You"ve worked with number on a number line. Friend know just how to graph numbers prefer 0, 1, 2, 3, etc. ~ above the number line. There are various other kinds of numbers that can be graphed ~ above the number line, too. Let"s watch what castle look like and also where lock are situated on the number line.

You are watching: Natural numbers their opposites and 0


In mathematics, it"s sometimes beneficial to talk about groups of things, which are called sets. Numbers can be grouped right into sets, and also a particular number have the right to belong to an ext than one set.

You most likely are familiar with the set of natural numbers, i m sorry are likewise called the counting numbers. These space the number 1, 2, 3, and so on—the numbers we use as soon as counting.

The adhering to illustration reflects the natural numbers graphed top top a number line.

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The number line proceeds in both directions. The collection of organic numbers only proceeds to the right, so girlfriend can encompass 6, 7, and so on, all the means up right into the hundreds, thousands, and also beyond. You have the right to only present so much on one picture!

When 0 is included to the set of 1, 2, 3, and also so on, it forms the collection of whole numbers. This are called “whole” because they have no fountain parts. (A trick to aid you remember which are herbal numbers and which are whole numbers is to think the a “hole,” which have the right to be stood for by 0. The totality ("hole") numbers encompass 0, the natural numbers execute not.)

The complying with illustration mirrors the whole numbers graphed top top the number line.

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Integers


When you job-related with something like temperature, you sometimes want to use numbers the are much less than zero, which are referred to as negative numbers. An adverse numbers are written using a an unfavorable sign in front, such together −1, −5, and also −30. This are read "negative one," "negative five," and "negative thirty." (The an adverse sign must not be read as "minus"; minus method subtraction.)

The numbers higher than 0 are dubbed positive numbers and also can it is in written through or without the “+” sign. Notice that 0 is neither hopeful nor negative!

Integers space the numbers: …, −3, −2, −1, 0, 1, 2, 3, …. An alert that all of the entirety numbers are likewise integers. The complying with illustration shows the integers graphed on the number line. The integers incorporate zero and continue come the right and also to the left.

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The number 0 belongs come which that the complying with sets of numbers?

natural numbers

whole numbers

integers

A) natural numbers only

B) totality numbers only

C) natural and whole numbers

D) integers only

E) entirety numbers and also integers


Show/Hide Answer

A) herbal numbers only

Incorrect. The natural numbers are 1, 2, 3, and so on. Lock don’t incorporate 0. The exactly answer is whole numbers and integers.

B) whole numbers only

Incorrect. If the entirety numbers include 0, so do the integers. The exactly answer is totality numbers and integers.

C) natural and also whole numbers

Incorrect. The natural numbers are 1, 2, 3, and also so on. Lock don’t incorporate 0. The exactly answer is whole numbers and integers.

D) integers only

Incorrect. While the integers encompass 0, so execute the totality numbers. The exactly answer is entirety numbers and integers.

E) totality numbers and integers

Correct. Both entirety numbers and integers include 0, but the natural numbers carry out not.

Absolute Value and the Number Line


The number line below shows all the integers between and also including −5 and also 5. An alert that the hopeful integers walk to the right: 1, 2, 3, and also so on. The an unfavorable integers walk to the left: −1, −2, −3, and so on.

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The distance between a number’s ar on the number line and 0 is called the number’s . To create the absolute value of a number, use brief vertical lines (|) top top either next of the number. Because that example, the absolute value of −3 is created |−3|.

Notice that street is constantly positive or 0.

|−3| = 3, together −3 is 3 systems away from 0 and also |3| = 3, together 3 is 3 devices away indigenous 0.

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Here space some other examples.


|0| = 0

|−23| = 23

|6| = 6

|817| = 817

|−3,000| = 3,000


Example

Problem

Find |x| as soon as x = 7.

|x|

|−7|

Substitute −7 for x in the expression.

Answer

|−7|= 7

Since −7 is 7 units from 0, the absolute value is 7.


To situate an creature on the number line, imagine standing on the number line at 0. If the number is 0, you’re there. If the number is positive, challenge to the right—numbers greater than 0. If the number is negative, confront to the left—numbers less than 0. Then, relocate forward the variety of units same to the absolute value of the number.


Example

Problem

Find 4 on the number line. Then recognize |4|.

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Imagine standing at 0. Since −4 is negative, face to the left.

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Move 4 units from 0 in the an adverse direction.

Draw a period on the number line at the location, i beg your pardon is −4.

Answer

|−4| = 4

Direction moved does not impact absolute value, only the distance moved.


Which point represents −2 top top this number line?

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Show/Hide Answer

A)

Incorrect. This allude is 4 systems to the left of 0, for this reason it to represent −4. The point that represents −2 is 2 devices to the left the 0. Suggest B is correct.

B)

Correct. Allude B is 2 devices to the left that 0, therefore it represents −2.

C)

Incorrect. This allude is 1 unit to the left the 0, for this reason it to represent −1. The point that represents −2 is 2 devices to the left that 0. Point B is correct.

D)

Incorrect. This allude is 1 unit come the appropriate of 0, so it to represent 1. A an adverse number must be come the left the 0. The allude that represents −2 is 2 systems to the left that 0. Allude B is correct.

E)

Incorrect. This allude is 2 systems to the ideal of 0. A an adverse number need to be come the left the 0. The point that represents −2 is 2 units to the left of 0. Allude B is correct.

Opposites


You may have actually noticed that, other than for 0, the integers come in bag of confident and an adverse numbers: 1 and −1, 3 and −3, 72 and −72, and also so on. Every number is the opposite of the various other number in the pair: 72 is opposing of −72, and −72 is the contrary of 72.

A number and also its opposite room the very same distance from 0, for this reason they have the same absolute value.

|72| = 72, and |−72| = 72

The collection of integers room all the entirety numbers and also their opposites.

What is the opposite of −29?


Show/Hide Answer

Answer: the opposite of −29 is 29.

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Summary


Some numbers are herbal numbers (1, 2, 3, ...) or totality numbers (0, 1, 2, 3, ...). Totality numbers are additionally integers. Over there are other integers which space the opposites that the entirety numbers (−1, −2, −3, ...). These an adverse numbers lie come the left the 0 top top the number line. Integers room the whole numbers and also their opposites. The absolute worth of a number is its distance to 0 on the number line. Absolute values are constantly positive or 0.