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Introduction To Integers

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

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.

  • 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 \(=-5°F\)

  • 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 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

Comparison of Integers

  • We can compare integers. For comparison, we use greater than (>), less than (<) or equal to (=) symbols.
  • If two integers are negative, the integer closer to zero is greater than the other.
  • If two integers are positive, the larger number is greater.

For example: Compare \(-9\) and \(-13\).

\(-9\;\Box-13\)

\(-9\) is closer to zero than \(-13\).

  • So, \(-9\) is greater than \(-13.\)  

\(\therefore\;-9>-13\)

  • A larger negative number is always less than a smaller negative number.

Example: \(-125<-75\)

  • Zero is always greater than the negative integers.

Example: \(0>-100,\;0>-500\)

Illustration Questions

Which one of the following statements shows the correct comparison between \(-12\) and \(-6\) ?

A \(-12>-6\)

B \(-12<-6\)

C \(-12=-6\)

D \(-12\leq-6\)

×

\(-6\) is closer to zero than \(-12\).

image

So, \(-6\) is greater than \(-12\).

\(\therefore\;-6>-12\)

Hence, option (B) is correct.

Which one of the following statements shows the correct comparison between \(-12\) and \(-6\) ?

A

\(-12>-6\)

.

B

\(-12<-6\)

C

\(-12=-6\)

D

\(-12\leq-6\)

Option B is Correct

Opposite Numbers

  • The same numbers which have different signs are known as opposite numbers.

For example: \(2\) and \(-2\) are opposite to each other.

  • The opposite numbers are also known as additive inverse.
  • To find the opposite of a number, just reverse the sign of the number.

For example: 

The opposite number of \(5\) is \(-5\) .

The opposite number of \(-4\) is \(4\) .

Properties of opposite numbers

  • The absolute values of a number and its opposite number are same.

For example: \(|2|=|-2|=2\)

  • The sum of a number and its opposite is always zero.

For example: \(2+(-2)=2-2=0\)

  • On a number line, a number and its opposite number, have the same distance from zero.

Example: Show the opposite of \(+2\) on the number line.

  • We can observe that \(+2\) is located at two units to the right of zero.

  • As, the opposite numbers are at the same distance from zero, so we move \(2\) units to the left of zero.
  • Thus, we reach at \(-2\).
  • Hence, the opposite number of \(2\) is \(-2\).

  • Thus, we reach at \(-2\).
  • Hence, the opposite number of \(2\) is \(-2\).

Illustration Questions

Identify the number that makes the statement true. \(\Box+5=0\)

A \(5\)

B \(-5\)

C \(-4\)

D \(4\)

×

Given: \(\Box+5=0\)

According to one of the properties of the opposite numbers:

"The sum of a number and its opposite is always zero".

The opposite of \(5\) is \(-5\).

Thus, \(-5+5=0\)

\(\therefore\;\Box=-5\)

Hence, option (B) is correct.

Identify the number that makes the statement true. \(\Box+5=0\)

A

\(5\)

.

B

\(-5\)

C

\(-4\)

D

\(4\)

Option B is Correct

Properties of Operation

Some properties for addition and multiplication of two or more integers are as follows:

1. Associative property of addition

The associative property states that changing the grouping of addends (terms) will not change the sum.

\(\text{(a + b) + c = a + (b + c)}\)

Thus, in the above equation, either we add \(\text{a}\) and \(\text{b}\) first and then add \(\text{c}\) to their sum or we add \(\text{b}\) and \(\text{c}\) first and then add \(\text{a}\) to their sum, the result is same.

Example: \((2+3)+5=2+(3+5)\)

\((5)+5=2+(8)\)

\(10=10\)

2. Commutative property of addition

The commutative property states that the numbers can be added in any order and we will still get the same result.

\(\text{a + b = b + a}\)

Example: \(7+1=1+7\)

\(8=8\)

3. Additive identity property of zero

The additive identity property states that a number does not change when zero is added to that number.

\(\text{a + 0 = 0 + a = a}\)

Example: \(2+0=0+2\)

\(2=2\)

4. Inverse property of addition

 The additive inverse of a number is the opposite of that number.

The inverse property of addition states that when we add the number to its additive inverse, it gives zero as the answer.

\(\text{a + (–a) = (–a) + a=0}\)

Example: \(7+(–7)=(–7)+7\)

\(0=0\)

5. Associative property of multiplication

The associative property states that changing the grouping of terms will not change the result.

\(\text{(a . b) . c = a . (b . c)}\)

Thus, in the above equation, either we multiply \(\text{a}\) and  \(\text{b}\) first and then multiply \(c\) or we multiply \(\text{b}\) and \(\text{c}\) first and then multiply \(\text{a}\), the result is same.

6. Commutative property of multiplication

The commutative property states that the numbers can be multiplied in any order and we will still get the same result.

\(\text{a . b = b . a}\)

Example: \(3×2=2×3\)

\(6=6\)

7. Multiplicative identity property of 1

The multiplicative identity property states that a number does not change when it is multiplied by \(1\).

\(\text{a . 1 = 1 . a = a}\)

Example: \(6×1=1×6=6\)

8. Inverse property of multiplication

The multiplicative inverse of a number is the reciprocal of that number.

The inverse property of multiplication states that when we multiply a number with its multiplicative inverse, it gives \(1\) as the answer.

\(\text{a . }\dfrac{1}{\text{a}}=\dfrac{1}{\text{a}}\text{ . a}=1\)

Example: \(3×\dfrac{1}{3}=\dfrac{1}{3}×3=1\)

9. Distributive property of multiplication over addition

The distributive property of multiplication over addition states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the product together.

\(\text{a . (b + c) = a . b + a . c} \)

Example: \(\begin{array}\\6×(3+2)&=&6×(3)+6×(2)\\ =6×5&&=18+12\\ =30&&=30\\ \end{array}\)

Illustration Questions

Choose the property that satisfies the given equation:  \((6+3)+7=6+(3+7)\)

A Commutative property of addition

B Distributive property

C Inverse property of addition

D Associative property of addition

×

Option (A):

We know that the commutative property of addition is:

\(\text{a + b = b + a}\)

On applying commutative property on the given equation, 

\((6+3)+7=7+(6+3)\)

Hence, option (A) is incorrect.

Option (B):

We know that the distributive property is:

\(\text{a . (b + c) = a . b + a . c}\)

The given equation is not in the form of distributive property.

Hence, option (B) is incorrect.

Option (C):

We know that the inverse property of addition is:

\(\text{a + (– a) = (– a ) + a = 0}\)

The given equation is not in the form of the additive inverse.

Hence, option (C) is incorrect.

Option (D):

We know that the associative property of addition is:

\(\text{(a + b) + c = a + (b + c)}\)

On applying the associative property for the given equation,

we get: \((6+3)+7=6+(3+7)\)

Thus, we obtain the same equation as given in the problem.

Hence, option (D) is correct.

Choose the property that satisfies the given equation:  \((6+3)+7=6+(3+7)\)

A

Commutative property of addition

.

B

Distributive property

C

Inverse property of addition

D

Associative property of addition

Option D is Correct

Ordering of Integers

  • We can order negative and positive integers using a number line.
  • To arrange the negative and positive integers in the order of least to greatest, remember the following points:
  1. Positive integers are always greater than the negative integers.
  2. The integer furthest to the left on the number line is always less than the integer furthest to the right. In other words we can say that on a number line, any integer is greater than the number on its left side and is smaller than the number on its right side. 
  3. For ordering from least to greatest, write the integers placed on the number line from left to right.

For example: Arrange \(5,\;-4,\;2\,and-5\) in the order of least to greatest.

First, draw a number line from \(-5\) to \(5\) and plot the integers on it according to their places.

Now, arrange the plotted numbers as per the number line.

Start from the furthest to the left and move the way up to the right.

\(-5<-4<2<5\)

Now, arrange the given numbers according to the number line. Start from the furthest to the left and move the way up to the right.

\(-5<-4<2<5\)

Illustration Questions

Which one of the following list of integers is in the order of least to greatest?

A \(-8,\;4,\;1\)

B \(4,-8,\;1\)

C \(-8,\;1,\;4\)

D \(1,\;4,-8\)

×

First, we draw a number line from \(-8\) to \(4\) and plot the numbers according to their places.

image

Now, we arrange the plotted numbers as per the number line.

We start from furthest to the left and move our way up to the right.

On a number line, any integer is greater than the number on its left side and is smaller than the number on its right side.

\(-8<1<4\)

Hence, option (C) is correct.

Which one of the following list of integers is in the order of least to greatest?

A

\(-8,\;4,\;1\)

.

B

\(4,-8,\;1\)

C

\(-8,\;1,\;4\)

D

\(1,\;4,-8\)

Option C is Correct

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