Integers Rule- Multiplication of Integers

Let’s find out some easy tips and tricks to multiply integers.

Multiplying integers:

Integers consist of zero, all natural numbers and the additive inverse of all natural numbers. Therefore, integers can either be positive or negative and thus they have a magnitude and a sign associated with them.

For example: -53697,0,7,3,-12,-41,-863655,3286600 etc.

Integers are represented using Z or I and they can be located on the real number line as shown below.

Number Line

Figure 1 Integers

The numbers lying on the right-hand side of zero are positive, and those lying on the left-hand side are negative integers. Thus, the position of any integer can be identified using the real number line.

Now, as we have a brief idea about what integers are, let us look at the mathematical operations associated with them.

The four basic operations viz. Addition, subtraction, multiplication and division are applicable to integers. Similar to natural numbers, the operation of multiplication is closed for all integers. It means that when integers are multiplied, the result is also an integer.

Multiplication implies repetitive addition. Suppose two numbers x and y are to be multiplied. Then, considering x as the multiplier and y as the multiplicand, the multiplication is given as:

Multiplication of Integers

If x and y are integers, then depending upon the sign of both these integers, following four cases can arise:

i.  x is positive(+) and y is positive(+)

ii. x is positive(+) and y is negative(-)

iii. x is negative(-) and y is positive(+)

iv. x is negative(-) and y is negative(-)

Let us now look into the multiplication process of integers.

i) Multiplication of two positive integers

If x and y are two positive integers, then the simple rule of multiplication is followed, i.e. y is repeatedly added x times. As both integers have positive sign, therefore, sign of the product (P) is also positive.

(+x) × (+y) = +P

Where x and y are integers and P is their product.

For example Multiply 7 and 5

Solution: (+7) × (+5) = 5 + 5 + 5 + 5 + 5 + 5 + 5 = 35

This can be understood with the help of the number line.

ii) Multiplication of a positive and a negative integer

If x is a positive integer and y is a negative integer, then their magnitudes are multiplied according to simple multiplication, and the negative sign is retained. Also, multiplication of integers is commutative i.e. order of the terms does not affect the result.

(-x) × (+y) = -P or (+x) × (-y) = -P

Where x and y are integers and P is their product.

For example: Multiply 7 and -5

Solution: (+7) × (-5) = (-5) + (-5) + (-5) + (-5) + (-5) + (-5) + (-5) = -35

iii) Multiplication of two negative integers:

If two integers x and y are both negative, then the magnitude of both integers is multiplied and the sign of their product is positive.

(-x) × (-y) = +P

Where x and y are integers and P is their product.

For example: Multiply-7 and -5

Solution: (-7) × (-5) = -[(-5) + (-5) + (-5) + (-5) + (-5) + (-5) + (-5)] = 35

To summarize, the product of two integers with similar sign will always be positive. This means, the product of two positive integers or two negative integers will always give a positive value. While the product of a positive integer and a negative integer will always give a negative value. This can be remembered using the following table:

S.No. Type of  Numbers Operation Result Example
1 Positive × Positive Multiplication Positive (+)  6 × 9 = 54
2 Negative × Negative Multiplication Positive (+)  -6 × -9 = 54
3 Positive × Negative Multiplication Negative (-)  6 × -9 = -54
4 Negative × Positive Multiplication Negative (-)  -6 × 9 = -54

Since multiplication follows the commutative law, therefore the result of rule 3 and 4 in the above table is same.

It is interesting to observe that in a case where multiple numbers are to be multiplied, then the sign of the product is negative if there are an odd number of negative terms and positive if the number of negative terms is even.

Practise This Question

Can a vector have zero component along a line and still have non-zero magnitude?