Addition

We call the process of totalling items “addition” and we use this to group together a number of items which we can represent by one number called “the total”. For example, if we have 3 of something and we then receive another 4 of something, we can at these together to make a total of 7 of something:

So for example, if initially we had 3 books but then we buy a further 4, we can say that in total we have 7 books:

we show this operation of “addition” by the simple mathematical expression shown below:

Where the plus symbol represents addition and the equal symbol represents the equality, in other words (as you will see much later in the book) the information on the one side of the = is equal to the information on the other, that is indeed 3 + 4 does equal 7!

Because a book is a whole item, that is you are unlikely to buy a fraction of a book, we are in fact dealing with “integer addition” which is the simplest addition we can have. Things start to become a little bit trickier when we are looking at negative numbers, or numbers with decimal or fractional parts. We will of course come to these in due course.

We are taught in primary school to add numbers together by putting one on top of the other with the + at the side, and a line underneath the bottom number where we would write our answer (in fact when I was at primary school we were taught to put two lines at the bottom to make a box without 2 of the sides, and the answer would be written between them):

In this example we are simply adding 3 to 5 to make a total of 8, as these are all “units” they sit underneath each other because the answer does not exceed 9 we don’t have to start on the next column to the left which would be the tens. In the next example we will create an addition which will go above 10 and I will show you what we have to do then.

Of course you’ve probably worked out that HTU means “hundreds, tens and units”

Okay, let’s look at another example where the answer will put us into the tens column as well:

This one is a bit more complicated and I have drawn a few lines all over it. We perform additions using the units first then we move across to the tens when our number of units is bigger than 9, and likewise when our number of tens is bigger than 9 we move across to the hundreds, and so on.

5+7 = 12 but we can’t simply write 12 in the box, because 12 represents one 10 and 2 units, so what we have to do is write the units in the box under the units column and write 10 just outside the box underneath under the tens column (we do this because we have to add it to the tens and putting it underneath the tens outside the box means that we won’t forget it) I have coloured it red to make it standout, and also to indicate that it is a by-product of the addition of the units column which has a red line through it. Taking this 10 and putting it outside the box ready to be added to the tens is a process called “carrying” and this particular example means we have added 2 to the units box but we have carried the 10.

Going back to the top of the sum, and now moving across to the tens column, you can see that we have one 10 but below it there is nothing to add, so we prepare to move it into the answer box, however we have carried 10 from our previous operation so we add it to the 1 making 2. There is nothing in the hundreds column so we have in fact finished our sum and can see that 15+7 is indeed 22.

The next example now to show you will involve some hundreds, but the principle is exactly the same, we start with units, then we move on to tens, then finally we move on to hundreds, remembering to “carry” where necessary from one column to the next. Here is our next example:

This time we have “carried” twice, and the carried numbers are coloured blue and red to indicate which columns they came from. In just the same way as before we first of all add up the units column:

We put the unit into the units column (below the number 6 in 96) and we carry the 10 which is coloured in red underneath outside the box and under the tens column. Next, we add up the tens column:

Remember though we have one extra 10 carried from the previous operation, so we add this to our answer here which makes it 11. Once again we put the one in the tens column, and we carry the hundred (which is coloured in blue) and put it outside the box underneath the hundreds column.

Finally, we add up the hundreds column, we can see that there is nothing below the 2 so we get ready to place the 2 directly into the answer box in the hundreds column, but remember that when we added up the tens we ended up with 100 leftover which we carried (in blue) so we add this to the 2 making 3.

Answer therefore is:

For the final time, I’m going to go through another example but this time I’m going to include tens of thousands as well, so we are going to jump from hundreds tens and units straight to “ten thousands, thousands, hundreds tens and units!”.

Let’s start by adding together 5+6, this of course is 11 so we put the one in the box (the rightmost number 1 in the answer box) and we carry the 10 (as a number 1, coloured red because it came from the units column).

Move into the tens column, 1+8 = 9, but we have to add the ‘red 10’ which makes the total in the tens column also 10, so we put a 0 in the answer box to the left of the 1 and we carry the hundred, coloured in blue because it came from the tens column.

We now move into the hundreds and we add 2 and 9 together making 11, we add the ‘blue 100’ from the previous addition to make 12, we then put the 2 in the answer box and we carry the one, this time coloured green because it came from the hundreds. Our last but one step is to look the thousand column where we add 4 and 9 to make 13, but don’t forget we carried the ‘green 1000’ from the previous addition so we add one to 13 making 14, put the 4 in the answer box and we carry the one this time in purple to show that it came from the thousands column. Our final step is to bring the 6 down, and add to it the one ‘10000 in purple’ making 7 and we have now arrived at our answer, that 64,215+9986 does indeed come to 74,201.

As you can probably see, addition isn’t that difficult provided you line things up in proper columns and take your time, not forgetting every time you ‘carry’ you have to add it on to the subsequent addition. Of course nowadays you would probably just use a calculator to do this sort of thing, and indeed you could use a calculator is to prove that you’ve actually done the manual addition correctly. In fact I would be inclined to do just that, have lots of practice in addition, and where possible check your answers with the calculator. It is fair to say that calculators have made us lazy, so these basic skills are very worthwhile (after all you might find yourself with a calculator or a mobile phone with a flat battery one day!).

In this “addition” section we have “carried” when the addition of each column has exceeded 10, in the next section that we are going to look at “subtraction” we will find that we have to use in some cases a not dissimilar action which we will call (for reasons which might become obvious later) “borrowing”, but like with all good “borrowing” we also pay back quite quickly.