Multiply by Zero

Dhristi JEE 2022-24

What is Multiplication by Zero?

What happens when you perform multiplication of a number by 0? Multiplying a number by 0 makes the product equal to zero. Remember that the product of any real number and 0 is 0. For any real number m, m⋅0 = 0. As per the zero property of multiplication, the product of any number and zero (0), is 0. Now, we have already learnt that zero is the additive identity, given that it can be added to any number without modifying the number’s identity.


Fundamental Properties of Zero

Zero and one are actually very special numbers and, thus, hold special properties. But zero also has some unique properties concerning multiplication and division.

  • Addition Property of Zero

Zero is the additive identity and since it can be added to any number without changing the number’s identity. But zero has some special properties when it comes to multiplication and division. When multiplying a number by 0 it makes the product equal zero, so the product of any real number and 0 is 0.

Adding 0 to a numerical digit leaves its same. 0 is known as the additive identity and the property is referred to as the additive identity property.

6 + 0 = 6

1 + 0 = 1


  • Multiplication Property of Zero

Zero is the additive identity and since it can be added to any number without changing the number’s identity. The zero property of multiplication states that when multiplying a number by zero, the product is always zero. The zero can come before or after the number, which means the position of zero does not affect the property.

So, 2 x 0 = 0. The zero property of multiplication applies to all types of numbers, whether they are integers, fractions, decimals, or even algebraic terms. It should not be considered the same as the identity property of multiplication, which involves 1 as the identity element and in which the product is the number itself.

Zero times any numerical digit is equal to zero, meaning that multiplying any number by 0 provides 0.

0 × 6 = 0

1 × 0 = 0


  • Exponent Zero

Any number raised to power 0 is one. For example,

290 = 1

-570 = 1


  • Exponent of Zero

The number 0 raised to any power remains 0. For example,

039 = 0

0-4 = 0


  • Zero as a Numerator

0 divided by any non-zero number is 0. For example,

0 ÷ 7 = 0

0 ÷ 45 = 0


  • Zero as a Denominator

Any division by 0 is undefined. For example,

51 ÷ 0 = not defined

12 ÷ 0 = not defined


Ways to Represent Multiplication

A number ‘a’ multiplied by a number ‘b’ can be represented in several ways as given below in the table:


a⋅b

Using a centred dot.

Juxtaposition is easier and preferred, for variables.

The centered dot is very much useful for constants: e.g., 2⋅3 = 6.

ab

Using juxtaposition 

Placing items side-by-side.

It is a standard format to write a constant before a variable.

For example, we write 3a, not a3.

(a)(b)

Using parentheses.

Juxtaposition is easier and preferred, for variables.

Parentheses are required in situations like this: (a + 1)(b + 3)


Note: In algebra and beyond, while taking the variable x, do not use the multiplication symbol ‘×’ to denote multiplication, since it can be confusing with the variable x.

(Exception: it is conventional to use an ‘×’ for scientific notation).


Division of Zero

Dividing any number by zero is undefined. Division means splitting something into equal parts or groups so that it can be shared evenly among everyone. Though, the value of zero as a number is nothing. If it is placed before one, it is an even number that is neither positive nor negative. 

For any real number m, except 0, 0/m = 0, and 0 ÷ m = 0.

Zero divided by any real number other than 0 is 0. 

Note that, we can always check division with the corresponding multiplication fact. So, we know that

0 ÷ 5 = 0 since 0⋅5 = 0


Dividing With Zero

What about dividing a number with 0? Just imagine a real example: if there are no candies in the jar and five kinds want to share them, how many candies would each kid get? There are 0 candies to share, so there is no point in sharing something which isn’t present.

5÷ 0 = not defined


Solved Examples on Zero Property of Multiplication

Here are a few examples of the zero property of multiplication. These will help you learn the property of multiplication, how to multiply by 0 and its outcomes.

Example:

Simplify the expression 7/9 × 0

Solution:

Since it involves multiplication by 0 and any number we multiply by 0 = 0

Thus, 7/9 × 0 = 0.


Fun Facts

  • If there is 0 in multiplication, then the answer will always be 0.

  • Even the larger to largest numbers multiplied by 0 is 0. For example 9012765 × 0 = 0.

  • It really does not matter if the 0 comes first or not in the equation. For example, x multiplied by 0 = 0. Or: 0 × 7 = 0 or: 9 × 0 = 0.

  • It really does not matter how many numbers there are. If there is only multiplication taking place and then there is 0, the result will be 0. For example, = 459 × 9 × 0 × 5 = 0.

  • It does not matter the number of operations happening. If there is only multiplication taking place and there is 0, the result will be simply a 0. For example, 459 + 7 - 6 × 0 = 0.

  • 0 × 0 = 0 (since we have zero 0).

FAQs on Multiply by Zero

1. Define the properties of zero.

The properties of zero are:

  • Zero is even, which means it is neither odd nor neutral.

  • Zero is neither positive nor negative, it is the only number with this property.

  • Zero is an integer and must be considered when the question limits choices to integers.

  • Zero is a multiple of all numbers that means x*0 = 0, so 0 is a multiple of any x.

  • Zero is not a prime number.

  • Zero is neither black nor red. (in the game of roulette only).

2. What is division by zero called?

In mathematical concepts, division by zero has no meaning because any number divided by zero is considered undefined. Multiplication and division both will go hand-in-hand. When a number is multiplied with zero the result is a zero itself and in the case of division, it is undefined. Division by zero explains that when any non-zero, positive, or negative number is divided by zero the result is always undefined. For example 2/0 = infinity or undefined.

3. What are the facts about division and zero?

There are various facts about division and zero like:

  • Any number that is divided by 1, answers the same as the dividend. In other words, it can be explained that 1 is the divisor and the quotient will be equal to the dividend. Example : 2 ÷ 1 = 2.

  • A number can’t be divided by zero and the result is undefined. For Example: 8 ÷ 0 = undefined (but 0 ÷ 8 = 0).

  • When the dividend is equal to the divisor, that means the same numbers but not 0, then the answer is always 1. For example: 6 ÷ 6 = 1.

  • Zero is a real number, an integer, a rational number, and a whole number.

  • Zero is always neutral, that is zero is never written like +0 or -0.

  • The power of any number which is raised by zero is always one.

4. What are the facts about multiplication?

There are various facts about multiplication like:

  • In multiplication, the number that is multiplied is called the multiplicand and the number by which it is being multiplied is called the multiplier. 

  • The product of any number that is multiplied by zero is zero.

  • The product of any number that is multiplied by one is equal to the number itself.

  • In multiplication, the product can be equal to the numbers being multiplied or greater than, but never the smaller, except when one of the numbers is zero.

  • For every multiplication, there are two division facts.

  • In multiplication, the change in the order of the numbers being multiplied does not change the product.

Comment