INTG 002: Zero/nothing
D. DeTurck
University of Pennsylvania
Zero and place value
• It’s remarkable that it took so long to realize the utility of having an actual symbol to represent zero.
• Place value notation was already standard in some places
(Babylonians, etc) but they simply left a column blank if there wasn’t anything there, causing confusion. It took nearly 15 centuries to begin using some indication of a zero in a column.
Why place value?
• “Now thou are an O without a figure. I am better than thou
are now. I am a fool, thou are nothing.” (Act I, Scene iv).
• The need for place value is apparent if you try do arithmetic with Roman numerals:
• What is MCCCLXXIV +MMCMXCVIII?
Be careful with zero (and nothing)
• Assume a=b Then: ab =b2 ab−a2 =b2−a2 a(b−a) = (b+a)(b−a) a=b+a a= 2a 1 = 2Be careful with zero (and nothing)
• Assume a=b Then: ab =b2 ab−a2 =b2−a2 a(b−a) = (b+a)(b−a) a=b+a a= 2a 1 = 2Be careful with zero (and nothing)
• Assume a=b Then: ab =b2 ab−a2 =b2−a2 a(b−a) = (b+a)(b−a) a=b+a a= 2a 1 = 2Be careful with zero (and nothing)
• Assume a=b Then: ab =b2 ab−a2 =b2−a2 a(b−a) = (b+a)(b−a) a=b+a a= 2a 1 = 2Be careful with zero (and nothing)
• Assume a=b Then: ab =b2 ab−a2 =b2−a2 a(b−a) = (b+a)(b−a) a=b+a a= 2a 1 = 2Be careful with zero (and nothing)
• Assume a=b Then: ab =b2 ab−a2 =b2−a2 a(b−a) = (b+a)(b−a) a=b+a a= 2a 1 = 2More trouble with nothing
• A dog has nine legs
• Proof: No dog has five legs, and any dog has four legs more than no dog, hence a dog has nine legs.
• A ham sandwich is better than complete happiness.
• Proof: Nothing is better than complete happiness, and a ham sandwich is certainly better than nothing.
More trouble with nothing
• A dog has nine legs
• Proof: No dog has five legs, and any dog has four legs more than no dog, hence a dog has nine legs.
• A ham sandwich is better than complete happiness.
• Proof: Nothing is better than complete happiness, and a ham sandwich is certainly better than nothing.
More trouble with nothing
• A dog has nine legs
• Proof: No dog has five legs, and any dog has four legs more than no dog, hence a dog has nine legs.
• A ham sandwich is better than complete happiness.
• Proof: Nothing is better than complete happiness, and a ham sandwich is certainly better than nothing.
More trouble with nothing
• A dog has nine legs
• Proof: No dog has five legs, and any dog has four legs more than no dog, hence a dog has nine legs.
• A ham sandwich is better than complete happiness.
• Proof: Nothing is better than complete happiness, and a ham sandwich is certainly better than nothing.
The rector’s stepson
• 1612: According to historian of science Richard Westfall: Barnabas Smith – rector of North Witham, “entered a grandly conceived set of theological headings” in a huge notebook, and “under these headings a few pertinent passages culled from his reading. If these notes represent the sum total of his lifetime assault on his library, it is not surprising that he left no reputation for learning. Such an expanse of blank paper was not to be discarded in the seventeenth century.”
• The notebook was seized by Smith’s stepson, Isaac Newton,
who called it his “Waste Book”, and it records the birth of the calculus and of mechanics.
Palimpsests and infinitesimals
• In medieval times, paper (or parchment) was even scarcer.
Old manuscripts were washed clean by monks so they could record religious documents. Imperfect erasure would leave traces of the original, called a palimpsest.
• 1906: J.L. Heiberg tracked down a mathematical palimpsest
in Constantinople, discovered works of Archimedes, including the first known copy of “The Method”.
• The Method gives some clues to Archimedes’ mathematical intuition. E.g., technique for slicing solids into infinitely many pieces of infinitesimal thickness and hanging these on the arms of an imaginary balance, where their sum could be compared with a known object. For instahce, he calculated the volume of the sphere in this way.
• The idea of infinitesimals plagued mathematicians for
centuries after Archimedes, right up until the mid-nineteenth century, when they were replaced by the notion of a limit.
Zeno of Elea
• But back to the Greeks: Zeno’s paradoxes (Zeno was a
follower of Parmenides of Elea) – which raised further challenges (beyond irrationality) to the Pythagorean ideal of number as the basis of everything.
• Four paradoxes about the nature of space and time.
1 “Dichotomy paradox”: attacks infinite divisibility of space. Before getting somewhere, an object must travel half as far, etc.
2 Achilles and the tortoise – for Achilles to catch up, he has to get to where the tortoise was, but by then the tortoise has moved on.
3 The “flying arrow is at rest” attacks discreteness
4 The “stadium” paradox attacks the idea of a maximum speed (hence discreteness of space and time) – presages special relativity!.
• These paradoxes pushed the Greeks even further away from
Zeno of Elea
• But back to the Greeks: Zeno’s paradoxes (Zeno was a
follower of Parmenides of Elea) – which raised further challenges (beyond irrationality) to the Pythagorean ideal of number as the basis of everything.
• Four paradoxes about the nature of space and time.
1 “Dichotomy paradox”: attacks infinite divisibility of space. Before getting somewhere, an object must travel half as far, etc.
2 Achilles and the tortoise – for Achilles to catch up, he has to get to where the tortoise was, but by then the tortoise has moved on.
3 The “flying arrow is at rest” attacks discreteness
4 The “stadium” paradox attacks the idea of a maximum speed (hence discreteness of space and time) – presages special relativity!.
• These paradoxes pushed the Greeks even further away from
Infinitesimals?
• Is the line composed of points strung together in order?
• What is the “first point to the right of 0”?
• Not 0.01, or 0.001, or 0.0001, etc.. (Achilles?)
• One modern view: there is no “next number”, so there are no
“ultimate atoms” or indivisibles. So a line is not a set of points strung together in order — but any particular position on the lineis a point. Hmm. . . .
• If you wanted to posit thatx is the smallest number greater
than zero (an infinitesimalnumber), then you have to answer
Infinitesimals?
• Is the line composed of points strung together in order?
• What is the “first point to the right of 0”?
• Not 0.01, or 0.001, or 0.0001, etc.. (Achilles?)
• One modern view: there is no “next number”, so there are no
“ultimate atoms” or indivisibles. So a line is not a set of points strung together in order — but any particular position on the lineis a point. Hmm. . . .
• If you wanted to posit thatx is the smallest number greater
than zero (an infinitesimalnumber), then you have to answer
Infinitesimals?
• Is the line composed of points strung together in order?
• What is the “first point to the right of 0”?
• Not 0.01, or 0.001, or 0.0001, etc.. (Achilles?)
• One modern view: there is no “next number”, so there are no
“ultimate atoms” or indivisibles. So a line is not a set of points strung together in order — but any particular position on the lineis a point. Hmm. . . .
• If you wanted to posit thatx is the smallest number greater
than zero (an infinitesimalnumber), then you have to answer
Useful infinitesimals
• But infinitesimals are useful – Nicholas of Cusa: area of a circle by infinitesimals:
Useful infinitesimals
• But infinitesimals are useful – Nicholas of Cusa: area of a circle by infinitesimals:
Useful infinitesimals
• But infinitesimals are useful – Nicholas of Cusa: area of a circle by infinitesimals:
Conclusion
• These pictures indicate that
A= (1 2C)r (so if C = 2πr then A= (πr)r =πr2).
• But is it rigorous?
Tangents
• Tangents: How to draw the tangent to a curve. Appolonius
solved this for conic sections by purely geometric means. Newton and Leibniz used functions and coordinate geometry.
• Parabola: y =x2 To find the slope of the tangent line atx,
pick a number near x, say x+o. Then two nearby points on
the parabola have coordinates (x,x2) and ((x+o),(x+o)2). The slope of the line connecting these two points, or the
average rate of change between x and x+o is
(x+o)2−x2
(x+o)−x =
2ox+o2
o = 2x+o
Ifo approaches zero, then the slope approaches 2x+ 0 = 2x. This is the slope of the tangent, or, as Newton put it, the
fluxion of the fluent x2. This agrees with Apollonius
• Going beyond Apollonius: the fluxion of the fluent x3 is 3x2, etc..
Bishop Berkeley
• Was this rigorous? Bishop George Berkeley, 1734: The
Analyst, Or a Discourse Addressed to an Infidel
Mathematician. Wherein It is examined whether the Object, Principles, and Inference of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith. ‘First cast out the beam in thine own Eye; and then shalt thou see clearly to cast out the mote out of thy brother’s Eye.’
• Berkeley’s objection: Either o is not exactly zero, in which case the answers are (slightly) wrong, or else it is zero, in which case you can’t divide by it so the calculation doesn’t make sense. The use of the little o rather than the big 0 isn’t
Ghosts of departed quantities
• Newton defined a fluxion as the “ultimate ratio of evanescent
increments’.
• Berkeley’s response: “And what are these fluxions? The
velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?”
• Berkeley objected to the calculation of a fluxion as a ratio between two quantities (2ox +o2 ando) which both vanish. The ratio 0/0 makes no sense.
• Although Berkeley was grinding an axe about the difference
(or lack thereof) between mathematical proof and religious belief, he had the analysts dead to rights.
Who won?
• On the other hand, Newton and his followers knew that what
they were doing made sense on physical grounds — Newton thought of o as a variable, which could approach 0 as closely as we please without actually vanishing. Newton was thinking
about the process where o shrinks to zero, but Berkeley
wanted Newton to fix the value of o at the instant before it disappeared forever. Newton knew that Achilles will catch the tortoise because he is running faster; Berkeley wanted Newton to isolate the instant immediately before the tortoise is overtaken, even though it is “clear” that there is no such thing.
Toward a rigorous definition
• Newton kept trying to get it right. In the third edition of his
Principia Matematica he wrote: “Ultimate ratios in which quantities vanish are not, strictly speaking, ratios of ultimate quantities, but limits to which the ratios of these quantities, without limit, approach, and which, though they can come nearer than any given difference whatever, they can neither pass over nor attain before the quantities have diminished indefinitely.”
• In other words, to find the limit which the ratio 2ox−oo 2
approaches aso diminishes towards 0, you donotset o = 0 to
get the ratio 0/0. Rather, you keep o non-zero, simplify to get 2x1+0 and then observe that aso approaches0, this ratio
approaches 21x.
• Newton knew what he was doing but didn’t have a precise
Weierstrass, epsilons and deltas
• Things improved only slightly until the 19th century. Cauchy
in his Cours d’Analyse(1821) came the closest, but finally Karl Weierstrass around 1850 settled it all by taking the phrase “as near as we please” seriously.
• A function f(x) approaches a limit Lasx approaches a value
a if, given any positive numberε, the difference|f(x)−L|is less than εwhenever|x−a|is less than some numberδ
depending on ε.
• It’s like a game: You tell me how close you wantf(x) to be to
L, then I’ll tell you how close x has to be to a. So Player Epsilon says how near hepleases, then Player Delta is free to seek a difference that works. If Delta always has a winning strategy, then f(x) tends to the limitL.
Weierstrass, epsilons and deltas
• Things improved only slightly until the 19th century. Cauchy
in his Cours d’Analyse(1821) came the closest, but finally Karl Weierstrass around 1850 settled it all by taking the phrase “as near as we please” seriously.
• A function f(x) approaches a limit Lasx approaches a value
a if, given any positive numberε, the difference|f(x)−L|is less than εwhenever|x−a|is less than some numberδ
depending on ε.
• It’s like a game: You tell me how close you wantf(x) to be to
L, then I’ll tell you how close x has to be to a. So Player Epsilon says how near hepleases, then Player Delta is free to seek a difference that works. If Delta always has a winning strategy, then f(x) tends to the limitL.
The birth of analysis
• This might seem cumbersome, but like the Greek method of
exhaustion, a competent professional gets good at it.
• In this definition, the physical ideas of motion (of x flowing toward a) are replaced by a set of static events, for each choice of ε.
• Weierstrass’s definition of limit freed calculus from