Ch. 10: What is a number?. MAIN DEFINITION OF THE COURSE: A symmetry of an object (in the plane or space) means a rigid motion (of the plane or space)

Ch. 10: What is a number?

MAIN DEFINITION OF THE COURSE: A symmetry of an object (in the plane or space) means a rigid motion (of the plane or space) that leaves the object apparently unchanged.


What is a rigid motion of the plane? Of space?

This doesn’t make sense until we decide:




What is the plane? What is space?







Since the plane is made of pairs (x,y) of numbersand space is made of triples (x,y,z) of numbers,this doesn’t make sense until we decide:

What is a number?






What is a number?

Since the plane is made of pairs (x,y) of numbersand space is made of triples (x,y,z) of numbers,this doesn’t make sense until we decide:

Let’s start our backtrack here!

What is a number?

What is a number?

N = {1, 2, 3, 4, 5, 6, …} “the natural numbers”

What is a number?


Good for counting sheep

What is a number?


Including the primes {2, 3, 5, 7, 11, 13, 17, 19…}

What is a number?



DEFINITION: A prime number is a natural number greater than 1 that cannot be expressed as a product of two smaller natural numbers.

What is a number?




For example, 20 is not prime because 20 = 5×4.

What is a number?





THEOREM: Every natural number greater than 1 is either prime or can be expressed in a unique way as a product of primes.

What is a number?






For example, the prime factorization of 300 is: 300 = 2×2×3×5×5.

What is a number?







We figured this out by breaking 300 down step-by-step until the pieces could not be further broken down…

…like this: 300 = 3×100

What is a number?








…like this: 300 = 3×100 = 3×4×25

What is a number?








…like this: 300 = 3×100 = 3×4×25 = 3×2×2×5×5.

What is a number?








…like this: 300 = 3×100 = 3×4×25 = 3×2×2×5×5.

…or like this: 300 = 10×30

What is a number?








…like this: 300 = 3×100 = 3×4×25 = 3×2×2×5×5.

…or like this: 300 = 10×30 = 2×5×3×10

What is a number?








…like this: 300 = 3×100 = 3×4×25 = 3×2×2×5×5.

…or like this: 300 = 10×30 = 2×5×3×10 = 2×5×3×2×5.

What is a number?







…like this: 300 = 3×100 = 3×4×25 = 3×2×2×5×5.

…or like this: 300 = 10×30 = 2×5×3×10 = 2×5×3×2×5.

Same answer afterre-ordering!

This is the hard part to prove!

The prime factorization of a square

The prime factorization of 300 is: 300 = 2×2×3×5×5.

What is the prime factorization of 3002 = 300×300 = 90000?




90000 = 300×300




90000 = 300×300 = 2×2×3×5×5×2×2×3×5×5




90000 = 300×300 = 2×2×3×5×5×2×2×3×5×5

(reorder) = 2×2×2×2×3×3×5×5×5×5.




90000 = 300×300 = 2×2×3×5×5×2×2×3×5×5

(reorder) = 2×2×2×2×3×3×5×5×5×5.

NOTE: 3002 has twice as many of each prime as 300 has.




90000 = 300×300 = 2×2×3×5×5×2×2×3×5×5

(reorder) = 2×2×2×2×3×3×5×5×5×5.

NOTE: 3002 has twice as many of each prime as 300 has.

FACT: The square of any natural number greater than 1 has an even number of occurrences of each prime in its prime factorization.

What is a number?


Good for counting your sheep

But has limits…

What is a number?


Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers”

What is a number?


Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers”

Good for counting positive and negative sheep…

But still has limits…

What is a number?


Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers”

Q = { all quotients “a/b” of integers with b≠0} “the rational numbers”

Examples: 1/2, -17/4, 7/1, -15/38

What is a number?


Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers”


synonyms: quotient, fraction, ratio, rational number

What is a number?


Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers”



Some fractions are the same as other, like 2/3 = 4/6.

What is a number?


Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers”



Some fractions are the same as other, like 2/3 = 4/6.

General rule: a/b = c/d whenever ad = bc.

This rule is an addendum to the definition of Q

What is a number?


Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers”

Q = { all quotients “a/b” of integers with b≠0} with the understanding that a/b = c/d whenever ad = bc. “the rational numbers”


Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers”


What is a number?

Good for counting parts of sheep…

But are the rational numbers enough?


Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers”


What is a number?

“All numbers arerational.”

Greek Mathematician, 500 BC


Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers”


What is a number?



SOME REASONS FOR THIS BELIEF:

(1) Finite divisibility of all matter(2) Rational numbers are gifts from

the gods(3) They can measure arbitrarily

small lengths.


Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers”


What is a number?




Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers”


What is a number?



“Which rational number squares to 2?”




THEOREM: There is no rational number that squares to 2.





“ὦ περίσσωμα”





Greek for “Oh shit.”

“ὦ περίσσωμα”



Today, we call this length “the square root of 2,”And we refer to it as an “irrational number.”

But to the Greek mathematicians in 500 BC,“number” meant “rational number”.



PROOF:



PROOF: Suppose that there are two integers, p and q, such that the fraction p/q squares exactly to 2.



PROOF: Suppose that there are two integers, p and q, such that the fraction p/q squares exactly to 2. That is:

(p/q)2 = 2 ↔ p2/q2 = 2 ↔ p2 = 2×q2




(p/q)2 = 2 ↔ p2/q2 = 2 ↔ p2 = 2×q2




(p/q)2 = 2 ↔ p2/q2 = 2 ↔ p2 = 2×q2




(p/q)2 = 2 ↔ p2/q2 = 2 ↔ p2 = 2×q2

This has an even number of 2sin its prime factorization.

(twice as many as p has)




(p/q)2 = 2 ↔ p2/q2 = 2 ↔ p2 = 2×q2

This has an even number of 2sin its prime factorization. This has an odd number of 2s

in its prime factorization.(twice as many as p has)

(one more than twice as many as q has)




(p/q)2 = 2 ↔ p2/q2 = 2 ↔ p2 = 2×q2


in its prime factorization.

But the prime factorization of a number is unique. If the left and right side of this equation really equaled each other, there would not be a difference between their prime factorizations.




(p/q)2 = 2 ↔ p2/q2 = 2 ↔ p2 = 2×q2


in its prime factorization.

But the prime factorization of a number is unique. If the left and right side of this equation really equaled each other, there would not be a difference between their prime factorizations.

SUMMARY: We assumed that there is a rational number that squares to 2, but this assumption lead to a contradiction; therefore, this assumption must be wrong!



“I need a new kind of number.”

Today, we put fractions like 2/3 and lengths like √2 on equal footingby them as decimal expressions.

2/3 = 0.66666666… and √2 = 1.41421356237…


2/3 = 0.66666666… and √2 = 1.41421356237…

indicates a continuation of the pattern (an unending string of 6s)

indicates some unending string of digits, which might not have a pattern that you understand.


2/3 = 0.66666666… and √2 = 1.41421356237…

DEFINITION: A real number means a “decimal expression”; that is, an expression formed from an integer followed by a decimal point followed by infinitely many digits. The set of all real numbers is denoted R.


2/3 = 0.66666666… and √2 = 1.41421356237…


COMMENT: It takes work to precisely define +,-,×,/ of real numbers. For example, what is 2/3 + √2?


2/3 = 0.66666666… and √2 = 1.41421356237…



GUESS: 12.75000000000… – 12.74999999999… = ???


2/3 = 0.66666666… and √2 = 1.41421356237…



GUESS: 12.75000000000… – 12.74999999999… = ???

The answer is less than 12.75 – 12.74 = 0.01


2/3 = 0.66666666… and √2 = 1.41421356237…



GUESS: 12.75000000000… – 12.74999999999… = ???

The answer is less than 12.75 – 12.74 = 0.01and less than 12.750 – 12.749 = 0.001


2/3 = 0.66666666… and √2 = 1.41421356237…



GUESS: 12.75000000000… – 12.74999999999… = ???

The answer is less than 12.75 – 12.74 = 0.01and less than 12.750 – 12.749 = 0.001and less than 12.7500 – 12.7499 = 0.0001


2/3 = 0.66666666… and √2 = 1.41421356237…



GUESS: 12.75000000000… – 12.74999999999… = ???

The answer is less than 12.75 – 12.74 = 0.01and less than 12.750 – 12.749 = 0.001and less than 12.7500 – 12.7499 = 0.0001and less than 12.75000 – 12.74999 = 0.00001

and so on…


2/3 = 0.66666666… and √2 = 1.41421356237…



GUESS: 12.75000000000… – 12.74999999999… = 0

The answer is less than 12.75 – 12.74 = 0.01and less than 12.750 – 12.749 = 0.001and less than 12.7500 – 12.7499 = 0.0001and less than 12.75000 – 12.74999 = 0.00001

and so on… THE ANSWER MUST BE ZERO!


2/3 = 0.66666666… and √2 = 1.41421356237…



GUESS: 12.75000000000… – 12.74999999999… = 0

CONCLUSION: 12.75000000000… = 12.74999999999…

If they differ by zero, they must be the same real number!

(This is similar to how 2/3 and 2/6 are two different ways of writing the same rational number)


2/3 = 0.66666666… and √2 = 1.41421356237…


CONCLUSION: 12.75000000000… = 12.74999999999…

REAL REDUNDANCY RULE: A digit (other than 9) followed by an unending string of 9s can be replaced by the next larger digit followed by an unending string of 0s. There are no other redundancies among real numbers.

REAL REDUNDANCY RULE: A digit (other than 9) followed by an unending string of 9s can be replaced by the next larger digit followed by an unending string of 0s. There are no other redundancies among real numbers.


Z = {…, -3, -2, -1, 0, 1, 2, 3, …} “the integers”

Q = {all quotients “a/b” of integers with b≠0} “the rational numbers” with the understanding that a/b = c/d whenever ad = bc.

R = {all decimal expressions} “the real numbers” with the understanding that the real redundancy rule determines which are the same.

What is a number?

Picture each real number as a point on the real number line.

0 1 2 3 4 5 6-1-2-3-4-5-6


0 1 2 3 4 5 6-1-2-3-4-5-6

-13/5 π 13/3


0 1 2 3 4 5 6-1-2-3-4-5-6

-13/5 π 13/3

The number line is like a yard stick, which matches the ancient Greekidea that numbers should represent all possible lengths…

…like this one:


The digits of a decimal expression locate the number on the real number line.

0 1 2 3 4 5 6-1-2-3-4-5-6



0 1 2 3 4 5 6-1-2-3-4-5-6

√2 = 1.41421356237…

How do these digits locate the number?



0 1 2 3 4 5 6-1-2-3-4-5-6

√2 = 1.41421356237…



1 2

√2 = 1.41421356237…



√2 = 1.41421356237…

1 2



√2 = 1.41421356237…

1.0 2.0



√2 = 1.41421356237…

1.0 2.00 1 2 3 4 5 6 7 8 9



√2 = 1.41421356237…

1.0 2.00 1 2 3 4 5 6 7 8 9



√2 = 1.41421356237…

4



√2 = 1.41421356237…

4



√2 = 1.41421356237…

1.4 1.5



√2 = 1.41421356237…

1.4 1.50 1 2 3 4 5 6 7 8 9



√2 = 1.41421356237…

1.4 1.50 1 2 3 4 5 6 7 8 9



√2 = 1.41421356237…

1.4 1.50 1 2 3 4 5 6 7 8 9

… and so on. Each next digit provides a ten fold increase in the accuracy with which we know the number’s location on the real number line.

Which real numbers are rational?


Q: Use long division to convert the fraction 3/7 into a decimal expression.



A: 3/7 = 0.428571428571428517….



A: 3/7 = 0.428571428571428517….

With long division, as soon as the remainder repeats, the digits always begin repeating.

How are you guaranteed that the remainder will repeat?



A: 3/7 = 0.428571428571428517….



FACT: Long division converts any fraction into aneventually repeating decimal expression.



A: 3/7 = 0.428571428571428517….



FACT: Long division converts any fraction into aneventually repeating decimal expression.

Like this: 103.45873065306530653065…

or like this: 4/5 = 0.80000000000…


Q: Convert N = 0.428571428571428517…. into a fraction.



TRICK: Since N has a 6-digit repeating string, we multiply it by 1,000,000 (which has 6 zeros), and then subtract:

1000000×N = 428571.428571428571…

N = 0.428571428571… subtract 999999×N = 428571.000000000000…

We learn that N = 428571/999999 (which reduces to N = 3/7).



TRICK: Since N has a 6-digit repeating string, we multiply it by 1,000,000 (which has 6 zeros), and then subtract:

1000000×N = 428571.428571428571…

N = 0.428571428571… subtract 999999×N = 428571.000000000000…

We learn that N = 428571/999999 (which reduces to N = 3/7).

FACT: This trick works to convert any eventually repeating decimal expression into a fraction.


FRACTIONEventually repeating decimal expression

Long division

Subtraction trick



Long division

Subtraction trick

THEOREM: A real number is rational precisely when its decimal expression is eventually repeating.



Long division

Subtraction trick


That is: Every rational number has a decimal expression that’s eventually repeating.Every irrational number has a decimal expression that’s not eventually repeating.



Long division

Subtraction trick



COROLLARY: √2 = 1.41421356237… is NOT eventually repeating!

(It’s surprising that we could prove this without understanding the digits)






Q: Is this rational or irrational: 0.01001000100001000001… (pattern continues)







A: Irrational. The pattern is more complicated than “eventually repeating”.








Make up your own irrational numbers.








The famous numbers π and e are irrational, but this is hard to prove.

Make up your own irrational numbers.

How many primes are there?


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …

How long until we reach the largest prime?Is there a largest prime?


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …

EUCLID’S THEOREM: There are infinitely many prime numbers.


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …


We know: (1) There are infinitely many natural numbers. (2) Each natural number is a product of primes.

But this does not prove Euclid’s Theorem.

After all, infinitely many things can be built from unlimited supplies of finitely many different building blocks.


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …


I think that these are ALL of the primes: 2, 3, 5, 7, 11, 13.

Andy


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …



Andy

How do we know Andy is wrong? Euclid found a concrete procedure for identifying a prime that is missing from Andy’s (or any) finite list of primes.


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …



Andy


In Andy’s list, the next prime, 17, is missing.

But “choosing the next prime” is not a good general procedure, since it begs the question of whether there always is a next prime.


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …



Andy


First multiply them and add 1: L = 2×3×5×7×11×13 + 1 = 30031


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …



Andy


First multiply them and add 1: L = 2×3×5×7×11×13 + 1 = 30031

This is bigger than anything on Andy’s listIt would be nice if numbers formed this way were always prime.


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …



Andy


First multiply them and add 1: L = 2×3×5×7×11×13 + 1 = 30031 = 59×509

But it’s not prime. Here is its prime factorization


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …



Andy



But here are two primesthat are missing from Andy’s list!


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …



Andy



But here are two primesthat are missing from Andy’s list!

59 and 509 divide evenly into 30031. Andy’s primes all leave a remainder 1 when divided into 30031. Thus, 59 and 509 are different from Andy’s primes!


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …


PROOF: Given any finite list of primes

p1, p2, p3, p4, …, pn

we can find a prime that is missing from the list as follows:


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …



p1, p2, p3, p4, …, pn


First multiply them and add 1:

N = p1×p2×p3×p4×…×pn + 1


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …



p1, p2, p3, p4, …, pn



N = p1×p2×p3×p4×…×pn + 1

Each prime factor of N is a prime that’s missing from the list!


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …



p1, p2, p3, p4, …, pn



N = p1×p2×p3×p4×…×pn + 1


(because they divide evenly into N, while everything on the list leaves a remainder 1 when divided into N.)


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …



p1, p2, p3, p4, …, pn



N = p1×p2×p3×p4×…×pn + 1


(because they divide evenly into N, while everything on the list leaves a remainder 1 when divided into N.)

Thus, no finite list of primes could ever be complete!


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …


After learning Euclid’s Theorem, we still might wonder: how frequently occurring are the prime numbers are among the natural numbers?

Are primes in abundance, or are they a rare breed?

Do most United States citizens have a prime social security number, or very few?


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …



N Fraction of numbers up to N that are prime101001,00010,000100,0001,000,000


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …



N Fraction of numbers up to N that are prime10 0.401001,00010,000100,0001,000,000

Up to N=10: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

4/10 = 0.40 (40% are prime)


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …



N Fraction of numbers up to N that are prime10 0.40100 0.251,00010,000100,0001,000,000

Up to N=100: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,

27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100.

25/100 = 0.25 (25% are prime)


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …



N Fraction of numbers up to N that are prime10 0.40100 0.251,000 0.16810,000 0.1229100,000 0.095921,000,000 0.078498


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …




What is the pattern?How does the pattern continue for larger and larger N?


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …




THE PRIME NUMBER THEOREM: The fraction of numbers up to

N that are prime is approximately .


2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …




THE PRIME NUMBER THEOREM: The fraction of numbers up to

N that are prime is approximately .

1/(twice 7) = 1/14 = 0.0714…7 digits

Famous unsolved questions about primes

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …

Q: Are there infinitely many “twin primes”?

consecutive odd numbers that areboth prime, like: 11 & 13,

29 & 31,41 & 43

…

Famous unsolved questions about primes

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …

Q: Are there infinitely many “twin primes”?

consecutive odd numbers that areboth prime, like: 11 & 13,

29 & 31,41 & 43

Q: (Goldbach) Can every positive even number larger than 2 be written as a sum of two primes?

4 = 2 + 26 = 3 + 38 = 3 + 510 = 5 + 512 = 7 + 514 = 7 + 7…

Documents

Ch. 10: What is a number?. MAIN DEFINITION OF THE COURSE: A symmetry of an object (in the plane or space) means a rigid motion (of the plane or space)