Lesson Video: Order of Operations: Negative Numbers | Nagwa Lesson Video: Order of Operations: Negative Numbers | Nagwa

# Lesson Video: Order of Operations: Negative Numbers Mathematics

In this video, we will learn how to evaluate numerical expressions involving negative numbers using the order of operations.

14:45

### Video Transcript

In this video, we’ll learn how to evaluate numerical expressions that involve negative numbers using the order of operations.

Let’s begin by recalling some of the key facts about operations on negative numbers, firstly, the rules for addition and subtraction. Adding a negative number is the same as subtracting a positive number. So for example, five plus negative seven is the same as five minus seven. It’s negative two. We say that subtracting a negative number is the same as adding a positive. So five minus negative seven is the same as five plus seven, which is 12.

And then, we move on to the rules for multiplication and division. Multiplying or dividing a positive number and a negative number gives a negative answer. So for example, five multiplied by negative seven gives negative 35, as does negative five multiplied by seven. And finally, multiplying or dividing two negative numbers, so multiplying a negative by a negative, gives a positive result. Negative five multiplied by negative seven is positive 35.

And next, we’ll recall what we mean by the order of operations. This is a way of ensuring that everyone gets the same answer when they perform a calculation. The acronyms PEMDAS or BIDMAS tell us the order in which to perform each part of the calculation. The first letter stands for parentheses or brackets. Whatever’s inside any set of parentheses is the first calculation we perform. Next is exponents or indices. We evaluate any numbers raised to a power. We then have multiplication and division. Now, these have the same weight, the same importance. So if there are two of these in the same sum, we go from left to right.

Finally, addition and subtraction, again, these have the same precedence. So if there are two of them in the same sum, we go from left to right.

Let’s have a look at how we can combine all this information to perform the order of operations on negative numbers.

Fill in the blank. Negative three minus two minus three is equal to what.

We’re looking to evaluate this number sentence. Whenever we’re performing calculations like this, we should always be considering the order of operations. Sometimes abbreviated to PEMDAS or BIDMAS, it’s a way of ensuring that everyone who performs this calculation gets the same answer. The first letter stands for parentheses or in BIDMAS brackets. We perform any calculations inside the pair of parentheses first. Note that when performing these calculations, we also use the order of operations on any problems inside the parentheses. In this case, we have negative three minus two. And we might use a number line to evaluate this. We start at negative three on the number line. Subtracting two means we move two down the number line in the negative direction. So that’s one, two, which leaves us at negative five. So negative three minus two is negative five.

The next part of our acronyms tell us to evaluate any exponents or indices. And in fact, our calculation doesn’t include any of these. We then perform any multiplication or division. Again, there are none of these in the calculation. There is, however, some further subtraction to perform. We replace negative three minus two, the bit we evaluated before, with negative five. Remember, we keep the calculation otherwise in the same order. So we’re going to evaluate negative five minus three.

Let’s go back to our number line. This time we start at negative five. And we move one, two, three places further down the number line. That takes us to negative eight. And so, the blank that is the solution to negative three minus two minus three is negative eight.

Now in fact in this question, we really didn’t need that pair of parentheses. We had only subtractions in our sum. And we know that when performing addition and subtraction in the same calculation, we simply go from left to right. So either way, we would have began by calculating negative three minus two and then subtracted three from that.

In our next calculation, we’ll look at performing some multiplication.

Calculate negative 19 multiplied by 19 multiplied by negative five multiplied by negative two.

In order to evaluate a calculation like this, we need to consider the order of operations. Remember, abbreviated often to PEMDAS or BIDMAS, this is a way of ensuring that everyone who performs this calculation gets the same answer. Each letter stands for a certain operation. We begin with P that stands for parentheses or in BIDMAS stands for brackets. We perform any calculations inside our bracket. Now, in this case, we have negative 19, negative five, and negative two all inside a pair of parentheses. This time, that doesn’t tell us to perform the calculation. There is no calculation to perform. It’s just a way of highlighting that we’re actually working with negative numbers.

Next, we have exponents or indices. This is followed by multiplication and division. Now, multiplication and division have the same precedence. So, when they’re performed in the same calculation, we simply move from left to right. The last two letters stand for addition and subtraction. And similarly, they have the same importance. So when we have them in the same calculation, we move from left to right.

Now, if we look at our calculation, we see we only have multiplication to perform. So we could move from left to right. But there’s an extra fact that can be useful. And that is multiplication is commutative. It can be done in any order. Now, were there any other operations in this problem, we wouldn’t be able to use this fact. But here, we can. And so what we’re going to do is we’re going to calculate negative 19 times 19 and negative five times two. Now the reason this is useful is because we know that five times two is 10. And it’s quite straightforward to multiply by 10. We also know that a negative number multiplied by another negative number gives a positive. So negative five times negative two is also positive 10.

Next, we’ll calculate negative 19 times 19. Well, let’s begin by calculating 19 times 19. And we use the grid method. 10 times 10 is 100, and nine times 10 is 90. Finally, nine times nine is 81. 19 multiplied by 19 is the sum of these four values; that’s 361. This means 19 multiplied by 19 is 361. We know that a negative multiplied by a positive gives a negative result. So negative 19 times 19 is negative 361. We calculated negative 19 times 19 to be negative 361 and negative five times negative two to be 10. So we’re now going to calculate negative 361 times 10. We know that to multiply by 10, we move the digits to the left one space. And when we do, we create a gap which we fill with the zero. So negative 361 times 10 is negative 3610. So the answer to negative 19 times 19 times negative five times negative two is negative 3610.

Evaluate four minus 13 plus negative 19.

Whenever we’re looking to evaluate a number sentence, we need to be thinking about the order of operations. Sometimes shortened to PEMDAS or BIDMAS, this is a way of ensuring that everyone who performs that calculation gets the same result. The first letter stands for parentheses or brackets. We evaluate any sum inside the parentheses. Here, that’s 13 plus negative 19.

Now, we mustn’t be thrown off that negative 19 is itself inside a pair of parentheses. In this case, it’s not a calculation; it’s just showing us that we indeed are working with a negative number. So, we calculate 13 plus negative 19. Adding a negative is the same as subtracting a positive. So 13 add negative 19 is the same as 13 minus 19. And that’s negative six. We replace the expression inside our parentheses with negative six, ensuring that we keep our question in the same order. And it becomes four minus negative six.

In fact, there is only now one operation left to perform. There are no exponents or indices nor multiplication or division. We just have a subtraction. We next recall that subtracting a negative is the same as adding a positive. So four minus negative six is the same as four plus six, which is 10. So four minus 13 plus negative 19 is 10.

In our next example, we look to include a few more operations.

Calculate 21 minus negative 19 divided by five times three.

Whenever we’re presented with a number sentence that we’re looking to evaluate the answer to, we need to be thinking about the order of operations. You might know that as PEMDAS or BIDMAS, where each letter tells us the operation that we do and in what order. So let’s recall what the letters stand for. The first letter stands for parentheses or brackets. We perform any calculations inside a pair of parentheses. Now, if there is quite a complicated problem inside those parentheses, we once again look to perform the order of operations on that bit itself.

Now in this case, we simply have 21 minus negative 19. And don’t be thrown off that negative 19 is itself inside a pair of parentheses. This is just a way of highlighting the fact that negative 19 is a negative number. And so, let’s calculate 21 minus negative 19. We know that subtracting a negative number is the same as adding a positive. So this is the same as 21 plus 19, which is 40. We now replace that expression, 21 minus negative 19, with 40 in our calculation. It’s important we keep the order the same, so we wouldn’t want to add 40 at the end. It needs to be 40 divided by five times three.

Let’s go back to PEMDAS and BIDMAS. E and I stand for indices or exponents, in other words, finding a power of a number. Well, we have no powers here. The next letters stand for multiply and divide. Now, if we have more than one multiplication or division or multiplication and division in the same calculation, we remember that they have the same precedence or the same importance. And we simply move from left to right. We’ll start by calculating 40 divided by five; that’s eight. And so, our calculation is now eight times three. Eight times three is of course 24. And so 21 minus negative 19 divided by five times three is 24.

In our very final example, we’ll look at some common misconceptions on a problem involving indices.

Calculate three times negative four squared minus seven plus negative two.

When presented with a problem like this, we need to think about the order of operations. We often use the acronyms PEMDAS or BIDMAS here, where each letter in the acronym tells us the order in which we perform the calculations. The first letter P or B stands for parentheses or brackets, respectively. We begin by performing any calculation inside a pair of parentheses. Here, that’s seven plus negative two. Now of course, adding a negative is the same as subtracting a positive. So seven plus negative two is just the same as seven minus two. And that of course is equal to five. We therefore replace seven plus negative two with five in our calculation. And it’s now three times negative four squared minus five.

The next letter E or I stands for exponents or indices, respectively, in other words, any power. Well, we do have one of these. We have negative four squared. So let’s evaluate negative four squared. A common misconception is to think that negative four squared is negative 16. But we know that squaring a number is the same as timesing it by itself. So negative four squared is negative four times negative four. And since multiplying a negative number by another negative number, we get positive 16. So we replace negative four squared with 16. And our calculation is now three times 16 minus five.

The next two letters stand for multiplication and division. Now, multiplication and division take equal precedence. So if more than one appears in the same calculation, we move from left to right. In fact, we only have one multiplication here. It’s three times 16. Three multiplied by 16 is 48. So we replace three times 16 with 48. And our calculation is now 48 minus five. Now another common mistake here is to switch the order of the calculation and to think that we need to perform five minus 48. In fact, it’s really important that we maintain the order of the calculation.

Our final two letters A and S stand for addition and subtraction. But actually, that’s all that’s left to perform anyway. We need to work out 48 minus five. It’s 43. And so three times negative four squared minus seven plus negative two is 43.

In this video, we’ve learned that we need to apply the order of operations to all calculations including those involving negative numbers. And when we do so, we recall the rules for multiplying, dividing, adding, and subtracting with negative numbers. We saw that the acronyms PEMDAS or BIDMAS can help us to remember the order in which to perform the calculations, where P or B stands for parentheses or brackets, respectively. E and I stand for exponents or indices. The next two letters tell us to multiply or divide. And our final two letters tell us to add or subtract.

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