Addition and subtraction

Addition and subtraction are the most basic mathematical operations that people need even in everyday life. Although we can add many different structures, here we will only look at adding whole numbers.

Addition

The operation of addition is denoted by the plus sign +. In addition, we add add addends, so if we want to add 2 + 3, then both two and three are called addends. The result is then the sum, so the five in the expression 2 + 3 = 5 is the sum.

We can add virtually any number, whether natural numbers like 1, 5, 157 or negative numbers like -4, -74. We can also add rational numbers, or fractions, real numbers such as π and finally complex numbers. We can also add other structures - matrices or vectors.

Addition is usually explained using a pile of apples - if you have three apples in your basket and someone gives you four more apples, how many apples will you have left? You will be left with 3 + 4 = 7.

How to add large numbers

If I gave you the task of adding 17564 and 3272, you probably wouldn't be able to tell the result off the top of your head. That's why you either use a calculator or a pencil and paper. I'm not going to teach you how to type it into a calculator, let's focus on the process on paper.

Write the numbers we want to add underneath each other and align them to the right - so that the same orders (tens, hundreds, ...) are underneath each other. Make a horizontal line under both numbers. So it would look like this:

$$\begin{array}{ccccc} 1&7&5&6&4\\ &3&2&7&2\\ \hline \end{array}$$

Now we will add the numbers that are underneath each other. If the sum of the numbers is less than ten, we simply write the sum below the line. We proceed from the right-hand side, starting with the smallest order. So we add 4 + 2 first, which equals six. Six is less than ten, so we write the number six below the line:

$$\begin{array}{ccccc} 1&7&5&6&4\\ &3&2&7&2\\ \hline &&&&6 \end{array}$$

Moving on, we add 6 + 7. That equals 13, which is already greater than ten. At this point, we just write the digit in the units position below the line, that is, we write a three (subtract the ten from the thirteen).

$$\begin{array}{ccccc} 1&7&5&6&4\\ &3&2&7&2\\ \hline &&&3&6 \end{array}$$

What happened to the one that we cut off? And what does it represent? At the moment we are counting the second column from the right, which represents the tens in the number. So we were actually adding tens, i.e. we were adding 60 + 70 = 130. We decided to keep the number 30 and put 100 away for later. For when? For the next step, the next column. In the next step, we get to add the digits that correspond to the hundreds. And we have one extra hundred that we got from adding the tens. So we don't do anything easier than add that hundred in the next step of the addition.

So instead of adding 5 + 2 in the next step, we're going to add 5 + 2 + 1, where the one represents the hundred that we got in the last step's addition. The total is eight, which is less than ten. So we write eight below the line.

$$\begin{array}{ccccc} 1&7&5&6&4\\ &3&2&7&2\\ \hline &&8&3&6 \end{array}$$

No extra hundred goes into the next step, so we only add 7 + 3 = 10. This number is not less than ten, so we write the last digit below the line and one goes next. So we write a zero below the line and keep an extra thousand until the next step.

$$\begin{array}{ccccc} 1&7&5&6&4\\ &3&2&7&2\\ \hline &0&8&3&6 \end{array}$$

In the last column, we only have the digit in the first line. So in the second line we imagine a zero and add 1 + 0 and we must not forget to add the one from the last step: 1 + 0 + 1 = 2. Below the line we write a two.

$$\begin{array}{ccccc} 1&7&5&6&4\\ &3&2&7&2\\ \hline 2&0&8&3&6 \end{array}$$

This completes the algorithm and we have the sum of the numbers 17564 + 3272.

Animation of the addition procedure on paper

You can put two natural numbers in the following boxes and then just enjoy the animation of the whole procedure.

Graphical representation of addition

We can express addition graphically on a number line. The number line is the straight line on which all the numbers are plotted. It can look like this:

If we wanted to illustrate the addition of two numbers, for example 3+4, on this number line, we would do the following: We would draw a line starting at the origin, zero, and ending at three. As we add, we point the line to the right of the point where we started. So, like this:

Now we plot a four on the number line. But at this point we start at number three and from there we take a line segment to the right of four. Here we go:

The point at which the second, green, line ended represents the final result: 3 + 4 = 7.

Subtraction

Subtraction is the inverse operation to addition. If you add the number b to some number a and subtract the number b, you get back the number a. We'll show the procedure with the example 158 748 − 99 57. As in the case of addition, we will write these numbers underneath each other:

$$\begin{array}{ccccccc} &1&5&8&7&4&8\\ -&&9&9&5&7&5\\\hline \end{array}$$

We will write the result below the line, just like in addition. Proceed from the right, starting with the smallest rows. We will count the units first. If the top number is not smaller than the bottom number, we write the difference of the top number − below the line. (Or vice versa - if the bottom number is smaller or the same, then we can safely subtract.) In place of units, the top number is not smaller, so we write the difference 8 − 5 = 3.

$$\begin{array}{ccccccc} &1&5&8&7&4&8\\ -&&9&9&5&7&5\\\hline &&&&&&3 \end{array}$$

Moving one number to the left, we're in the tens place. Here the top number is already smaller than the bottom number. At this point, we add a ten to the top number and then do the same calculation: top number + 10 − bottom number. The result will be equal to 4 + 10 − 7 = 14 − 7 = 7.

$$\begin{array}{ccccccc} &1&5&8&7&4&8\\ -&&9&9&5&7&5\\\hline &&&&&7&3 \end{array}$$

At this point, we've helped ourselves with the subtraction of tens by including ten tens, so we need to think about that in the next step. Ten tens is one hundred, so in the next step we subtract one hundred more. That means we add the one from the previous step to the bottom five.

So in place of the hundreds, we solve for whether seven (the top number) is less than six (the bottom number + 1 ). It isn't, so we can do the standard top minus bottom difference. But even with the difference, we have to add one to the bottom number, so we end up with 7−(5 + 1) = 7 − 5 − 1 = 1.

$$\begin{array}{ccccccc} &1&5&8&7&4&8\\ -&&9&9&5&7&5\\\hline &&&&1&7&3 \end{array}$$

We don't carry anything over to the next step, we didn't need to borrow any higher orders in this step. So we don't add anything to anything in place of thousands. However, the eight in the top number is smaller than the nine in the bottom number, so we will have to add a ten to the top number. This gives us the difference 8 + 10 − 9 = 9.

$$\begin{array}{ccccccc} &1&5&8&7&4&8\\ -&&9&9&5&7&5\\\hline &&&9&1&7&3 \end{array}$$

In the next step, we convert the one, add it to the bottom number. Nine plus one is ten, the top five is less than ten. We add ten to the top number and make the difference: 5 + 10−(9 + 1) = 5.

$$\begin{array}{ccccccc} &1&5&8&7&4&8\\ -&&9&9&5&7&5\\\hline &&5&9&1&7&3 \end{array}$$

Again, we've helped ourselves to a higher order, so we carry the one to the next step. There we are missing a number at the bottom, so we assume zero. When we add the one, we get the result of one. The top number is not less than the bottom number (we now have the top number equal to the bottom number), so we can simply subtract: 1 − 1 = 0.

$$\begin{array}{ccccccc} &1&5&8&7&4&8\\ -&&9&9&5&7&5\\\hline &0&5&9&1&7&3 \end{array}$$

And we have the result. We can remove the zero at the beginning and write: 158 748 − 99 575 = 59 173.

Graphical representation of subtraction

Subtraction, like addition, can be expressed on the number line. Only when we plot the line, we change its direction. We do not apply the line to the right of the current point, but to the left. So if we were to subtract 2 − 5, we would do the following. First we would apply a two to the number line, in the positive direction, because two is positive, We subtract four when it is reversed.

Now we'll put a second line on the axis to represent minus five. We'll start at the point where we left off, which is two. But we'll draw the line backwards, to the left.