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.