WEBVTT

00:00:07.044 --> 00:00:10.294
In the 16th century, the mathematician
Robert Recorde

00:00:10.294 --> 00:00:13.044
wrote a book called 
"The Whetstone of Witte"

00:00:13.044 --> 00:00:15.967
to teach English students algebra.

00:00:15.967 --> 00:00:21.115
But he was getting tired of writing
the words "is equal to" over and over.

00:00:21.115 --> 00:00:22.626
His solution?

00:00:22.626 --> 00:00:27.238
He replaced those words with
two parallel horizontal line segments

00:00:27.238 --> 00:00:32.265
because the way he saw it,
no two things can be more equal.

00:00:32.265 --> 00:00:34.954
Could he have used four line segments
instead of two?

00:00:34.954 --> 00:00:36.196
Of course.

00:00:36.196 --> 00:00:38.289
Could he have used vertical line segments?

00:00:38.289 --> 00:00:40.704
In fact, some people did.

00:00:40.704 --> 00:00:44.995
There's no reason why the equals sign
had to look the way it does today.

00:00:44.995 --> 00:00:48.202
At some point, it just caught on,
sort of like a meme.

00:00:48.202 --> 00:00:50.728
More and more mathematicians
began to use it,

00:00:50.728 --> 00:00:55.568
and eventually, 
it became a standard symbol for equality.

00:00:55.568 --> 00:00:56.967
Math is full of symbols.

00:00:56.967 --> 00:00:57.742
Lines,

00:00:57.742 --> 00:00:58.562
dots,

00:00:58.562 --> 00:00:59.301
arrows,

00:00:59.301 --> 00:01:00.257
English letters,

00:01:00.257 --> 00:01:01.212
Greek letters,

00:01:01.212 --> 00:01:02.189
superscripts,

00:01:02.189 --> 00:01:03.348
subscripts.

00:01:03.348 --> 00:01:05.959
It can look like an illegible jumble.

00:01:05.959 --> 00:01:09.819
It's normal to find this wealth
of symbols a little intimidating

00:01:09.819 --> 00:01:13.048
and to wonder where they all came from.

00:01:13.048 --> 00:01:16.608
Sometimes, as Recorde himself
noted about his equals sign,

00:01:16.608 --> 00:01:21.508
there's an apt conformity
between the symbol and what it represents.

00:01:21.508 --> 00:01:25.200
Another example of that 
is the plus sign for addition,

00:01:25.200 --> 00:01:30.487
which originated from a condensing
of the Latin word et meaning and.

00:01:30.487 --> 00:01:33.840
Sometimes, however, the choice of symbol
is more arbitrary,

00:01:33.840 --> 00:01:36.571
such as when a mathematician
named Christian Kramp

00:01:36.571 --> 00:01:40.181
introduced the exclamation mark
for factorials

00:01:40.181 --> 00:01:44.683
just because he needed a shorthand
for expressions like this.

00:01:44.683 --> 00:01:48.058
In fact, all of these symbols 
were invented or adopted

00:01:48.058 --> 00:01:51.972
by mathematicians who wanted
to avoid repeating themselves

00:01:51.972 --> 00:01:57.022
or having to use a lot of words
to write out mathematical ideas.

00:01:57.022 --> 00:01:59.683
Many of the symbols used
in mathematics are letters,

00:01:59.683 --> 00:02:03.819
usually from the Latin alphabet
or Greek.

00:02:03.819 --> 00:02:08.029
Characters are often found
representing quantities that are unknown,

00:02:08.029 --> 00:02:11.191
and the relationships between variables.

00:02:11.191 --> 00:02:15.251
They also stand in for specific numbers
that show up frequently

00:02:15.251 --> 00:02:21.020
but would be cumbersome or impossible
to fully write out in decimal form.

00:02:21.020 --> 00:02:26.351
Sets of numbers and whole equations
can be represented with letters, too.

00:02:26.351 --> 00:02:29.489
Other symbols are used 
to represent operations.

00:02:29.489 --> 00:02:32.193
Some of these are especially valuable
as shorthand

00:02:32.193 --> 00:02:36.882
because they condense repeated operations
into a single expression.

00:02:36.882 --> 00:02:41.553
The repeated addition of the same number
is abbreviated with a multiplication sign

00:02:41.553 --> 00:02:44.482
so it doesn't take up more space
than it has to.

00:02:44.482 --> 00:02:47.922
A number multiplied by itself
is indicated with an exponent

00:02:47.922 --> 00:02:51.212
that tells you how many times
to repeat the operation.

00:02:51.212 --> 00:02:54.252
And a long string of sequential terms
added together

00:02:54.252 --> 00:02:57.213
is collapsed into a capital sigma.

00:02:57.213 --> 00:03:01.403
These symbols shorten
lengthy calculations to smaller terms

00:03:01.403 --> 00:03:05.024
that are much easier to manipulate.

00:03:05.024 --> 00:03:07.954
Symbols can also provide 
succinct instructions

00:03:07.954 --> 00:03:10.637
about how to perform calculations.

00:03:10.637 --> 00:03:13.965
Consider the following set
of operations on a number.

00:03:13.965 --> 00:03:15.924
Take some number that you're thinking of,

00:03:15.924 --> 00:03:17.394
multiply it by two,

00:03:17.394 --> 00:03:18.964
subtract one from the result,

00:03:18.964 --> 00:03:21.397
multiply the result of that by itself,

00:03:21.397 --> 00:03:23.235
divide the result of that by three,

00:03:23.235 --> 00:03:26.645
and then add one to get the final output.

00:03:26.645 --> 00:03:32.186
Without our symbols and conventions,
we'd be faced with this block of text.

00:03:32.186 --> 00:03:35.796
With them, we have a compact,
elegant expression.

00:03:35.796 --> 00:03:37.496
Sometimes, as with equals,

00:03:37.496 --> 00:03:40.754
these symbols communicate meaning
through form.

00:03:40.754 --> 00:03:43.607
Many, however, are arbitrary.

00:03:43.607 --> 00:03:46.678
Understanding them is a matter
of memorizing what they mean

00:03:46.678 --> 00:03:52.017
and applying them in different contexts
until they stick, as with any language.

00:03:52.017 --> 00:03:54.616
If we were to encounter
an alien civilization,

00:03:54.616 --> 00:03:58.757
they'd probably have a totally
different set of symbols.

00:03:58.757 --> 00:04:04.367
But if they think anything like us,
they'd probably have symbols.

00:04:04.367 --> 00:04:08.636
And their symbols may even correspond
directly to ours.

00:04:08.636 --> 00:04:10.767
They'd have their own multiplication sign,

00:04:10.767 --> 00:04:12.127
symbol for pi,

00:04:12.127 --> 00:04:14.906
and, of course, equals.