You"ve functioned with numbers on a number line. You know exactly how to graph numbers like 0, 1, 2, 3, etc. On the number line. There are various other kinds of numbers that deserve to be graphed on the number line, too. Let"s view what castle look like and where they are located on the number line.

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In mathematics, it"s sometimes valuable to talk about groups the things, i m sorry are called sets. Numbers have the right to be grouped into sets, and a certain number deserve to belong to more than one set.

You probably are acquainted with the set of natural numbers, i m sorry are additionally called the counting numbers. These are the numbers 1, 2, 3, and also so on—the numbers we use as soon as counting.

The adhering to illustration shows the herbal numbers graphed ~ above a number line.

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The number line continues in both directions. The collection of organic numbers only continues to the right, so girlfriend can incorporate 6, 7, and so on, all the method up right into the hundreds, thousands, and beyond. You have the right to only present so lot on one picture!

When 0 is included to the set of 1, 2, 3, and so on, it creates the set of whole numbers. This are referred to as “whole” because they have actually no spring parts. (A trick to assist you remember i beg your pardon are herbal numbers and also which are whole numbers is come think that a “hole,” which have the right to be represented by 0. The totality ("hole") numbers encompass 0, the natural numbers carry out not.)

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

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Integers


When you occupational with something like temperature, you occasionally want to use numbers that are less than zero, i beg your pardon are called negative numbers. An unfavorable numbers are written using a an unfavorable sign in front, such as −1, −5, and also −30. These are review "negative one," "negative five," and also "negative thirty." (The an adverse sign should not be check out as "minus"; minus means subtraction.)

The numbers greater than 0 are dubbed positive numbers and also can be written v or there is no the “+” sign. An alert that 0 is neither optimistic nor negative!

Integers room the numbers: …, −3, −2, −1, 0, 1, 2, 3, …. Notification that all of the entirety numbers are additionally integers. The following illustration shows the integers graphed ~ above the number line. The integers encompass zero and also continue come the right and to the left.

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The number 0 belongs to which the the following sets the numbers?

natural numbers

whole numbers

integers

A) natural numbers only

B) totality numbers only

C) natural and also whole numbers

D) integers only

E) totality numbers and integers


Show/Hide Answer

A) natural numbers only

Incorrect. The herbal numbers space 1, 2, 3, and also so on. They don’t encompass 0. The exactly answer is totality numbers and also integers.

B) entirety numbers only

Incorrect. While the totality numbers include 0, so carry out the integers. The exactly answer is entirety numbers and integers.

C) natural and also whole numbers

Incorrect. The herbal numbers room 1, 2, 3, and so on. Lock don’t encompass 0. The correct answer is totality numbers and also integers.

D) integers only

Incorrect. If the integers encompass 0, so carry out the entirety numbers. The exactly answer is whole numbers and also integers.

E) whole numbers and integers

Correct. Both whole numbers and also integers incorporate 0, yet the organic numbers do not.

Absolute Value and the Number Line


The number line below shows all the integers between and also including −5 and also 5. Notification that the hopeful integers go 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 location on the number line and also 0 is dubbed the number’s . To write the absolute worth of a number, use short vertical currently (|) on either next of the number. Because that example, the absolute worth of −3 is composed |−3|.

Notice that street is constantly positive or 0.

|−3| = 3, as −3 is 3 systems away native 0 and |3| = 3, together 3 is 3 systems away indigenous 0.

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Here space some various 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 because that x in the expression.

Answer

|−7|= 7

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


To find an integer on the number line, imagine was standing on the number heat at 0. If the number is 0, you there. If the number is positive, face to the right—numbers higher than 0. If the number is negative, confront to the left—numbers much less than 0. Then, move forward the number of units equal to the absolute worth of the number.


Example

Problem

Find 4 top top the number line. Then identify |4|.

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Imagine standing at 0. Because −4 is negative, confront to the left.

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Move 4 units from 0 in the an unfavorable direction.

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

Answer

|−4| = 4

Direction moved does not influence absolute value, just the street moved.


Which allude represents −2 top top this number line?

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Show/Hide Answer

A)

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

B)

Correct. Suggest B is 2 systems to the left that 0, for this reason it to represent −2.

C)

Incorrect. This point is 1 unit to the left that 0, for this reason it represents −1. The suggest that represents −2 is 2 systems to the left that 0. Suggest B is correct.

D)

Incorrect. This point is 1 unit to the right of 0, so it to represent 1. A an adverse number need to be come the left the 0. The allude that to represent −2 is 2 units to the left the 0. Point B is correct.

E)

Incorrect. This allude is 2 systems to the appropriate of 0. A an unfavorable number must be come the left of 0. The allude that to represent −2 is 2 units to the left of 0. Suggest B is correct.

Opposites


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

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

|72| = 72, and |−72| = 72

The set of integers space all the entirety numbers and also their opposites.

What is the opposite of −29?


Show/Hide Answer

Answer: the contrary of −29 is 29.

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Summary


Some numbers are natural numbers (1, 2, 3, ...) or entirety numbers (0, 1, 2, 3, ...). Entirety numbers are likewise integers. There are various other integers which are the opposites of the totality numbers (−1, −2, −3, ...). These an adverse numbers lie come the left the 0 on the number line. Integers are the entirety numbers and their opposites. The absolute worth of a number is its street to 0 ~ above the number line. Absolute values are always positive or 0.