## Numbers

Where do you start counting from? zero? Think again! Calculating with numbers is a basic skill for everyday life. Watch these videos to see how the number system of integers (counting numbers) works.

- Multiplying numbers
- Dividing numbers
- Order of operations
- Negative numbers: addition and subraction
- Negative numbers: multiplication and division

### Multiplying numbers

How do you multiply and divide large numbers? This tutorial will teach you how to multiply (times) and use division (how many, ÷) with larger numbers of 3 digits or more.

Hi, I’m Martin Lindsay from the Study and Learning Centre at RMIT University. This is a short movie on multiplying numbers.

Let’s start by multiplying two numbers by one number, or two digits by one digit. Look at the left hand side, 52 times four. We start with the four and say four times two is eight, place the eight under the four. Then we say four times five is 20 and we place the 20 next to the eight. So 52 times four is 208. Not let’s look on the right hand side example, 49 times 8. We start as before with the eight and we say eight times nine is 72, place the two under the eight and we carry the seven, shown in red. We then go from eight to four, eight times four is 32, and we add the seven to give us 39. So our answer is 392.

Now let’s multiply two numbers by two numbers or two digits by two digits. 34 times 87. Start on the left hand side and say seven times four is 28, carry the two, shown in red at the top. Then we go from seven to three, seven times three is 21. Now we add the two shown in red to the 21, that gives us 23. So our first row is 238. Now we move to the next row, but before that we place a zero on the right hand side, shown in blue. Start our multiplication by multiplying eight by four, eight fours are 32, two carry three, shown in red at the top. Then we say eight times three is 24, plus three is 27. So our next row is 2720. We now add the two rows up, eight and zero is eight, three plus two is five, two plus seven is nine, and two. So our multiplication is equal to 2958.

Now let’s multiply number by 10, 100 and 1000. Here’s a number 52 times 10. And 52 times 10 gives us 520. The thing to notice about this multiplication is that because 10 has got one zero we simply add the one zero to the 52 to give us our answer, 520. So we’re adding the zero. What about 52 times 100? The same idea applies, 100 has got two zeros, so we add the two zeros to the 52 to give us 5200. Finally, let’s multiply 52 by 1000. 1000 has three zeros, so we add the three zeros to the right hand side, to give us 52000.

Now try some questions for yourself. The answers to these questions are on the next slide. Thanks for watching this movie.

### Dividing numbers

Hi, this is Martin Lindsay from the Study and Learning Centre at RMIT University. This is a short movie on dividing numbers.

Let’s start by dividing 74,400 by 600. You set it out in the usual way, in other words we say 600 into 74,400. Notice with this question that 600 and 74,400 both have two zeros, so we can in fact cross off the two zeros in both numbers. This simplifies matters a lot, because now what we’re doing is we’re dividing six into 744. So let’s carry out the division. We start by saying six into seven goes once, shown in red, and that gives us one left over. So we place the one next to the four. So the new number becomes 14. So we now say six into 14 goes two, remainder two. The two goes next to the four, giving us a new number of 24. So we now say six into 24 goes four times. So the answer is 124.

Now let’s divide 9,682 by 47. First of all we set it out in the usual way. We say 47 into 9,682. We now notice that there are no zeros in either number so we can’t cross off zeros, so we go ahead with the division. So we say 47 into nine won’t go, 47 into 96 goes two times, shown in red. So 47 times two is 94. We place the 94 under the 96 and subtract. That gives us two left over. We then bring the eight down as shown with the arrow alongside the two, giving us a number 28. So we now say 47 into 28 won’t go, so we put the zero in red at the top, then we bring down the two shown with the red arrow. So we now have 282. So we divide 47 into 282, and that goes exactly six times. Six times 47 is 282. Do the subtraction, as long as you get zero left over, that’s the end of the division. So the answer is 206 times.

Let’s look at our final example, 12,065 divided by 19. Let’s rewrite that as 19 into 12,065. So we now carry out the long division. Notice there are no zeros at the end so we have to carry on as normal. 19 into one won’t go, 19 into 12 won’t go, 19 into 120 will go six times. So we say six times 19 is 114. Do the subtraction, 120 take away 114 gives us six. Bring the six down shown with the red arrow, so the new number is now 66. So we now say 19 into 66 goes three times, shown in red. 19 times three is 57, write the 57 under the 66, subtract, that gives us nine left over. Now we bring down the five, shown with the red arrows, to give us our new number of 95. Carry out the long division, 19 into 95 goes five times, five 19s are 95. Carry out the subtraction, that gives us zero. In other words, the answer is 635.

Now carry out some division questions for yourself. The answers to these questions are on the next slide. Thanks for watching this movie.

### Order of operations

What is BODMAS or BIDMAS? That is another name for ‘Order of Operations’. These are the rules about which operation (+ - x or ÷) takes priority when you have more than one in a mathematical expression. Watch this video to see how to apply these rules.

Hi, I’m Martin Lindsay from the Study and Learning Centre at RMIT University. This is a short movie on order of operations.

When we work through a maths problem we have to be aware of following a set of rules. These rules are called order of operations. The first thing to look out for are brackets and if a problem’s got brackets in it, then what’s inside those brackets must be done first. Secondly, if the problem has got multiplication and division, then they must be done next and we carry out the operation from left to right as we work through the problem. And finally, if there are adding and subtracting parts to the problem they must be done at the end, and again they must be done from left to right.

Let’s do an example. Here’s the question two plus seven minus six divided by two times three. Notice there are no brackets so we now move down to the second level, which is multiplication and division. Here we have six divided by two times three. So this part of the question must be done first, six divided by two is three, times three. Notice we’re moving from left to right, and finally we multiply three by three which gives us nine. So we end up with two plus seven minus nine. Now we have just adding and subtracting to do, which is the last operations. Moving from left to right, two plus seven is nine, then we take away nine and finally we have our answer, zero. So the answer to this question is zero.

Let’s look at this problem. 16 divided by eight plus four times brackets seven minus three close brackets. Notice here we have brackets, so that’s the first operation that we carry out. So seven take away three is worked out to give us four. Now we have 16 divided by eight plus four times four. The division and multiplication must be carried out next, and we move from left to right, 16 divided by eight is two plus four times four is 16. So we have an answer of 18.

Now do some questions for yourself. The answers to these questions are on the next slide. Thanks for watching this short movie.

### Negative numbers: addition and subraction

How do you minus a negative? Ideas about profit / loss and many STEM areas need an understanding of adding and subtracting positive and negative numbers. Watch this video to get the concepts.

Hi, I’m Martin Lindsay from the Study and Learning Centre at RMIT University. This is a short movie on adding and subtracting negative numbers.

Let’s start by adding negative numbers. Here’s the example. Plus two minus four, I place the number line at the bottom of the screen, zero at the centre, numbers to the right are positive; numbers to the left are negative. So let’s start with a plus two, plus two means moving two units to the right on the number line from zero, so now we’re at positive two. Now we have to take away four, so starting at the two we move back four places because it’s negative. Notice the answer here will be from where we started to where we finished, in other words from zero to negative two. In other words plus two, take away four, is equal to minus two.

Let’s do another one. Minus three plus six, notice I’m starting with a negative here, so by going negative I’m moving to the left from the number line, from zero, so I’m now at negative three. Now I have to plus six, being a plus I have to move to the right six units, so starting at negative three, moving six places to the right we end up at positive three. So again the answer is from where we started, which is a zero, to where we finished, which in this place is positive three, so our answer is positive three. So negative three plus six is positive three.

Now let’s add two negative numbers. Negative one, negative two, so because they’re all negative I’m moving to the left on the number line, negative one and then negative two and the answer is from where we start to where we finish, which will be negative three, so negative one negative two is equal to negative three.

Let’s look at another one. Here we’re starting with plus one minus negative four. Notice again that the two negatives are next to each other so when they’re next to each other we add, so this problem becomes positive one plus four, which is quite a simple operation to do on the number line, plus one is moving one place to the right, adding four will be adding four more, giving us an answer of five. In other words, positive one minus negative four is positive five.

Now try some questions for yourselves. The answers to these questions are on the next slide. Thank you for watching this short movie.

### Negative numbers: multiplication and division

How do you multiply and divide negative numbers? There are rules when multiplying and dividing negative numbers. The numbers work the same way but the sign may change. Watch this video to understand how.

Hi, I’m Martin Lindsay from the Study and Learning Centre at RMIT University. This is a short movie on multiplying and dividing positive and negative numbers.

Let’s multiply positive and negative numbers first. At the table red pluses and minuses are what we start with, the blue pluses and minuses are the answers. So for instance if I multiply a plus by a plus I trace down from the plus in the top row and I trace across from the plus in the left hand column. Where they intersect is my answer, in other words, a plus times a plus is a plus shown in blue. Similarly do the same thing with the negative signs, the negative in the top row, trace it down, the negative in the left hand column trace it across, where they intersect is the answer and as you can see the answer is a plus. In other words if the signs are the same the answer is positive.

Now let’s look at the other signs. Let’s multiply a plus by a minus. So we trace down from the plus in the top and we trace across from the minus in the left hand column, where they intersect is our answer in blue and as you can see they intersect at the negative sign. Similarly if I multiply a minus by a plus a minus in the top row by a plus in the left hand column, trace the minus down from the top and the plus sign from the left across until they intersect, where they intersect is the answer and as you notice it’ll be a negative sign again. In other words when the signs are different, in other words plus and minus, the answer is negative.

So let’s look at this in more detail. Let’s go through each one step by step. The first one is positive by positive is equal to a positive. Plus times plus equals plus. Here’s an example; plus four times plus four, plus times plus gives us a positive answer, four times seven is 28, in other words the answer is plus 28. So let’s look at this in more detail. Let’s go through each one step by step, the first one is positive by positive is equal to a positive, plus times plus. Now let’s look at negative times negative, minus times a minus is a plus. [Unclear] minus five times minus nine, two negatives gives us a positive answer, notice if the signs are the same the answer is always positive, so minus times a minus is a plus, five times nine is 45, so our answer is plus 45.

Now we have negative times a positive is a negative, minus times a plus is a minus, notice the signs are different the answer is always negative, so with an example, minus six times plus 11, the signs are different so the answer is negative, six times 11 is 66 so the answer is minus 66.

Finally positive times a negative is a negative. Plus times a minus is a minus. The signs are different therefore the answer is always negative. Example, plus 12 times minus seven, the signs are different so the answer is negative, 12 sevens are 84 so the answer is negative 84.

Now let’s divide positive and negative numbers. Look at the table, and look again at the similarity between this table and the table we had for multiplication. For instance if the signs are the same the answer is positive, if the signs are different the answer is negative. In other words the rules that we use for division are exactly the same as the rules that we use for multiplication.

So let’s look at them one by one. Plus divided by a plus is a plus, 36 divided by nine is the same as positive 36 divided by positive nine which would give us an answer of positive four. Negative divided by a negative is a positive, minus divided by a minus is a plus, notice if the signs are the same the answer is always positive. So for instance minus 56 divided by minus eight, the signs are the same, the answer must be positive, 56 divided by eight is seven so our answer is plus seven.

Now let’s look at negative divided by positive, the answer is negative because if the signs are different the answer is always negative. For instance minus 78 divided by plus three, minus divided by a plus is a negative, 78 divided by three is 26, so the answer is negative 26. Finally positive divided by a negative is a negative, plus divided by minus is a minus. Example, plus 96 divided by minus six, the signs are different, therefore the answer must be negative, 96 divided by negative six is 16, so the answer is negative 16.

Now do some questions for yourself. The answers to these questions are on the next slide. Thanks for watching this short movie.