负整数

Negative Number Problems and Extensions

We're going to dive into some practice problems and extensions that revolve around negative numbers. First, if you haven't done so, make sure you've covered the basics of negative numbers by looking through some explanations.

Don't worry if you get some of these questions wrong, they are here for practice, and each problem has an explanation immediately after.

Example 1: Theory

Some questions will test you on your knowledge of numbers and number sets. Let's take a look.

This requires us to think back to the definitions of our number sets. Whole numbers start at $0$ and increase, so there are no negative whole numbers. Natural numbers start at $1$ and increase, so there are no negative natural numbers. All of the other sets listed have no restriction on positive or negative, so they contain negative numbers.

Example 2: Word Problems

Word problems return! They are always going to be around to challenge your understanding by giving you some extra context.

You find the goal differential by subtracting the total number of goals scored against a team from the total number of goals scored by the team. In this case the team scored $100$ goals and had $125$ scored against them. $$100-125=-25$$ The team has a negative goal differential.

Example 3: Identification

Some problems will ask you to identify negative numbers from a list of numbers.

We are looking for the numbers with negative signs in this problem. The correct answers are $-2$ and $-4$.

Wrap-Up

Negative numbers can be approached from many different directions as they are part of the building blocks of math. Getting a solid grasp of negative numbers and how they interact with various number sets will set up you up for the future. Each of these problems can be tackled once you have a base level understanding of negative numbers.

When you count forward from zero, $0$, $1$, $2$, $3$, ... you get positive numbers. Actually, you get the counting numbers. These are the numbers we usually use to count objects. This same list, except for zero, is the set of positive integers.

But what if you count back from zero? What if you are to subtract one number from another, but what if the minuend is less than the subtrahend? Guess what—you get a negative number!

What kind of numbers are part of the negative numbers?

Negative numbers are real numbers that are less than zero. They are usually denoted by a little minus sign "$-$" in front of them. $-13, -1.8, - \sqrt{62}, - \pi$ are all negative numbers. The tick marks represent integers. Here, $-1, -2, -3$ are all negative integers.

What does the "$-$" sign mean? Why isn't there a minuend?

If we see a minus sign in front of a number, but did not see any number to subtract it from, it is conventional to assume that the number is subtracted from zero. $-8$ represent the value of the expression $0-8$, and $-(-1)$ represent the value $0-(0-1)$, which is $1$.

Another way to think of this minus sign is take it as a shorthand of $(0-1)$, that is, see $-8$ as $(0-1)\times8$ and see $-11.9$ as $(0-1) \times (11.9)$. We define $(0-1)$ as the number such that its square, $(0-1)\times(0-1)$, is $1$.

These two ways of understanding the little minus sign are the same. Well, really? Or what does it even mean to be "the same"? We need to learn the $+-$ $\times \div$ of negative numbers before we can talk about the meaning of "Same". In short, if we have two numbers $a,b$, we write them as $a_1, b_1$ in one way and $a_2, b_2$ in another, if $a_1+b_1 = a_2+b_2$, $a_1 \times b_1 = a_2 \times b_2$, these two notations are the same.

What does it mean to be "less than" zero?

It means if we subtract it from zero, we get a positive number. A positive number is a number such that when multiplied with itself, we get another positive number.

(Video) Negative Numbers

by Math Antics

This video by Math Antics explains negative numbers, their connection to other number sets, and some real life uses for negative numbers.

$\large{\text{Summary}}$

When looking at a number line, negative numbers are found to the left of $0$.

The negative numbers look exactly like the positive numbers, only they have a negative sign in front of them. $$\color{#29A603}{-}5$$

As you go further and further to left, the numbers get smaller. So, for example, $-5$ is smaller than $-2$. There are negative fractions in between the negative integers, and these are still numbers but we don't label them on the number line.