Numbers: Real, Imaginary, Complex, and Beyond ...

Roger HouseScientific Buzz Café

Coffee CatzSebastopol, CA2010 February 3

Copyright © 2010 Roger House

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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•131,048,576 = 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 (1690-1764) 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 1 2 3

0 ¼ ½ ¾ 1 9/8 4/3 13/8 5/3 2

0 1/8 ¼ ½ 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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1

0 1 √2 = x

√2 is on the number line

1 1x

x2 = 12 + 12

x2 = 1 + 1

x2 = 2

x = √2

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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 demon-strandum = “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., not a ratio of two integers.

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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 multiply subtract divide take

roots

yes yes

yes yes yes

yes yes yes yes

yes yes yes yes yes

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, , , 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 • a3 • 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 +1 and -1.

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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 number

among the real numbers.

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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 (1501-1576): “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-dimension-al, 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

-2 -1 0 1 2

i

i

-i

2i

-2i

2+i

2-i

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is algebraically closed

• The complex plane looks a lot like the usual 2-dimensional 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 mathe-matics, 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-dimen-sional, 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 re-enacted 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 non-associative, 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 8-dimensional number system.

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The end of the line

• Speaking somewhat loosely:

, , and are the only real finite-dimensional 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