www.mathcentre.ac.uk/.../mathematical%20language.mp4

Mathematics has its own
language, most of which which

were very familiar with. For
example, the digits 012.

345678 and nine are part
of our everyday lives, and

whether we refer to this
digit as zero.

Nothing.

Note
Oh oh, as in the

telephone code 0191 we
understand its meaning.

There are many symbols in
mathematics and most of them are

used as a precise form of

shorthand. We need to be
confident using the symbols and

to gain that confidence, we need
to understand their meaning.

To understand that meaning,
we've got two things to help us,

there's context. What else is
with the symbol? What's with it?

What's it all about?

And this convention and
convention is where

Mathematicians and
scientists have decided that

particular symbols will
represent particular things.

Let's have a look at some

symbols. That sign words
associated with it

a plus. That

Increase.

And positive. As
it stands, it has some form of

meaning, but we really need it
in a context. For example, 2 + 3

or two ad three. We know the
context, we add the two numbers

and we get 5.

Let's have a look at another
context. Now, if you've gone

abroad, say to Europe, and you'd
gone to a conference, you might

hand out some business cards,
and on your business card so

that people could contact you.
You might have your telephone

number, so chances are it will
be written as plus 44, and let's

have a phone number 1911234567.

And in this context, that plus
means that the person dials the

necessary code for an
international line from their

country, then 441911234567.

So it still means In addition
to, but not in the same way as

that plus sign were not actually
adding the number on. We're

doing it In addition.

Let's have a look

now. At this
sign.

Well, it's associated with this
one and minus.

Subtract. Take

away.

Negative.

And decrease.
And again, we need a context.

Let's six takeaway four or six.
Subtract 4 and we all know the

answer to be 2.

In a different context, if we
have minus 5 degrees C for

example. That means a
temperature of five

degrees below 0.

How about this one?

Words associated with
this symbol multiply.

Lots of. Times Now
this one is really just a

shorthand for adding.

Let's have a look at the
context. For example, six at 6,

at 6, at 6 at 6.

What we've actually got a

123456 is. And in short
hand we want to write 5 sixes.

On the symbol that's used is the
multiply sign, so we've got five

lots of 6 or 5 * 6.

Now this shorthand becomes even
shorter when we use some letters

instead of numbers.

So for example, instead of six,
let's say we were adding AA

again, will have 5 days.

But because we've used the
letter, if we use the multiply.

It could be confused
with the letter X.

So we don't write it in.
Basically we miss it out and we

just write 5, eh? So the
shorthand becomes shorter, so

when letters are involved, the
multiply sign is missed out is

the only symbol that we miss

out. Let's look
at division.

Now this symbol can actually be
written in three different ways.

We could have 10 / 5.

10 / 5 as it's written in a
fraction or 10 with a slash

and five now this ones come
about mainly because of

printing and typing and using
a computer. When these

symbols are not readily
available on the keyboard.

So how many times does 5 go
into 10 or 10 / 5?

Another very common symbol that
we use all the time without

thinking is the equals sign.

Again, it doesn't mean anything
on its own. We need it in some

form of context. So if we have
the sum 1 + 2 equals 3.

What we're saying is whatever we
have on this side is exactly

equal to. Whatever we have on

this side. Or it's the same as?

Variations on the

equals sign. Of the
not equal sign, a line through

the equals which is not equal

to. For example, X is not
equal to two.

So we know that X does not take
that value of two.

Two WAVY lines without
equals means approximately

equal to.

So it may not be
exact, but it's approximately

equal to that value.

Are greater than
or equal sign?

An example here
might be X

is greater than
or equal to

two. That means X could take the
value of two, but it could be

any number larger than two.

And are less

than. Or equal
2.

I'm here for example. Why would
be less than or equal to 7?

Why could equals 7? But it could
be any number less than 7.

And an easy way of remembering
which is the greater and the

less than part is the greater
is the wider part of the

symbol, whatever's on the
wider part of the symbol,

whatever's on that side is the
greater than the one where the

point is on the other side.

Other forms
of mathematical

symbols are

variables. I'm
basing used where things

take different values.

For example, imagine your
journey to work in your car.

And imagine the speed that
you're traveling at as you go

along your journey, your
speed will change, so we

might refer to the speed with
a letter. Let's say the

letter V, because throughout
your journey the actual

numerical value is changing.

We usually use letters for
variables, letters of the

alphabet. You can call them

anything. But we try to use
letters of the alphabet and we

try to have those letters giving
us some indication of what our

variable is all about.

For example, we might use
deep for distance.

And T for time.

By convention, we use you to
be initial speed.

And V to
be our final

speed. But of course, we
might also refer to volume.

And This is why context is
important and we look to see

what a variable is with as to
what it might mean. So, for

example, if we were to CV is D
divided by T.

And we're told that Diaz
distance and T is time. Then we

know that that V is speed.

Whereas if we saw V
with Four Thirds Paillard cubed.

Where are is the radius of a
sphere. We know that that V is

also the volume of a sphere.

Now going back to our example
again of our journey to work, we

might want to record the time it
took us to go to work, and we

might want to do that over a
number of days.

The most sensible variable to
use would be a T for our time.

But of course, if we want to
record it for each day, how do

we distinguish between the TS?

Well, in this case we
use a subscript.

So for day

one. We might

use T1. The
next day to

223-2425. Or we might want
to be a little bit clearer and

actually put perhaps TM for the
time on Monday TT for Tuesday

TW. For Wednesday we would have
to do TH for Thursday so it

doesn't get confused with
Tuesday and TF for Friday, so we

can use these subscripts small
numbers or letters at the bottom

right of the variable to
distinguish between them.

Now, by convention,
mathematicians have decided that

we're going to use some
letters of the Greek alphabet

for some of our mathematical

symbols. For example,
we have pie.

Empires being chosen to equal
the number 3.14159 and it

goes on and on forever,
never repeats itself.

And so that we can precisely say
what we mean, rather than having

to round the value.

We use the letter Pi knowing
that it's that number.

From the Greek alphabet we also
use some other letters such as

Alpha. 's

pizza. And

feature. Now, these
are often used as variables to

represent angles. So you might
see an angle marked in that way,

and the Greek letter Theta.

Another one that is commonly
used is a capital Sigma.

Now you should be familiar with
this one from any spreadsheet

program on a computer, because
you'll find it somewhere on the

toolbar because it means the sum

of. And it's a shorthand for
adding up a column or a row of

numbers on a spreadsheet. So it
means the sum of.

The positioning of where we put
letters or figures without

symbols. Has gives meaning to
it, just like our subscripts we

can have superscripts now

superscripts. Instead of going
at the bottom right, go at the

top right. So for example, for
the little two at the top right

hand side, that's a superscript.

And again, it's a shorthand as
most of our symbols are and that

means 4 squared.

4 *
4.

If we have 4 cubed.

We've got three of them,
4 * 4 * 4.

Alright, a number that means to
the power of.

So we've raised 4 to the power
of three 4 * 4 * 4.

32

With the 0 as a superscript now
this is actually got two

meanings. And we need the
context to be able to know which

meaning it is. It could
be 32 degrees.

And that would mean an angle.

Of 32 degrees.

But it could mean 32.

To the power.

Of 0.

Which is actually one.

Which are very different meaning
to an angle. So you need to know

the context to be able to decide
which one it means.

Staying without superscript,
we could have 32 degrees

again, but this time with a
capital C after it, and that

would mean a temperature of
32 degrees Celsius.

Let's have a look at a couple of

numbers now. It could be 6,

three. Or it could be 63.

Let's put a little bit more

context there. If I put a comma

between them. Then I know it's
not 63, but I'm still not sure

what the meaning is. It could be
a number of things, but if I

then put brackets around it, I
know straight away that it's a

pair of coordinates.

And what it represents is
the position 6, three on

my coordinate axis.

So I would go six
in the X Direction, 3

in the Y direction and
that is the position that

that coordinate is representing.

Now brackets may be used in a

different way. If for example,
I saw a pee and

brackets H equals 1/2.

I know that's actually nothing
to do with coordinates at all,

and that is most likely to be

probability. And that is
saying the probability of

event H happening.

Is equal to 1/2.

I don't know what that event is,

I could. Surmise that it
could be the probability,

perhaps of scoring ahead when
I toss a coin is equal to

half, but I don't know I'd
need more information, but

certainly that it's something
to do with probability.

Another symbol we use.

The percentage symbol.

This is familiar to us because
here we've got our divide sign.

And this means out

of 100. So if we
have for example 90%, it means

90 out of 100.

Let's look at
a few more

symbols, for example.

R square root sign.

An example here might be the
square root of 16.

And it's the number that when we
multiply it by itself gives us

16. So the answer could be full.

Or it could be minus four,
because minus four times minus

four gives us 16.

Now, often in printed material,
you might just see the square

root is just the tick without
the line across the top.

If you're writing it.

It's clearer if you put the line
across the top to include, so

it's clear what you're including
in that square root sign. But be

aware that you might just see

the tick sign. Another symbol
that's used as an X with a bar

across the top and it said X

Bar. And it means

the mean. So it's the mean
of a set of numbers. Instead

of writing the word mean, we
write X Bar.

Another symbol, let's say we
have one point 3.

And we see a dot over the three.

And this is our recurring

decimal sign. And what that
means is we've got 1.3 and

three 333 and it goes on
forever and ever reccuring, so

the dot over the decimal place
means it goes on forever.

We might have 1.317
for example.

And dots over the three on the

7. Similarly, it means it goes

on forever. But what it means
this time slightly differently?

Is this the 317 that's repeated
and goes on forever?

In summary, mathematical
symbols are precise

form of shorthand.

They have to have meaning for
you. You need to understand them

and to help with that
understanding. You have context.

What else is with them and you
have convention which is what

mathematicians and scientists
have decided. Certain symbols

will represent. And that's
all you need to know.