1
00:00:04,420 --> 00:00:07,221
Imagine we are living in prehistoric times.
2
00:00:07,221 --> 00:00:09,468
Now, consider the following:
3
00:00:09,468 --> 00:00:12,721
How did we keep track of time without a clock?
4
00:00:12,721 --> 00:00:15,315
All clocks are based on some repetitive pattern
5
00:00:15,315 --> 00:00:18,890
which divides the flow of time into equal segments.
6
00:00:18,890 --> 00:00:20,688
To find these repetitive patterns,
7
00:00:20,688 --> 00:00:22,918
we look towards the heavens.
8
00:00:22,918 --> 00:00:24,902
The sun rising and falling each day
9
00:00:24,902 --> 00:00:26,184
is the most obvious [pattern].
10
00:00:26,184 --> 00:00:28,760
However, to keep track of longer periods of time,
11
00:00:28,760 --> 00:00:30,811
we looked for longer cycles.
12
00:00:30,811 --> 00:00:32,512
For this, we looked to the moon,
13
00:00:32,512 --> 00:00:33,853
which seemed to gradually grow
14
00:00:33,853 --> 00:00:36,578
and shrink over many days.
15
00:00:36,578 --> 00:00:37,894
When we count the number of days
16
00:00:37,894 --> 00:00:38,978
between full moons,
17
00:00:38,978 --> 00:00:40,910
we arrive at the number 29.
18
00:00:40,910 --> 00:00:42,833
This is the origin of a month.
19
00:00:42,833 --> 00:00:45,873
However, if we try to divide 29 into equal pieces,
20
00:00:45,873 --> 00:00:49,227
we run into a problem: it is impossible.
21
00:00:49,227 --> 00:00:51,676
The only way to divide 29 into equal pieces
22
00:00:51,676 --> 00:00:54,819
is to break it back down into [29] single units.
23
00:00:54,819 --> 00:00:57,102
29 is a 'prime number.'
24
00:00:57,102 --> 00:00:59,061
Think of it as unbreakable.
25
00:00:59,061 --> 00:01:00,879
If a number can be broken down into
26
00:01:00,879 --> 00:01:02,814
equal pieces greater than one,
27
00:01:02,814 --> 00:01:04,621
we call it a 'composite number.'
28
00:01:04,621 --> 00:01:06,608
Now if we are curious, we may wonder,
29
00:01:06,608 --> 00:01:08,450
"How many prime numbers are there?
30
00:01:08,450 --> 00:01:10,398
– and how big do they get?"
31
00:01:10,398 --> 00:01:13,744
Let's start by dividing all numbers into two categories.
32
00:01:13,744 --> 00:01:15,611
We list the primes on the left
33
00:01:15,611 --> 00:01:17,648
and the composites on the right.
34
00:01:17,648 --> 00:01:20,379
At first, they seem to dance back and forth.
35
00:01:20,379 --> 00:01:23,017
There is no obvious pattern here.
36
00:01:23,017 --> 00:01:24,439
So let's use a modern technique
37
00:01:24,439 --> 00:01:26,077
to see the big picture.
38
00:01:26,077 --> 00:01:29,047
The trick is to use a "Ulam spiral."
39
00:01:29,047 --> 00:01:32,011
First, we list all possible numbers in order
40
00:01:32,011 --> 00:01:34,043
in a growing spiral.
41
00:01:34,043 --> 00:01:37,164
Then, we color all the prime numbers blue.
42
00:01:37,164 --> 00:01:41,290
Finally, we zoom out to see millions of numbers.
43
00:01:41,290 --> 00:01:42,860
This is the pattern of primes
44
00:01:42,860 --> 00:01:45,365
which goes on and on, forever.
45
00:01:45,365 --> 00:01:47,967
Incredibly, the entire structure of this pattern
46
00:01:47,967 --> 00:01:50,314
is still unsolved today.
47
00:01:50,314 --> 00:01:51,843
We are onto something.
48
00:01:51,843 --> 00:01:52,987
So, let's fast forward to
49
00:01:52,987 --> 00:01:55,526
around 300 BC, in ancient Greece.
50
00:01:55,526 --> 00:01:58,183
A philosopher known as Euclid of Alexandria
51
00:01:58,183 --> 00:01:59,411
understood that all numbers
52
00:01:59,411 --> 00:02:02,607
could be split into these two distinct categories.
53
00:02:02,607 --> 00:02:04,896
He began by realizing that any number
54
00:02:04,896 --> 00:02:07,078
can be divided down – over and over –
55
00:02:07,078 --> 00:02:10,599
until you reach a group of smallest equal numbers.
56
00:02:10,599 --> 00:02:12,921
And by definition, these smallest numbers
57
00:02:12,921 --> 00:02:15,760
are always prime numbers.
58
00:02:15,760 --> 00:02:17,148
So, he knew that all numbers are
59
00:02:17,148 --> 00:02:20,542
somehow built out of smaller primes.
60
00:02:20,542 --> 00:02:23,317
To be clear, imagine the universe of all numbers –
61
00:02:23,317 --> 00:02:25,674
and ignore the primes.
62
00:02:25,674 --> 00:02:28,037
Now, pick any composite number,
63
00:02:28,037 --> 00:02:30,518
and break it down,
64
00:02:30,518 --> 00:02:33,354
and you are always left with prime numbers.
65
00:02:33,354 --> 00:02:34,774
So, Euclid knew that every number
66
00:02:34,774 --> 00:02:37,675
could be expressed using a group of smaller primes.
67
00:02:37,675 --> 00:02:40,221
Think of these as building blocks.
68
00:02:40,221 --> 00:02:41,996
No matter what number you choose,
69
00:02:41,996 --> 00:02:46,157
it can always be built with an addition of smaller primes.
70
00:02:46,157 --> 00:02:48,032
This is the root of his discovery,
71
00:02:48,032 --> 00:02:50,759
known as the 'Fundamental Theorem of Arithmetic' –
72
00:02:50,759 --> 00:02:52,013
as follows:
73
00:02:52,013 --> 00:02:53,934
Take any number – say 30 –
74
00:02:53,934 --> 00:02:55,501
and find all the prime numbers
75
00:02:55,501 --> 00:02:57,233
it [can be divided into] equally.
76
00:02:57,233 --> 00:02:59,763
This we know as 'factorization.'
77
00:02:59,763 --> 00:03:01,624
This will give us the prime factors.
78
00:03:01,624 --> 00:03:05,811
In this case 2, 3, and 5 are the prime factors of 30.
79
00:03:05,811 --> 00:03:07,906
Euclid realized that you could then multiply
80
00:03:07,906 --> 00:03:10,714
these prime factors a specific number of times
81
00:03:10,714 --> 00:03:12,739
to build the original number.
82
00:03:12,739 --> 00:03:13,780
In this case, you simply
83
00:03:13,780 --> 00:03:16,178
multiply each factor once to build 30.
84
00:03:16,178 --> 00:03:20,158
2 × 3 × 5 is the prime factorization of 30.
85
00:03:20,158 --> 00:03:23,153
Think of it as a special key or combination.
86
00:03:23,153 --> 00:03:24,887
There is no other way to build 30,
87
00:03:24,887 --> 00:03:27,110
using some other groups of prime numbers
88
00:03:27,110 --> 00:03:28,792
multiplied together.
89
00:03:28,792 --> 00:03:31,276
So every possible number has one –
90
00:03:31,276 --> 00:03:34,046
and only one – prime factorization.
91
00:03:34,046 --> 00:03:36,299
A good analogy is to imagine each number
92
00:03:36,299 --> 00:03:38,017
as a different lock.
93
00:03:38,033 --> 00:03:39,722
The unique key for each lock
94
00:03:39,722 --> 00:03:42,054
would be its prime factorization.
95
00:03:42,054 --> 00:03:43,937
No two locks share a key.
96
00:03:43,937 --> 00:03:47,889
No two numbers share a prime factorization.