Where do math symbols come from? - John David Walters

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Math is full of symbols: lines, dots, arrows, English letters, Greek letters, superscripts, subscripts ... it can look like an illegible jumble. Where did all of these symbols come from? John David Walters shares the origins of mathematical symbols, and illuminates why they’re still so important in the field today.

Lesson by John David Walters, directed by Chris Bishop.

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In the 16th century, the mathematician
Robert Recorde

wrote a book called
"The Whetstone of Witte"

to teach English students algebra.

But he was getting tired of writing
the words "is equal to" over and over.

His solution?

He replaced those words with
two parallel horizontal line segments

because the way he saw it,
no two things can be more equal.

Could he have used four line segments
instead of two?

Of course.

Could he have used vertical line segments?

In fact, some people did.

There's no reason why the equals sign
had to look the way it does today.

At some point, it just caught on,
sort of like a meme.

More and more mathematicians
began to use it,

and eventually,
it became a standard symbol for equality.

Math is full of symbols.

Lines,

dots,

arrows,

English letters,

Greek letters,

superscripts,

subscripts.

It can look like an illegible jumble.

It's normal to find this wealth
of symbols a little intimidating

and to wonder where they all came from.

Sometimes, as Recorde himself
noted about his equals sign,

there's an apt conformity
between the symbol and what it represents.

Another example of that
is the plus sign for addition,

which originated from a condensing
of the Latin word et meaning and.

Sometimes, however, the choice of symbol
is more arbitrary,

such as when a mathematician
named Christian Kramp

introduced the exclamation mark
for factorials

just because he needed a shorthand
for expressions like this.

In fact, all of these symbols
were invented or adopted

by mathematicians who wanted
to avoid repeating themselves

or having to use a lot of words
to write out mathematical ideas.

Many of the symbols used
in mathematics are letters,

usually from the Latin alphabet
or Greek.

Characters are often found
representing quantities that are unknown,

and the relationships between variables.

They also stand in for specific numbers
that show up frequently

but would be cumbersome or impossible
to fully write out in decimal form.

Sets of numbers and whole equations
can be represented with letters, too.

Other symbols are used
to represent operations.

Some of these are especially valuable
as shorthand

because they condense repeated operations
into a single expression.

The repeated addition of the same number
is abbreviated with a multiplication sign

so it doesn't take up more space
than it has to.

A number multiplied by itself
is indicated with an exponent

that tells you how many times
to repeat the operation.

And a long string of sequential terms
added together

is collapsed into a capital sigma.

These symbols shorten
lengthy calculations to smaller terms

that are much easier to manipulate.

Symbols can also provide
succinct instructions

about how to perform calculations.

Consider the following set
of operations on a number.

Take some number that you're thinking of,

multiply it by two,

subtract one from the result,

multiply the result of that by itself,

divide the result of that by three,

and then add one to get the final output.

Without our symbols and conventions,
we'd be faced with this block of text.

With them, we have a compact,
elegant expression.

Sometimes, as with equals,

these symbols communicate meaning
through form.

Many, however, are arbitrary.

Understanding them is a matter
of memorizing what they mean

and applying them in different contexts
until they stick, as with any language.

If we were to encounter
an alien civilization,

they'd probably have a totally
different set of symbols.

But if they think anything like us,
they'd probably have symbols.

And their symbols may even correspond
directly to ours.

They'd have their own multiplication sign,

symbol for pi,

and, of course, equals.