** Zero vs Null
**

Zero is a number in the set of real numbers, also an integer with an interesting history and properties. Seemingly unimportant, because carries no value; or more precisely an empty magnitude or a null value.

Of all the numbers in mathematics, zero has an important place in the history. It was one of the most intriguing and significant ideas in the development of mathematics. Mathematics is based on the numbers, and in the early days only the countable were used as numbers; therefore the set of numbers was limited to the set of natural numbers; as we call it today.

However, the introduction of the concept of zero spawned a newer set of numbers, which helped to extend the usage of mathematics. It is not a positive number or a negative number, therefore, the only real number neither negative nor positive. It is the additive identity. Also, in positional number systems, zero is used as a digit too.

The first rule regarding the mathematical properties of zero was presented first by an Indian mathematician Brahmaguptha in his book Brahmasputha Siddhanta, and they are as follows:

*The sum of zero and a negative number is negative.**The sum of zero and a positive number is positive.**The sum of zero and zero is zero.**The sum of a positive and a negative is their difference; or if their absolute values are equal, then zero.**A positive or negative number, when divided by zero, is a fraction with zero as the denominator.**Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as the numerator and the finite quantity as denominator.**Zero divided by zero is zero.*

In contrast to the modern mathematical definition, his views allow division by zero, which is considered as an undefined state in the modern mathematics. This clearly states the importance of zero as an additive identity. Properties of the commonly used operations are as follows:

*Addition: x + 0 = 0 + x = x*

*subtraction: x – 0 = x and 0 – x = -x*

*Multiplication: x × 0 = 0 × x = 0*

*Division: 0/x = 0 and x/0 is not defined*

*Exponentiation: x ^{0 }= x^{1-1 }= x/x = 1 but when x = 0 i.e. 0^{0} is not defined at times*

*Factorial: 0! = 1 : Factorial of zero is defined as 1*

Null is a term in mathematics implying the empty / void value or quantity. It is synonymous with zero, but it may differ based on the context.

Null vector is a vector with all the elements as zero, and null is also applied in the same sense to the matrices with all zero elements. An empty set is often known as a null set while an empty graph is known as a null graph. Many definitions like these can be found with the term ‘null’ implying the emptiness or whole zero composition of the entity.

**What is the difference between Null and Zero?**

• Zero is a number in the set of real numbers with empty magnitude while null is a term used to denote the empty nature of a quantity or an entity.

• Zero is a number representing a null quantity and the additive identity.

• Null is often used synonymously with zero when used to represent emit nature of the variable or mathematical entity (e.g. Null vector or null graph), but in the set theory, null set is an empty set, i.e. it’s a set without any elements in it, but the cardinality of the set is zero.

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