Multiply by Zero

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Multiplication By 0

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.


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Math Fundamentals Properties of 0

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

  • Addition Property of Zero

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

0 × 6 = 0

1 × 0 = 0

  • Exponent Zero

Any number raised to power 0 is one

290 = 1

-570 = 1

  • Exponents of Zero

The number 0 raised to any power remains 0.

039 = 0

0-4 = 0

  • Zero as a Numerator

0 divided by any non-zero number is 0.

0 ÷ 7 = 0

0 ÷ 45 = 0

  • Zero as a Denominator

Any division by 0 is undefined

51 ÷ 0 = not defined

12 ÷ 0 = not defined


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


Ways to Represent Multiplication

a⋅b

Using a centred dot

Juxtaposition is easier and preferred, for variables.


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

a⋅b

Using juxtaposition

Placing items side-by-side

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


For example, 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 0

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.


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 each kid gets 0 candies.

0 ÷ 5 = 0

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

0 ÷ 5 = 0 since 0⋅5 = 0 because 0⋅5 = 0.


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 irrespective.

  • 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 are there. 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 numbers of operations happening. If there is only multiplication taking place and then 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) lots of nothing.

FAQ (Frequently Asked Questions)

Q1. What is Zero?

Answer: 0 also written as zero is a mathematical number. 0 is the numerical digit used to represent this number in numerals. Zero (0) plays a cardinal role in mathematics as real numbers, the additive identity of the integers, and many other algebraic structures. As a numerical digit, 0 is also used as a placeholder in place value systems.

Q2. What are the Basic Properties of Zero (0)?

Answer: The number zero (0) can be used in countless ways to fiddle with students and change seemingly wholesome questions into head-scratchers. That being said, let’s review some of the basic properties of zero:

  1. Zero is a multiple of all numbers (x × 0 = 0, thus a multiple of any x).

  2. Zero is an even number (not odd, not neutral).

  3. Zero is not positive not negative (the only number with such property).

  4. Zero is an integer (and must be regarded when question restricts choices to integers).

  5. Neither Zero nor 1 is a prime number (the smallest prime number is 2).

  6. Zero is neither red nor black in colour (pertains to roulette only).

Q3. What is the ‘Additive Inverse of Zero (0)?

Answer: There are names for the number zero.

The numbers 5 and −5 are opposites; they are the same distance from 0 but on opposite sides of 0. Any number added to its opposite is 0. For all real numbers.


a, a + (−a) = (−a) +a=0. Note that the ‘opposite of a’ is also referred to as the ‘additive inverse of a’: it is the number which, when added to a, gives zero as an outcome.