Informative line

# Multiplication on a Number Line

• Number line is quite easy way of understanding operations on whole numbers.
• Suppose we want to perform multiplication operation of $$4$$ and $$3.$$
• To show multiplication of $$4$$ and $$3$$ on the number line, start from $$0$$ and move $$4$$ points forward at a time.

• Similarly, make $$2$$ more moves forward (total $$3$$ moves) of $$4$$ points each.

We reach the number $$12$$.

Thus, $$4×3=12$$

#### Which one of the following number lines represents $$7×3=?$$

A

B

C

D

×

To show multiplication of $$7$$ and $$3$$ on the number line, start from $$0$$ and move $$7$$ points forward at a time.

Similarly, make $$2$$ more moves forward (total $$3$$ moves) of $$7$$ points each.

We reach the number $$21.$$ Thus, $$7×3=21$$

Hence, option (A) is correct.

### Which one of the following number lines represents $$7×3=?$$

A
B
C
D

Option A is Correct

# Closure Property

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

whole number × whole number = whole number

• Let us take a pair of whole numbers ($$12$$ and $$5$$) and multiply them.

$$12×5=60$$

Thus, the result $$60$$ 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 the closure property for multiplication of whole numbers?

A $$4×5=20$$

B $$4-5=-1$$

C $$4 × \dfrac{3}{4}=3$$

D $$4+5 = 9$$

×

According to closure property for multiplication, product of two whole numbers is always a whole number.

Only option (A) shows product of two whole numbers (4 and 5) and their result (20) is also a whole number.

Hence, option (A) is correct.

Since option (B) involves subtraction of whole numbers, option (C) involves the product of a fraction and a whole number and option (D) involves the addition of whole numbers.

So, all of these are incorrect.

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

A

$$4×5=20$$

.

B

$$4-5=-1$$

C

$$4 × \dfrac{3}{4}=3$$

D

$$4+5 = 9$$

Option A is Correct

# Commutative Property

• Commutative property states that even if we change the order of numbers in the product, the answer still remains the same.

$$a×b=b×a$$

• For example: Multiplying $$4×2$$ will give us the same answer as multiplying $$2×4$$.

$$4×2\;=\;2×4\\\;\;\;\,\,\,\,\,8\;=8$$

• This also works for more than two numbers.

Example:

$$5×2×6\;=\;6×5×2$$

$$60\;=\;60$$

• Thus, we can say that whole numbers are commutative under multiplication. This property is known as commutative property for multiplication of whole numbers.

#### Which equation shows the commutative property for multiplication of whole numbers?

A $$8×7=56$$

B $$8×7=7×8$$

C $$8+7=7+8$$

D $$7+8=15$$

×

According to commutative property for multiplication, the numbers can be multiplied in any order and we still get the same answer.

$$a×b=b×a$$

Option (A) shows product of two whole numbers ($$8$$ and $$7$$) and their result ($$56$$) is also a whole number. Thus, this represents closure property of multiplication.

Hence, option (A) is incorrect.

Option (B) represents that even if we change the order of numbers in the product, the answer still remains the same.

$$8×7=7×8$$

$$56=56$$

Thus, this represents commutative property of multiplication.

Hence, option (B) is correct.

Option (C) represents that if we change the order of addends in addition, the answer still remains the same.

$$8+7=7+8$$

$$15=15$$

Thus, this represents commutative property of addition.

Hence, option (C) is incorrect.

Option (D) represents sum of two whole numbers and their result is also a whole number.

Thus, this represents closure property of addition.

Hence, option (D) is incorrect.

### Which equation shows the commutative property for multiplication of whole numbers?

A

$$8×7=56$$

.

B

$$8×7=7×8$$

C

$$8+7=7+8$$

D

$$7+8=15$$

Option B 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

# Associative Property

• The associative property states that it does not matter how we group the numbers (i.e. where we put parenthesis) but on multiplying them, the product remains the same.

$$a×(b×c)=(a×b)×c$$

where $$a,\,b$$ and $$c$$ are whole numbers

For example:

$$4×(5×6)=(4×5)×6$$

$$4×30=20×6$$

$$120=120$$

In both the groupings, answer remains the same.

#### Which one of the following illustrates the associative property of multiplication?

A $$7×(8×9)=(7×8)×9$$

B $$7×(8+9)=(7×8)+9$$

C $$(7+8)×9=7×(8+9)$$

D $$(7-8)×9=7-(8×9)$$

×

The associative property of multiplication states that it does not matter how we group the numbers (i.e. where we put parenthesis) but on multiplying them, the product remains the same.

$$a×(b×c)=(a×b)×c$$ ... (1)

Given numbers are $$7,\,8$$ and $$9$$.

Let $$a=7,\,b=8$$ and $$c=9$$

According to equation (1),

$$7×(8×9)=(7×8)×9$$

Hence, option (A) is correct.

### Which one of the following illustrates the associative property of multiplication?

A

$$7×(8×9)=(7×8)×9$$

.

B

$$7×(8+9)=(7×8)+9$$

C

$$(7+8)×9=7×(8+9)$$

D

$$(7-8)×9=7-(8×9)$$

Option A is Correct

# Multiplication Property of Zero

• The multiplication property of zero states that a number gets changed(becomes zero) when multiplying zero to that number.

$$a×0=0\,;\,b×0=0$$

• It doesn't matter what the number is, when we multiply it with zero, we always get zero as the answer.

Example:

$$2×0=0$$

$$100×0=0$$

$$52627×0=0$$

#### Which one of the following illustrates multiplication property of zero?

A $$11×7=7×11$$

B $$18×0=0$$

C $$57+41=41+57$$

D $$64+0=64$$

×

According to multiplication property of zero, when any number is multiplied with zero, we always get zero as the answer.

$$a×0=0\,;\,b×0=0$$

Among the choices only equation $$18×0=0$$ involves multiplication with $$0.$$

Also, it illustrates zero property of multiplication.

Hence, option (B) is correct.

### Which one of the following illustrates multiplication property of zero?

A

$$11×7=7×11$$

.

B

$$18×0=0$$

C

$$57+41=41+57$$

D

$$64+0=64$$

Option B is Correct