Informative line

• Number line is quite useful in understanding operations on whole numbers.
• Suppose we want to perform addition operation of $$7$$ and $$4$$ on the number line.
• We start from $$0$$ (zero) and move $$7$$ points forward at a time.

• We reach the number $$7.$$
• Now to add $$4$$ to the number $$7,$$ we move $$4$$ points forward by taking $$1$$ point at a time.

• We reach the number $$11$$.

Thus, $$7+4=11$$

#### Which one of the following number lines represents $$2+4\,?$$

A

B

C

D

×

To represent $$2+4$$ on the number line, start from zero and move $$2$$ points forward at a time. We reach the number $$2$$.

Now to add $$4$$ to $$2$$, move $$4$$ points forward by taking $$1$$ point at a time.

We reach the number $$6$$.

Thus, $$2+4=6$$

Hence, option (C) is correct.

### Which one of the following number lines represents $$2+4\,?$$

A
B
C
D

Option C is Correct

# Closure Property

• The sum of any two whole numbers is always a whole number. This is known as closure property for addition of whole numbers.

Whole number + Whole number = Whole number

• Let us take a pair of whole numbers (9 and 8) and add them.

$$9+8=17$$

Here, the result $$17$$ is also a whole number.

• If $$a$$ and $$b$$ are any two whole numbers, then $$(a+b)$$ is also a whole number.

#### Which one of the following represents closure property of addition for whole numbers?

A $$12+3=15$$

B $$12.5+3=15.5$$

C $$\dfrac{2}{3}+2=2\dfrac{2}{3}$$

D $$12.5+3.5=16$$

×

According to closure property of addition, sum of two whole numbers is always a whole number.

In option (A), sum of two whole numbers is a whole number.

$$12+3=15$$

So, it represents closure property of addition.

In option (B), $$12.5$$ is not a whole number.

So, it does not represent closure property of addition for whole numbers.

In option (C), $$\dfrac{2}{3}$$ is not a whole number.

So, it does not represent closure property of addition for whole numbers.

In option (D), $$12.5$$ and $$3.5$$ are not whole numbers.

So, it does not represent closure property of addition for whole numbers.

Hence, option (A) is correct.

### Which one of the following represents closure property of addition for whole numbers?

A

$$12+3=15$$

.

B

$$12.5+3=15.5$$

C

$$\dfrac{2}{3}+2=2\dfrac{2}{3}$$

D

$$12.5+3.5=16$$

Option A is Correct

• The additive identity property says that a number does not change when adding zero to that number.

$$a+0=a$$

• Thus, in arithmetic, the additive identity is $$0$$ (zero).
• For example, if we are adding $$5$$ and zero, then we will get $$5$$ as the answer.

$$5+0=5$$

#### Which one of the following illustrates the additive identity property for whole numbers?

A $$0+4=4$$

B $$3+4=7$$

C $$2+4=6$$

D $$2+4=4+2$$

×

According to additive identity property, a number does not change when adding zero to that number.

$$a+0=a$$

Amongst the choices, only equation $$0+4=4$$ involves addition with zero.

Also it illustrates additive identity property.

Hence, option (A) is correct.

### Which one of the following illustrates the additive identity property for whole numbers?

A

$$0+4=4$$

.

B

$$3+4=7$$

C

$$2+4=6$$

D

$$2+4=4+2$$

Option A is Correct

# Number Line

• A number line is a pictorial representation of real numbers.
• We can plot whole numbers on a number line. Number line can be drawn as long as we want but we can never find infinity on it.
• Zero lies at the middle of the number line.

• The positive numbers always lie on the right side of zero and negative numbers always lie on the left side of zero.

• A number which is farther to the right on the number line is greater than its previous numbers.

$$-4<-3<-2<-1<0<1<2<3<4$$

• The section of the number line between two numbers is called an interval.

• If the numbers are placed in their correct order, then the interval of numbers is always equal.

#### Which point represents whole number $$6$$ on the number line? [Each interval is equal]

A Point $$A$$

B Point $$B$$

C Point $$C$$

D Point $$D$$

×

Since whole numbers start from zero $$(0)$$, so starting from zero and moving $$6$$ points forward, the point which comes on the number line represents $$6.$$

So, point $$C$$ represents the whole number $$6$$ on the number line.

Hence, option (C) is correct.

### Which point represents whole number $$6$$ on the number line? [Each interval is equal]

A

Point $$A$$

.

B

Point $$B$$

C

Point $$C$$

D

Point $$D$$

Option C is Correct

• Commutative property says that the numbers can be added in any order and we will still get the same answer.

$$a+b \;= \; b+a$$

• For example, if we are adding $$3$$ and $$4$$, the commutative property of addition says that we will get the same answer whether we add $$3$$ to $$4$$ or $$4$$ to $$3$$.

$$3+4=7\;\;\text{or}\;\;4+3=7$$

• This also works for more than two numbers.

Example:

$$4+5+6=15$$

$$6+5+4=15$$

• We can observe from above example that though the order of addends is changed, the sum is still same.

#### Which of the following equations shows the commutative property of addition for whole numbers?

A $$5+4=4+6$$

B $$5+4=4+5$$

C $$5+4=5+6$$

D $$6+3=5+4$$

×

According to commutative property of addition, the numbers can be added in any order but the sum remains the same.

In option (B), the order of the addends is changed,

5+4=9   or   4+5=9

but still the answer is same.

Hence, option (B) is correct.

### Which of the following equations shows the commutative property of addition for whole numbers?

A

$$5+4=4+6$$

.

B

$$5+4=4+5$$

C

$$5+4=5+6$$

D

$$6+3=5+4$$

Option B is Correct

# Associative Property

In a classroom, there are 30 students.

The teacher first makes two groups of 15 and counts them; there are 30 students.

Then she makes 3 groups of 10 each and counts them; there are still 30 students.

She then makes groups of 10,15 and 5; there are still 30 students.

It means no matter how she groups them, the total sum is always the same.

Similarly, the associative property also says that changing the grouping of the addends (students) will not change the sum.

Example: a+b+c=(a+b)+c=a+(b+c)

3+2+4=(3+2)+4=3+(2+4)= 9

#### Complete the equation below to demonstrate the associative property of addition for whole numbers. $$3+(2+4)=(\_+\_)+4$$

A $$3+(2+4)=(2+3)+4$$

B $$3+(2+4)=9$$

C $$3+(2+4)=(3+2)+4$$

D $$3+(2+4)=3+(2+4)$$

×

According to associative property of addition, changing the grouping of the addends (terms) will not change the sum.

$$(a+b)+c=a+(b+c)$$...(1)

On comparing the given equation with equation (1), we get

$$a=3$$

$$b=2$$

$$c=4$$

Thus, by associative property

$$3+(2+4)=(3+2)+4$$

Hence, option (C) is correct.

### Complete the equation below to demonstrate the associative property of addition for whole numbers. $$3+(2+4)=(\_+\_)+4$$

A

$$3+(2+4)=(2+3)+4$$

.

B

$$3+(2+4)=9$$

C

$$3+(2+4)=(3+2)+4$$

D

$$3+(2+4)=3+(2+4)$$

Option C is Correct