 = { m / n | m, n  , n ≠ 0 }
 stands for quotient.
• The rational numbers include the integers
and all ratios of integers (but don’t ever
divide by zero or you won’t go to heaven).
• Rational numbers arose because whole
numbers won’t solve all problems; we
need fractions.
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2x + 3 = 4
• Solving this equation:
2x + 3 = 4
2x = 4 - 3
2x = 1
x=½
• To get a solution we need a number
system bigger than the integers.
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The number line
-3
-2
-1
0
0
¼
½
¾
1 9/8
0
1/8
¼
1
2
4/3
3
13/8 5/3
½
2
1
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The unit interval 
0
1/9
1/3
1
The unit interval is the part of the number
line between 0 and 1 inclusive.
•These rational numbers appear in :
½ ¼ 1/8 1/16 1/32 1/64 1/128 ...
1/3 1/9 1/27 1/81 1/243 1/729 ...
1/5 1/25 1/125 1/625 1/3125 ...
•How many rational numbers are in
?
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
 is dense
• Between every two distinct rational
numbers, there is another rational number.
• If q and r are rational numbers with q < r,
then (q + r) / 2 is a rational number exactly
half way between q and r.
• What this really means is that between
any two distinct rational numbers there are
infinitely many rational numbers.
 is said to be dense.
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x2 – 2 = 0
• Since the rational numbers are dense, at
first glance it seems that every point on
the number line must correspond to a
rational number; but is this true?
• Solve this equation:
x2 - 2 = 0
x2 = 2
x = √2 = 1.41421356237...
• Is √2 a point on the number line?
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√2 is on the number line
x2 = 12 + 12
x2 = 1 + 1
x2 = 2
x = √2
1
x
1
0
1
√2 = x
1
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Let’s be rational ...
• If the number line consists solely of
rational numbers, then since √2 is on the
number line, √2 must be a rational number.
• But, does the number line consist solely of
rational numbers?
• Might there be some other kind of number
lurking on the number line?
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Odd facts (even ones too)
• If m is an even integer, then m = 2q for
some integer q.
• If m is an odd integer, then m = 2q + 1 for
some integer q.
• An even integer times any integer is even.
• An odd integer times an odd integer is
odd.
• If m2 is even, then m is also even.
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Theorem. √2 is not rational
• Proof. Assume that √2 is rational.
• Then √2 = m / n for some integers m and
n, which we can choose to have no
common divisor.
• Square both sides: (√2)2 = (m / n)2.
• So 2 = m2 / n2.
• Multiply both sides by n2 to get 2n2 = m2.
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Theorem. √2 is not rational
• Thus m2 is even, so m is even, so m = 2q
for some integer q.
• So 2n2 = (2q)2, or 2n2 = 4q2, so n2 = 2q2.
• So n2 is even, so n is even, so n = 2p for
some integer p.
• Notice that n = 2p and m = 2q, so m and n
have a common divisor, 2.
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Theorem. √2 is not rational
• But this is a contradiction because we
began with m and n having no common
divisor.
• So our initial assumption that √2 is rational
is not true, so √2 is not rational. 
• In the old days: QED (quod erat demonstrandum = “that which was to be
demonstrated”).
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Irrational numbers
• If √2 is not rational, then what is it?
• It’s irrational.
• Non-mathematical usage: "not endowed
with reason, incoherent, marked by a lack
of accord with reason or sound judgment".
• Mathematical usage: not rational, i.e.,
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not a ratio of two integers.
Irrational numbers
• The ancient Greeks discovered that √2 is
irrational.
• There is a proof in Euclid’s Elements, but it
was known long before Euclid’s time
(about 300 B.C.).
• The discovery that not all numbers are
rational came as a great shock and
caused a certain amount of panic in the
world of Greek mathematics.
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Irrational numbers
• Are there other irrational numbers?
YES!
• Some examples:
√n for any natural number n which is not a
perfect square (√3, √5, √6, √7, ...)
the cube root of any natural number n that is
not a perfect cube.
, the ratio of the circumference of a circle to
its diameter.
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The real numbers 
• We combine the sets of rational numbers and
irrational numbers to get the real numbers, 
 =   { the irrationals }
 stands for real (surprise, surprise).
• But why the term real?
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The real numbers 
• A bit of hand-waving is going on here.
• To properly define the real numbers
requires limits.
• The reals are quite subtle.
• They weren’t well-understood or
properly defined until 1880 or so.
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The real numbers 
• Now the number line is complete.
• Henceforth we will refer to the
number line as the real line.
• Note we can solve yet more
equations, for example: x2 = 2.
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1001 things to do with numbers
number
system
add

yes
yes

yes
yes
yes

yes
yes
yes
yes

yes
yes
yes
yes
multiply subtract divide
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take
roots
yes
, , , and 
irrational numbers

fractions

negative whole
numbers and zero
positive whole
numbers


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Break time
• Coming up:
Complex numbers
Hypercomplex numbers
• To be continued ...
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Number system properties
Ordered sets
Commutative laws
Associative laws
Distributive law
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The real numbers are ordered
Let a and b be any two real numbers;
then exactly one of these is true:
a<b
a=b
a>b
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All subsets of reals are ordered
 is ordered
 is ordered
 is ordered
 is ordered
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Commutative Laws
• Addition is commutative:
a+b=b+a
3+4=4+3=7
• Multiplication is commutative:
a•b=b•a
3 • 4 = 4 • 3 = 12
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Associative Laws
• Addition is associative:
(a + b) + c = a + (b + c)
(3 + 5) + 2 = 3 + (5 + 2) = 10
• Multiplication is associative:
(a • b) • c = a • (b • c)
(3 • 5) • 2 = 3 • (5 • 2) = 30
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Distributive Law
• Multiplication distributes over addition:
a • (b + c) = a • b + a • c
3 • (5 + 2) = 3 • 5 + 3 • 2 = 21
3 • (5 + 2) = 3 • 7 = 21
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It’s the Law!
, , , and  are ordered, and in each of
them addition and multiplication are
commutative and associative, and
multiplication distributes over addition.
• What’s the big deal? Why the laws?
• Because there exist number systems in
which these laws don’t hold (gasp!)
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x2 – 1 = 0
• Consider the equation
x2 – 1 = 0
x2 = 1
x = √1
x = 1 and x = -1
• Check: 12 = 1•1 = 1 and (-1)2 = (-1)(-1) =
1.
• Remember: negative times negative is
positive.
• So the roots of x2 – 1 = 0 are +139and -1.
x2 + 1 = 0
• Now consider the equation
x2 + 1 = 0
x2 = -1
x = √(-1)
x = ???
• We seek x such that x•x = -1, but
negative•negative = positive
positive•positive = positive
0•0 = 0
• So there cannot possibly be such
a
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number among the real numbers.
They’re not real!
• Can’t we just ignore √(-1), act like we
never saw it?
• This won’t work; square roots of negative
numbers start popping up everywhere.
• In the 16th century they appeared in
solutions of the cubic equation.
• We’ll look at an example (simpler than a
cubic) but first some notation:
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i = √(-1)
• It’s a bit cumbersome to write √(-1), so the
great Leonhard Euler (1707-1783) began
using i to stand for the square root of -1.
• This notation is used universally among
mathematicians to this day.
• BUT, electrical engineers use j (because
they use i for current).
• This leads to even more confusion later.
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x(4 – x) = 5
• Problem: Split 4 into two parts whose sum is 4
and whose product is 5: x(4 – x) = 5.
• Using the quadratic formula, we find that the two
numbers are 2+i and 2-i.
• Adding them together: (2+i) + (2-i) = 4.
• Multiplying them together:
(2+i)(2-i) = (2+i)2 + (2+i)(-i)
= 4 + 2i - 2i - i2
= 4 - i2 = 4 - (-1) = 4 + 1
=5
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Somehow it works ...
• It looks like everything works fine if we
follow the usual rules of arithmetic, but
replace i2 by -1.
• (It’s okay if you feel uneasy about this ...)
• Paraphrasing Gerolamo Cardano (15011576): “Putting aside the mental tortures
involved, multiply 2+i by 2-i making 4-(-1)
= 5 ... This is truly sophisticated ...”
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Are they real?
• John Napier (1550-1617) called these
strange numbers “ghosts” of real numbers.
• Around 1637 Descartes first used the term
imaginary -- in a derogatory sense.
• The term real was used to distinguish the
“usual” numbers from imaginary numbers.
• But really, aren’t all numbers imaginary?
• Or, just imagine, maybe they’re all real?
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“It’s so complicated ...” (The Rolling Stones)
• Doing arithmetic with numbers involving i
always results in a number of this form:
b + ci, where b and c are real
• Such numbers are called complex numbers,
with real part b and imaginary part c.
• If b is zero the number is just ci, a real
number c times i (e.g., -2i, 10i, i); such
numbers are called pure imaginary numbers
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The complex numbers 
 = { b + ci | b, c  , i = √(-1) }
 stands for complex.
• A complex number can be thought of as a
pair of real numbers, so  is 2-dimensional, as was noted by:
• Caspar Wessel (1745-1818)
• Jean Robert Argand (1768-1822)
• Carl Friedrich Gauss (1777-1764)
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The complex plane
2i
i

2+i
i
-2
-1
0
1
2
 2-i
-i
-2i
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 is algebraically closed
• The complex plane looks a lot like the usual 2dimensional Cartesian coordinate system.
• It’s essentially the same, but with an extra
property: The points in the plane can be
multiplied together, e.g., (2+i)(2-i).
• In , every polynomial equation in one unknown
has a solution
z4 - z3 + 4z2 + iz - 1 = 0.
• The end of a theme: There is no longer any
need to look for a bigger number system in
which to find solutions of equations.
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Laws, but no order
• In  we can add, subtract, multiply, divide, and
take roots to our heart’s content, just as in ,
simply replacing i2 by -1 whenever it appears.
• The commutative, associative, and distributive
laws all hold.
• BUT, we do give up something:  is not
ordered.
• We cannot say that c > d or c < d for c and d
complex numbers.
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A cautionary note
• Don’t get hung up on terminology: real,
imaginary, and complex as used in
mathematics are technical terms with
precise definitions.
• Do not confuse these mathematical terms
with the ordinary day-to-day words found
in the dictionary.
• In some sense they aren’t even related,
except, perhaps, historically.
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Three is better than two?
• Complex numbers and the complex plane
proved to be extremely powerful tools for
creating precise mathematical models in
two dimensions.
• A natural question: What about a number
system for three dimensions?
• William Rowan Hamilton (1805–1865)
spent 15 years looking for such a system.
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A more imaginative system...
• Basic idea: Two imaginary units, i and j,
with i2 = -1 and j2 = -1, and each number of
the form
b + ci + dj with b, c, and d in 
• Note that this looks a lot like complex
numbers, but 3-dimensional.
• Of course, we want all the usual laws of
arithmetic to hold.
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Four is better than three
• Unfortunately, no such number system exists.
• BUT, if we’re willing to throw in a third imaginary
unit, k with k2 = -1, and consider numbers of the
form
b + ci + dj + ak with b, c, d, and a in 
then we get a 4-dimensional number system
which works.
• This was Hamilton's flash of insight after 15
years.
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The quaternions 
 = { b + ci + dj + ak | b, c, d, a   } with
i2 = j2 = k2 = ijk = -1
 stands for Hamilton
Hamilton called the new numbers quaternions
which means “a set of four persons or items”.

•Just as a complex number can be thought of as a
pair of real numbers, a quaternion can be thought
of as a quadruple of real numbers.
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The flash of insight
“... on the 16th of October, 1843, as I was
walking with Lady Hamilton to Dublin, and
came up to Brougham Bridge
... I ... then and there felt the galvanic circuit
of thought closed, and the sparks which fell
from it
were the fundamental equations between i,
j, k exactly such as I have used them ever
since.”
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Mathematical vandalism
“Nor could I resist the impulse unphilosophical as it may have been to cut with a knife on a stone of
Brougham Bridge the fundamental
formula with the symbols i, j, k:
i2 = j2 = k2 = ijk = -1.”
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Breaking the law...
• From the relationships
i2 = j2 = k2 = ijk = -1
it is fairly easy to deduce that
ij = -ji
• This is a bit shocking and perhaps scary:
Multiplication of quaternions is non-commutative.
• Actually this was a break-through in mathematics, a step towards more general structures,
leading to the vast abstractions of modern
mathematics.
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The war you never heard of
• Quaternions are 4-dimenional, not 3-dimensional, but they can be used to create precise
mathematical models in three dimensions.
• Ignoring the real component, ci + dj + ak
looks very much like a 3-dimensional vector.
• The Great Quaternion-Vector War.
• Quaternions lost.
• BUT, quaternions are back!
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If you ever get to Dublin...
• On October 16 each year, Hamilton’s
fateful walk across Brougham Bridge is reenacted by various mathematicians.
• (Somehow this is not nearly as popular as
the Bloomsday walk in Dublin on June 16.)
• Brougham, Broom, or Broome?
• Hamilton’s scratchings on the bridge are
long gone, but there does exist a plaque:
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Brougham Bridge today
• “Here as he walked by on the 16th of
October 1843 Sir William Rowan Hamilton
in a flash of genius discovered the
fundamental formula for quaternion
multiplication i² = j² = k² = ijk = −1 & cut it
on a stone of this bridge.”
• Look up “Broom Bridge” in Wikipedia and
follow links.
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Eight is better than four(?)
• Mathematicians just can’t stop: Generalize
• After , , and , what might there be?
• Number systems of dimensions 5? 6? 7?
• How about 8? Just cook up 4 more
imaginary units, E, I, J, and K.
• John T. Graves (1806-1870) 1843 1848
• Arthur Cayley (1821–1895)
1845
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The octonions 
 = { b + ci + dj + ak + BE + CI +DJ + AK | b, c, d,
a, B, C, D, A   } with i, j, k, E, I, J, K as
imaginary units.
 stands for octonion, a term probably based on
quaternion but meaning “a set of eight items”.
• To rigorously define , a 49-element
multiplication table must be presented.
• We leave this as an exercise for the
perspicacious student.
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Scofflaw
• Multiplication of octonions is nonassociative, e.g.,
(ij)E ≠ i(jE)
• But, the set of octonions is a real division
algebra (like , , and ) in which we can
do arithmetic.
• Note that the octonions are an 8dimensional number system.
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The end of the line
• Speaking somewhat loosely:
, , and  are the only real finitedimensional associative division algebras.
 is the only non-associative one.
• Ferdinand Georg Frobenius (1849-1917)
• Adolf Hurwitz (1859-1919)
• Heinz Hopf (1894–1971)
• Max August Zorn (1906-1993)
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, , , and 
 + { E, I, J, K }

 + { j, k }

+{i}

real numbers 

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Number systems: A summary
John Baez says:
The real numbers are the dependable breadwinner of the
family, the complete ordered field we all rely on.
The complex numbers are a slightly flashier but still
respectable younger brother: not ordered, but
algebraically complete.
The quaternions, being noncommutative, are the eccentric
cousin who is shunned at important family gatherings.
But the octonions are the crazy old uncle nobody lets out of
the attic: they are nonassociative.
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Thats all, folks!
 - natural numbers
 - integers
 - rational numbers
 - real numbers
 - complex numbers
 - quaternions
 - octonions
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