Download Week 10: The Work of Modern Mathematics

Week 10: The Work of Modern Mathematics
Mathematicians of recent years have also favored abstraction, which,
though related to generalization, is a somewhat different tack. In the latter
part of the nineteenth century mathematicians observed that many classes
of objects… possess the same basic properties.
Morris Kline
Clock Arithmetic
Consider the way we think about time on a 12-hour clock. For example, 7 hours after 10
o’clock, it is 5 o’clock. So in the arithmetic of a 12-hour clock, we might say that 10 + 7 = 5.
Similarly, we could say that 8 + 3 ⋅ 9 = 11. In other words, if it is 8 o’clock, then after three
9-hour periods of time, it will be 11 o’clock. Another way to look at this is that 3 ⋅ 9 = 3.
It is also interesting to think about subtraction in the arithmetic of a 12-hour clock. For
example, 8 hours before 3 o’clock, it is 7 o’clock. So we might say that 3 − 8 = 7.
We begin with examples of “clock arithmetic” on a 12-hour clock, because that is a familiar
setting to most people. However, we can imagine hypothetical clocks with other numbers of
hours on them.
Definition: Let 𝑛 be a natural number greater than 1. In the arithmetic of an 𝑛-hour clock,
there are only 𝑛 numbers: the natural numbers from 0 up to 𝑛 − 1. When these numbers are
added or multiplied, the result is obtained by taking the “remainder” of the result in everyday
arithmetic when divided by 𝑛.
To illustrate how this definition relates to the previous examples, consider 10 + 7. The result
in everyday arithmetic would be 17, but when this is divided by 12, the remainder is 5, which
is the proper result in 12-hour arithmetic. Likewise, consider 8 + 3 ⋅ 9. In everyday
arithmetic, 3 ⋅ 9 is 27, whose remainder (when divided by 12) is 3. So 8 + 3 ⋅ 9 should be the
same as 8 + 3, which is 11.
Now, what about 3 − 8 in 12-hour arithmetic? Following the usual definition of subtraction,
3 − 8 must be a number that, when added to 8, gives 3 as a result. In 12-hour arithmetic, the
only “numbers” are 0 through 11, so 3 − 8 must be 7 because 7 + 8 is 3.
Notation: To avoid confusion, when working in the arithmetic of an 𝑛-hour clock, we will
use the symbol “=𝑛 ” in place of the everyday “=” sign. So, for example, instead of writing
8 + 3 ⋅ 9 = 11, we will write 8 + 3 ⋅ 9 =12 11 (while, on the other hand, 8 + 3 ⋅ 9 = 35).
The idea of clock arithmetic, as silly as it may seem, has some extremely important real-life
applications related to internet security. But for the purposes of this course, clock arithmetic
provides an illustration of a theme that runs throughout modern mathematics.
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Clock arithmetic obeys virtually all of the most important properties of everyday arithmetic
with whole numbers. In the arithmetic of an 𝑛-hour clock:








The sum of two numbers is always a number between 0 and 𝑛 − 1. (The technical way
to say this is that addition is “closed” in clock arithmetic.)
The difference of two numbers is always a number between 0 and 𝑛 − 1. (In other
words, subtraction is “closed” in clock arithmetic.)
The product of two numbers is always a number between 0 and 𝑛 − 1. (Multiplication
is “closed” in clock arithmetic.)
The order in which two numbers are added does not matter. In other words, 𝑥 + 𝑦 is
always the same as 𝑦 + 𝑥. (The technical way to say this is that addition is
“commutative” in clock arithmetic.)
𝑥 ⋅ 𝑦 is always the same as 𝑦 ⋅ 𝑥. (Multiplication is “commutative” in clock arithmetic.)
(𝑥 + 𝑦) + 𝑧 is always the same as 𝑥 + (𝑦 + 𝑧). That is, addition is “associative.”
Multiplication is “associative.” (𝑥 ⋅ 𝑦) ⋅ 𝑧 is always the same as 𝑥 ⋅ (𝑦 ⋅ 𝑧).
Multiplication “distributes” over addition. That is, 𝑥 ⋅ (𝑦 + 𝑧) is always the same as
𝑥 ⋅ 𝑦 + 𝑥 ⋅ 𝑧.
There are many other properties that 𝑛-hour arithmetic shares with elementary school
arithmetic. The most surprising general fact about 𝑛-hour arithmetic is that addition,
subtraction, and multiplication are closed. But even more surprising is that under the right
circumstances, clock arithmetic can obey one important property that the whole number
arithmetic fails to uphold – that division is closed!
In 𝑛-hour arithmetic, let’s use the usual definition of division: For any numbers 𝑥 and 𝑦 (as
long as 𝑦 ≠ 0), the quotient 𝑥/𝑦 is a number that, when multiplied by 𝑦, gives 𝑥 as a result.
For example, in 12-hour arithmetic, we could say that 8/5 =12 4, since 4 ⋅ 5 =12 8.
Exercise 1: It is fairly clear that addition, multiplication, and subtraction are closed in 12hour arithmetic. Prove that division (for nonzero numbers) is not closed in 12-hour
arithmetic.
Exercise 2: The additive inverse of a number 𝑥 would be a number 𝑦 such that 𝑥 + 𝑦 is 0.
Every number in whole number arithmetic has an additive inverse. Prove that every number
in 12-hour arithmetic has an additive inverse.
Exercise 3: The multiplicative inverse of a number 𝑥 would be a number 𝑦 such that 𝑥 ⋅ 𝑦
is 1. In rational numbers arithmetic, every nonzero number has a multiplicative inverse.
Prove that not every nonzero number in 12-hour arithmetic has a multiplicative inverse.
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Under certain circumstances, clock arithmetic obeys all of the fundamental properties of the
usual arithmetic of rational numbers. While 12-hour arithmetic does not have some of those
properties, it turns out that 7-hour arithmetic, for example, does have them. Most crucially,
every nonzero number in 7-hour arithmetic has a multiplicative inverse:






The multiplicative inverse of 1 is 1, since 1 ⋅ 1 =7
The multiplicative inverse of 2 is 4, since 2 ⋅ 4 =7
The multiplicative inverse of 3 is 5, since 3 ⋅ 5 =7
The multiplicative inverse of 4 is 2, since 4 ⋅ 2 =7
The multiplicative inverse of 5 is 3, since 5 ⋅ 3 =7
The multiplicative inverse of 6 is 6, since 6 ⋅ 6 =7
1.
1.
1.
1.
1.
1.
Once we know all the multiplicative inverses in a certain clock arithmetic, it is easy to
“divide” any two numbers. For example, to find 6/5 in 7-hour arithmetic, we note that
1/5 =7 3, so 6/5 =7 6 ⋅ (1/5) =7 6 ⋅ 3 =7 4.
Once we can divide any two (nonzero) numbers, we can solve many simple equations. For
example, let’s solve the equation 3𝑥 + 4 =7 2, following the same steps we would take in
everyday arithmetic. Subtracting 4 from each side, we have 3𝑥 =7 2 − 4 =7 5. Multiplying
each side by 1/3, which is 5, we have 5 ⋅ 3𝑥 =7 5 ⋅ 5, so 𝑥 =7 4.
Exercise 4: Prove that every nonzero number in 13-hour arithmetic has a multiplicative
inverse.
Exercise 5: Prove that division (for nonzero numbers) is closed in 13-hour arithmetic.
Exercise 6: Solve the equation 10𝑥 − 8 =13 9, explaining each step.
Abstraction
We see now that, if the number of “hours” is chosen appropriately, clock arithmetic can
satisfy all the essential properties of rational numbers:






Addition and multiplication are closed.
Addition and multiplication are associative.
Addition and multiplication are commutative.
Multiplication distributes over addition.
Everything has an additive inverse.
Everything (other than 0) has a multiplicative inverse.
Many systems of mathematical objects satisfy the properties above. In some cases, the
“objects” are very similar to numbers (as in clock arithmetic), but in other cases, the objects
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may be quite different. A large portion of modern mathematics consists of thinking about the
properties of everyday mathematical objects, and studying the inevitable consequences for
any other hypothetical system of objects with some, or all, of the same essential properties.
Many mathematicians find this kind of creative tinkering very enjoyable. It has also proved
to be deeply useful in real life, from time to time.
As an example of a system of objects that obey just some of the properties listed above, let’s
think about the set of all 2 × 2 real matrices. A 2 × 2 real matrix is an array of real numbers
with two rows and two columns. For example, consider the matrices
1
5
𝐴=(
3
0
) and 𝐵 = (
2
1
−2
).
1
In most real-life applications, matrices behave very much like numbers. We can add and
multiply 2 × 2 matrices according to the following rules:
𝑎
𝑐
(
(
𝑎
𝑐
𝑤
𝑏
) + (𝑦
𝑑
𝑤
𝑏
) ⋅ (𝑦
𝑑
𝑥
𝑎+𝑤
𝑧) = ( 𝑐 + 𝑦
𝑥
𝑎𝑤 + 𝑏𝑦
𝑧 ) = (𝑐𝑤 + 𝑑𝑦
𝑏+𝑥
),
𝑑+𝑧
𝑎𝑥 + 𝑏𝑧
).
𝑐𝑥 + 𝑑𝑧
So, using the matrices above to illustrate, we have
𝐴+𝐵 =(
1
6
1
)
3
and 𝐴 ⋅ 𝐵 = (
3
1
).
2 −8
Exercise 7: Which of the “essential properties of rational numbers” are satisfied by the
system of 2 × 2 matrices? Do your best to prove or disprove each of the properties.
Geometries
The proof of any interesting fact in high-school-level geometry (also known as Euclidean
geometry) is built upon dozens of assumptions and basic theorems. These assumptions and
properties of geometrical objects have been an active area of mathematical tinkering for
thousands of years.
One question of interest in geometry is how many assumptions must be made in order to
prove the things we would like to think of as true. Or, to look at it another way, how many
things can be proved based on only the most basic assumptions? This question eventually
arises in every field of mathematics, but it started to be raised thousands of years ago in
geometry.
To work at a very simple level, let’s throw out everything we may have ever learned about
geometry and start with only the following assumptions.
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Geometric Assumptions:
A1: There is a set 𝑃 called the plane, and its elements are called points.
A2: Some of the subsets of 𝑃 may be called lines, but lines must obey the following rules.
A3: For any two points, there is a unique line that contains the points.
A4: Every line contains at least two points.
A5: There are at least three points that are not all elements of the same line.
For convenience, we will make the following definition.
Definition: Lines 𝐴 and 𝐵 are parallel iff 𝐴 ∩ 𝐵 = ∅. Otherwise, we say the lines intersect.
Note carefully that neither the assumptions nor the definition listed above contradicts
anything you may have once learned in high school geometry. But it is interesting to see what
can be proved from only these most basic ideas about geometry, without using any other
preconception whatsoever.
Exercise 8: Prove that if 𝐴 is a line, then there is a point that is not an element of 𝐴.
Exercise 9: Prove that if 𝑝 is a point, then there are two different lines that each contain 𝑝.
Exercise 10: Prove that if 𝑝 is a point, then there is a line that does not contain 𝑝.
Exercise 11: Prove that if 𝐴 and 𝐵 are parallel lines, then there is a line that intersects 𝐴 and
also intersects 𝐵.
Remember that the “essential properties of rational numbers” mentioned before can be
obeyed by other systems of objects that may bear only a slight resemblance (if any) to the
rational numbers. The same is true of the geometric assumptions listed above!
As an example, let’s say we think of the “plane” as the set of all pairs of numbers (𝑥, 𝑦), with
both 𝑥 and 𝑦 being elements of 7-hour arithmetic. So there are 49 “points” in total. And let’s
say we think of a “line” as a set of all the (𝑥, 𝑦) number pairs that solve a certain equation of
the form 𝑎 ⋅ 𝑥 + 𝑏 ⋅ 𝑦 =7 𝑐, with 𝑎, 𝑏, and 𝑐 being elements of 7-hour arithmetic (except that
𝑎 and 𝑏 cannot both be 0). So, for example, one of the many lines in this version of geometry
is given by the equation 2𝑥 + 3𝑦 =7 1. Some of the points that are part of this line are (2, 6)
and (5, 4), because 2 ⋅ 2 + 3 ⋅ 6 =7 1 and 2 ⋅ 5 + 3 ⋅ 4 =7 1. It would be understandable for
someone to think that the equation 𝑥 + 5𝑦 =7 4, which looks so very different from the one
just mentioned, gives us an entirely different line, but in fact it turns out that these two lines
are the same, because the number pairs that solve the two equations are identical. However,
the equation 2𝑥 + 5𝑦 =7 3 does give an entirely different line. This can be seen by the fact
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that (2, 6) is not a solution of this last equation, and so the line given by this equation does
not have the same points as the line given by the other equations.
Let’s refer to what is described in the previous paragraph as “the finite geometry that comes
from 7-hour clock arithmetic.” It is possible to prove that this weird conception of
“geometry” satisfies all the geometric assumptions listed before.
Exercise 12: Geometric assumptions A1 and A2 are clearly satisfied in the finite geometry
that comes from 7-hour clock arithmetic. Prove that A3 is also satisfied. That is, for any two
points, there is one and only one line that contains the points.
Exercise 13: Prove that in the finite geometry that comes from 7-hour clock arithmetic,
every line contains at least two points.
Exercise 14: Prove that in the finite geometry that comes from 7-hour clock arithmetic,
there are at least three points that are not all elements of the same line.
Exercise 15: The following statement cannot be proved from the geometric assumptions A1
through A5, but in high school geometry, it is usually used as additional assumption:
For any line 𝐴, and any point 𝑝 ∉ 𝐴, there is a line that is parallel to 𝐴 and contains 𝑝.
Prove that the statement above is true in the finite geometry that comes from 7-hour clock
arithmetic.