The set of the integers

The integers are made up of positive numbers, negative numbers and zero. The positive numbers are like the naturals, but with a "plus" before: $$+1, +2, +3, +4, \ldots$$. Nevertheless, the "plus" of the positive numbers does not need to be be written. On the other hand, the negative numbers are like the naturals but with a "minus" before: $$-1, -2, -3, -4,\ldots$$ The number zero is special, because it is the only one that has neither a plus nor a minus, showing that it is neither positive nor negative.

For example, the following numbers are integers: $$3, -76, 0, 15, -22.$$

Although they may seem a bit strange, the negative numbers are used every day.

For example, someone gets into an elevator on the ground floor. Nevertheless, he does not want to go up, rather he wants to go down because that is where the parking is. Then he pushes the button for the floor $$-1$$, the floor beneath the ground floor. If he had pushed the button for the first floor, he would have gone to the first floor: and this is not what he wanted!

The integers can be drawn on a line as follows:

  1. A line is drawn and it is divided into equal segments.
  2. The zero is drawn.
  3. The positive numbers are drawn on the right of the zero in order: first $$1$$, then $$2, 3$$, etc.
  4. The negative numbers are drawn on the left of the zero as follows: first $$-1$$, then $$-2$$, $$-3$$, etc.

In the following drawing you can see an example of the integers from $$-5$$ to $$5$$ drawn on a line:


It is said that an integer is smaller than another one if when we draw it, it is placed on its left. In the previous drawing, we can see, for example, that: $$-2$$ is smaller than $$4$$, that $$-5$$ is smaller than $$-1$$, and that $$0$$ is smaller than $$3$$. To write this we will use the following symbol: $$<$$. This symbol means: the number that is on the left is smaller than the one that is on the right. In the previous example we would write: $$-2<4, -5<-1$$ and $$0<3$$.

Let's see a couple of exercises:

Say which of the following numbers are integers, and of these, which are positive and which are negative: $$5, -31, -11.2, 80, 6.2$$

$$5$$ is a natural number, therefore it is also an integer. Also, since it does not have a minus in front of it, it is positive. $$-31$$ is $$31$$ with a minus before it. As $$31$$ is natural, $$-31$$ is integer. And since it has a minus before, it is negative. $$-11.2$$ is $$11.2$$ with a minus before. But $$11.2$$ is not a natural number, therefore it is not an integer. $$80$$ is a natural number and therefore it is integer. Since it is not preceded by a minus, it is positive. $$6.2$$ is not natural, therefore it is not an integer.

The integers are: $$5, -31$$ and $$80$$. The only negative is $$-31$$, the other two are positive.

Sort the following numbers from smallest to greatest: $$12, -2, -6, 2, -7, 9$$

We draw the zero in a line and put the positive numbers on the right and the negative numbers on the left:


As $$-7$$ is the one on the far left, then we can see that it is the smallest. Then there comes $$-6$$, then $$-2$$. Next $$2$$, later $$9$$ and when we reach the top right, there is $$12$$, and therefore this is the largest number. Using the symbol $$<$$ we have that: $$-7<-6<-2< 2< 9< 12$$