Lesson 2.7: Operator Precedence - MATLAB Video Lecture | Introduction to Programming with MATLAB - IT & Software

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We've seen many of MATLAB's operators now, but
we haven't yet
seen how to determine the order in
which they're applied in an expression.
Well, the order is determined by two sets
of rules: precedence and associativity.
We'll look at precedence first.
We ran into precedence earlier in this
lesson when we used both transposition and
the colon operator in the same expression.
We had a surprise then.
You may remember it.
The transposition operator, which was
written after the colon operator,
was actually applied
before the colon operator.
It just transposed the 5,
which did nothing.
Transposition went first, because it
ranks higher than the colon operator.
To make the colon go first,
we had to add parenthesis, like this.
Those parenthesis force the colon
operator to make the row vector first, so
the transposition operator could
then make it into a column vector.
Well let's look at a more familiar
example, one involving arithmetic.
Here the multiplication operator is
written after the addition operator, but
it's applied before the addition operator.
So 3 is multiplied times
4 first to get 12.
And then 2 is added to get 14.
It's not the left to right
order shown on the right,
which would result in 5 times 4 equals 20.
Here again, the non-left to right order
happens because the operator on the right
outranks the one on the left.
This time it's multiplication,
which is ranked higher than addition,
and it goes first.
In programming,
we call this ranking “precedence” and
we say that multiplication has
a higher precedence than addition.
Having multiplication go before addition
seems like the obvious thing to do because
that's the way we've always done it.
With other operators,
it's not always so obvious.
So, how do we find out the order?
Well, we use a precedence table.
This is a precedence table for all the
other operators that we've covered so far.
Starting in the top, we find the
parentheses, which have a precedence of 0.
Next we have exponentiation and
transposition, which both
have a precedence of 1.
Before we look further, it's
important to note that lower numbers
actually mean higher precedence.
So parentheses with their
0 precedence always
operate before all the other
operators in an expression.
Coming in at 3, both multiplication and
division come before addition and
subtraction, which are at 4.
Down at the very bottom is
the poor old colon operator,
which explains why it
came after the transpose.
At the number 2 position,
you may be wondering what “unary plus” and
“unary minus” are.
Well “unary” means that the operation
involves only one operand.
Here's some examples of unary operators,
up there at the top.
Plus 2, -x, and H transpose.
Unary operators tend to rank high
in any programming language, and
in MATLAB they're at positions 1 and 2.
But the more common operators
are the binary operators,
which means that they take two operands,
and here you see some examples.
3 minus 4, X dot star Y, which, you
will remember, is array multiplication.
X star Y,
which is matrix multiplication, and
A to the 3rd power,
which is exponentiation.
The binary operators rank below the unary
operators with the notable exception of
exponentiation which is way up there at 1.
Well we've seen how precedence works.
And we've seen that unary operators
tend to come before binary operators.
But there's one other determinate
of order of operations.
Associativity.
Associativity is simpler
than precedence, and
we'll start with a very simple example:
6 plus 2 plus 3.
A computer can carry out only
one operation at a time, so
it adds the 6 to the 2 to get 8, and
then it adds the 3 to 8 to get 11.
The order is left to right, simple.
But what about this, 6 minus 2 plus 3.
Which goes first?
The minus or the plus?
We need to know because the results
will be different, 7 or 1.
Well, maybe the precedence
table will give us the answer.
Hm, no help.
They both have the same precedence, 4, so
here comes associativity to the rescue.
In programming,
the order in which operators of the same
precedents are executed
is called “associativity”.
In MATLAB, it is always left-to-right.
So we say that MATLAB uses
left-to-right associativity.
As a matter of fact, so
most all other languages do.
So this expression evaluates to 7.
Here are a couple of other examples.
6 divided by 2 equals 3.
3 times 3 gives 9.
Both of these operators
have a precedence of 3.
In the last expression,
6 to the 2 power is 36,
and 36 to the 3 power, well,
we'll have to ask MATLAB.
Whatever it is, I'm sure it's a pretty
big number, so let's do that, in fact.
Let's ask MATLAB to evaluate some
expressions that require it to use
these rules.
Okay, let's start with the first
example we did to illustrate precedence.
2 plus 3 times 4.
As promised,
multiplication has the higher precedence,
so the actual order is like this.
[CLICKING] The redundant parentheses
show that 3 times 4 goes first.
That gives 12, and
we add 2 to 12 and get 14.
And if we force addition to
go first with parentheses,
[CLICKING] we get a different result.
2 plus 3 equals 5, and 5 times 4 gives 20.
As I've mentioned before, we're used
to carrying out multiplication before
addition, even when addition is
written first in the expression.
Let's look at something that's not so
familiar, the colon
operator with addition.
So I'm going to type 1 colon 3 plus 10.
Surprised?
Well, remember that the plural colon
operator has a lower precedence than all
of the arithmetic operators.
To be specific, the colon is at 5 while
addition and subtraction are at 4.
And smaller numbers go first.
So, this is what we did.
[CLICKING] The 3 plus 10 went first,
so we're given the colon
operator 1 colon 13.
If you want to add the 10 after the colon
operator has made its row vector,
you need to force it with
parentheses like this.
[CLICKING] There, we got 1, 2, 3, and
then we added 10 to each element,
and got 11, 12, 13.
And by the way, note that we're
adding a scaler to a row vector.
So this is an example of
a scalar being added to a matrix.
In this case, the matrix has just one row.
But as for any matrix, the scalar
is added to each of its elements.
If you don't know which operator
goes first, you have three options.
One, use parenthesis to force the order.
Two, try an example and learn the order.
Or three, look at MATLAB's
precedence table and read the order.
So, where is this precedence table?
Well, it's right at your fingertips.
Just type help precedence.
[CLICKING] And
there's MATLAB's precedence table.
You'll note that there's
no parenthesis here.
Well MATLAB just figures it's easy for
you to remember that
parenthesis always go first.
We showed that they always go first
by giving them a precedence of 0.
You'll also note that there are some
operators here that we haven't
looked at yet.
Well, we'll get into
them in later lessons.
Now, let's look at some
examples of associativity.
Once again, we'll start with the same
example that we used when we introduced
the subject: 6 plus 2 plus 3.
The actual order is like this,
6 plus 2 in parenthesis plus 3.
Because as we've seen,
MATLAB always evaluates from
left to right when the operators
have the same precedence.
And of course, since we have two pluses,
they have to have the same precedence.
But you can't tell with addition,
if we force right to left order
with parenthesis like this,
[CLICKING] we get the same result,
as you knew we would.
When the right-to-left order gives the
same result as the left-to-right order,
the operator is said to be “associative” in
both mathematics and computer science.
So addition is associative.
[CLICKING].
But what about this?
[CLICKING]
The actual
order [CLICKING]
is like this.
The 6 minus 2 goes first,
because at the risk of being boring, or
perhaps I should say even more boring,
MATLAB always evaluates
from left to right when the operators
have the same precedence.
If we force right-to-left order
with parentheses like this,
[CLICKING].
the result is different.
So subtraction is not
an associative operation.
And when we mix subtraction and addition,
which have the same precedence, like this,
[CLICKING] since they have
the same precedence,
the order is, well, you know.
I just can't bear to say it again.
Now let's look at multiplication and
division
with this example, which we've seen before.
[CLICKING] And let's use a pair of redundant
parenthesis to make the order explicit.
[CLICKING] This one sometimes
surprises people because
it might look like
multiplication should go first.
It doesn't.
That's because division and
multiplication have the same precedence.
So the order is determined by MATLAB's
left to right associativity rule.
Oops, said it again.
If we force right to left
order with the parenthesis,
[CLICKING] the result is of course, different.
And finally, I sense that you are burning
with curiosity to find out what 6 to
the 2 to the 3 equals.
Well, as promised,
I'm going to ask MATLAB right now.
6 to the 2 to the 3.
[CLICKING] Since the left hand carat goes
first, this is 36 to the 3rd power.
And as I guessed before,
it's a pretty big number.
46,656.
But just out of curiosity,
what would the answer be if the 2 were
raised to the 3 power first and
then 6 were raised to that power?
Well, we'll have to use
parentheses to find out.
6 to the 2 to the 3 power.
[CLICKING] The result is different,
which shows that exponentiation
is not associative, and
the result is a really big number.
Let's say, that's 1,679,616.
Kind of amazing you can get such a big
number with just a 6, a 2, and a 3.
Well, I guess I'd have to say
that's the power of exponentiation.
Get it?
The power of exponentiation.
Well, on that hilarious note,
we've reached the end of lesson two.
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