Informative line

Integers As Positive And Negative Numbers

Identification of an Integer

  • Integers are a group of numbers that consists of a set of negative and positive numbers.

\((...\;-5,\;-4,\;-3,\;-2,\;-1,\;0,\;1,\;2,\;3,\;4,\;5\,...)\)

  • They do not include any fractional or decimal part.
  • They are whole numbers and their opposites.
  • The opposite of an integer can be written by changing its sign.
  • For example: Opposite of the integer \(5\) is \(-5\) which is obtained by changing its sign (positive changes to negative).

  • A group of numbers that consists of a set of whole numbers except zero are called positive integers.

Example: \((1,\;2,\;3,\;4,\;5\,...)\)

Positive integers are either represented by '+' sign or without any sign.

Example: \((+1,\;+2,\;+3\,...)\;\text{or}\;(1,\;2,\;3\,...)\)

  • A group of numbers that consists of a set of opposite of whole numbers except zero are called negative integers.

Example: \((-1,\;-2,\;-3,\;-4\,...)\)

Negative integers are represented by '–' sign.

Example: \((-1,\;-2,\;-3,\;-4,\;-5\,...)\)

  • Zero is an integer but it is neither positive nor negative.

Illustration Questions

Which one of the following numbers is an integer?

A \(-10\)

B \(\dfrac{5}{2}\)

C \(-10.5\)

D \(2.5\)

×

An integer is a whole number that can be positive, negative or zero.

It does not include any fractional or decimal part.

 

Hence, option (A) is correct.

Which one of the following numbers is an integer?

A

\(-10\)

.

B

\(\dfrac{5}{2}\)

C

\(-10.5\)

D

\(2.5\)

Option A is Correct

Representation of Integers on the Number Line

  • We can plot negative and positive integers on the number line.
  • Negative integers are to the left side of zero and positive integers are to the right side of zero.

For example: If we wish to represent \(2\) and its opposite \((-2)\) on the number line then

for \(2,\) start from \(0\) and move \(2\) points forward (right side of zero).

  • Since, we have moved \(2\) points forward to show \(2,\) so for the opposite of \(2,\) we move \(2\) points backward from \(0\) (left side of zero).

Illustration Questions

Which number represents the location of point \(Q\) on the number line?

A \(-3\)

B \(3\)

C \(2\)

D \(7\)

×

To find the location of point \(Q\) on the number line, we start from zero and move one point forward.

image

Now, keep on making such moves till we reach the point \(Q.\)

image

We made \(3\) moves to the right side of zero. So, point \(Q\) represents the number \(3\) on the number line.

Hence, option (B) is correct.

Which number represents the location of point \(Q\) on the number line?

image
A

\(-3\)

.

B

\(3\)

C

\(2\)

D

\(7\)

Option B is Correct

Absolute Value

  • The absolute value of an integer is the distance of that integer from zero.
  • Suppose, we want to find the absolute value of \(-2\).

Absolute value of \(-2\) is the distance of \(-2\) from zero.

  • Since the distance of \(-2\) from zero is \(2\) units, so the absolute value of \(-2\) is \(2.\) 

  • Distance can never be negative. It is always positive. Thus, the absolute value of any integer is always positive.
  • The absolute value is shown by enclosing the number between two vertical bars \((||)\), such as:

\(|10|=10\)

The absolute value of \(|10|\) is \(10\).

\(|-10|\) is equivalent to \(10\).

Note: The absolute values of an integer and its opposite are always same.

Illustration Questions

Find the absolute value of \(-5\).

A \(5\)

B \(0\)

C \(-5\)

D \(151\)

×

The absolute value of \(-5\), i.e. \(|-5|\), is the distance of \(-5\) from \(0\).

image

The distance of \(-5\) from zero is \(5\) units.

So, the absolute value of \(-5\) is \(5.\)

Hence, option (A) is correct.

Find the absolute value of \(-5\).

A

\(5\)

.

B

\(0\)

C

\(-5\)

D

\(151\)

Option A is Correct

Elevation as Integers

  • Integers can be used to represent the elevation.
  • The sea level represents the \(0\) (zero) integer and is taken as the point of reference.
  • Any location above the sea level represents a positive integer.
  • Any location below the sea level represents a negative integer.

Example: \(50\) feet below the sea level.

The sea level represents zero and any location which is below the sea level, is represented by a negative integer.

So, \(50\) feet below the sea level represents \(-50\) feet.

Illustration Questions

Which one of the following represents \('\text{11 feet below the sea level}'\)?

A 11

B -11

C 0

D 12

×

Any location which is below the sea level is represented by a negative integer.

Here, \(11\) feet below the sea level is represented by the integer \(-11\)

Hence, option (B) is correct.

Which one of the following represents \('\text{11 feet below the sea level}'\)?

A

11

.

B

-11

C

0

D

12

Option B is Correct

Opposite of an Integer

  • Integers are wholes and their opposites.

Example: Opposite of \(a\) is \(–a\).

  • If two integers are placed at the same distance from zero on both sides of the number line, then they are opposite of each other.
  • Thus, the opposite of \(-3\) is \(3.\)  

  • Opposite of a whole number \((a)\) is a negative integer \((-a)\).

Opposite of \((1)\) is \((-1)\).

  • Opposite of a negative integer \((-1)\) is a positive integer \((1)\) or \(-(-1)\).

Example: (1) Opposite of \(5\) is \(-5\).

(2) Opposite of \(-6\) is \(-(-6)\) or \(6\).

(3) Opposite of \(10\) is \(-10\).

(4) Opposite of \(-(-2)\) is \(-2\).

(5) Opposite of \(-8\) is \(-(-8)\) or \(8\).

Note: Opposite of the opposite of a number is the number itself.

Illustration Questions

Which one of the following number lines represents the locations of \(-6\) and its opposite?

A

B

C

D

×

Integers are wholes and their opposites.

So, the opposite of \(-6\) is \(6.\)

On a number line, the opposite of an integer has the same distance from zero on the other side.

image

Hence, option (C) is correct.

Which one of the following number lines represents the locations of \(-6\) and its opposite?

A image
B image
C image
D image

Option C is Correct

Temperature as Integers

  • Integers can be used to represent temperatures.
  • Zero is used as a reference point.

(1) Any temperature above zero is represented as a positive integer.

   \(15°F\) above zero \(=15°F\)

(2) Any temperature below zero is represented as a negative integer.

   \(15°F\)  below zero \(=-15°F\)

  • Smaller the integer, colder the temperature is and greater the integer, warmer the temperature is.

\(-15°F\)  is colder than \(-13°F\).

\(15°F\)  is warmer than \(13°F\).

Examples:

\(10°F\)  above zero \(=10°F\)

\(5°F\)  below zero \(=-5°F\)

Illustration Questions

Which one of the following temperatures represents \(11°F\) below zero?

A \(11°F\)

B \(-11°F\)

C \(0°F\)

D \(12°F\)

×

Any temperature below zero is represented by a negative integer.

Thus, \(11°\) below zero \(=-11°F\)

Hence, option (B) is correct.

Which one of the following temperatures represents \(11°F\) below zero?

A

\(11°F\)

.

B

\(-11°F\)

C

\(0°F\)

D

\(12°F\)

Option B is Correct

Money as Integers

  • Integers can be used to represent loss and gain of money.
  • Loss of money would be a negative integer.
  • Gain (profit) of money would be a positive integer.
  • An account (bank) is credited when money is deposited in it, which means gain of money, so this would be a positive integer.
  • An account is debited when money is taken out of it, which means loss of money, so this would be a negative integer.

Example: Ms. Wendy deposits $200.

When money is deposited into an account, it means gain of money, so this would be a positive integer.

Thus, it represents the integer $200.

Illustration Questions

Which one of these represents \('Sam \;earns\; $20'\)?

A 2

B -20

C 20

D 0

×

Gain of money would be a positive integer.

Here, Sam earns \($20\) means he gains money.

So, the integer is \(20\).

Hence, option (C) is correct.

Which one of these represents \('Sam \;earns\; $20'\)?

A

2

.

B

-20

C

20

D

0

Option C is Correct

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