Zero in Maths: Imagine, just for a moment, what Mathematics would be like without zeros. Mathematical and algebraic calculations would have been difficult because there would be no tens, thousands, or lakhs. Take a tin box and put a few pebbles inside. Shake the box, can you hear the sound? Now, remove the pebbles and shake the box again. Can you hear the sound? If the answer is no, then you got it right. Due to the absence of pebbles inside the box, no sound can be heard, meaning there are zero pebbles in the box.

Even though Zero is nothing, it is the most important element in all mathematical calculations. A zero (0) placed before any number has no value, but a zero placed after any number has a significant impact on its value. In this article, let’s understand the number 0 and its history.

History of Zero

Historically, the first recorded zero appeared in Mesopotamia around /(3/) B.C. The Mayans independently devised it around /(4/) A.D. It was then introduced to India in the middle of the fifth century, reached Cambodia near the end of the seventh century, and arrived in China and the Islamic countries at the end of the eighth century.

Definition of Zero

Zero is a special number. It is neither positive nor negative. The number zero is denoted with the ‘\(0\)’ symbol. Zero appears to be as follows:    

In simple words, Zero number means nothing or empty. Let’s understand Zero number with some real-life examples given below:

The white plate has \(7\) ladoos and the pink plate has no ladoos. It means there are \(0\) ladoos.

Now count the number of each fruit given below.

Number System

Mathematicians rely heavily on numbers since numbers play important role in mathematics. In lower grades, we learned about counting numbers. However, in this article, we will review them and broaden our knowledge to include a large number of concepts.

Notation

Writing a number in a figure is called notation.

Numeration

Writing a number in word is called numeration.

Hindu-Arabic System of Writing a Number

we use the symbols namely \(0,{\rm{ }}1,{\rm{ }}2,{\rm{ }}3,{\rm{ }}4,{\rm{ }}5,{\rm{ }}6,{\rm{ }}7,{\rm{ }}8\) and \(9\) called digits or figures, to represent any number.

Place Value Chart

We use the place-value chart to represent a number. For a given numeral, we start from the extreme right as Ones, Tens, Hundreds, Thousands, Ten Thousands, Lakhs and Ten Lakh etc.

Example: Given below a number arranged in a place-value table,

The given numeral is \(‘9038’.\)

It is ‘nine thousand thirty-eight’.

In the expanded form we can write, \(9038 = 9 \times 1000 + 0 \times 100 + 3 \times 10 + \left( {8 \times 1} \right).\)

Use of Zero in the Formation of Numbers

We can say that a number is the arranged group of digits formed while forming numbers with the given digits. Numbers can be formed with or without digit repetition. Let’s look at how zero is used in number formation.

Example: Let us say that the given digits are \(1,{\rm{ }}2,{\rm{ }}0,{\rm{ }}4\) and we need to form a \(4\)-digit number.

If we will keep \(0\) on the extreme left side, we will get a \(3\)-digit number. As \(0\) has no value on the left side of a number.

The formed number is \(0124\) or \(124\). 

So, we cannot use \(0\) at the extreme left to form a \(4\)-digit number.

If we keep \(0\) on any place (except extreme left), then we can form four-digit numbers.

Example: \(1024,{\rm{ }}1204,{\rm{ }}4021, \ldots \) etc.

Types of Numbers

The whole of the number system is divided into real and imaginary (or complex) numbers. Imaginary or complex numbers are those that cannot be represented on the number line. Real numbers are those that can be represented on a number line. 

What are Real Numbers?

Real numbers are the combination of rational numbers and irrational numbers. Real numbers can be both positive and negative which is denoted as ‘R’. Natural numbers, fractions, decimals all come under this category.

Types of Real Numbers

The types of real numbers are as given below:

1. Natural numbers N: Natural numbers are the counting numbers from \(1\). They are \(1,{\rm{ }}2,{\rm{ }}3,{\rm{ }}4,{\rm{ }}5,{\rm{ }} \ldots \)  Natural numbers are a subset of real numbers. It is denoted by N.
2. Whole numbers W: Numbers \(0,{\rm{ }}1,{\rm{ }}2,{\rm{ }}3 \ldots \) are called whole numbers. It is a natural number including \(0\) . Whole numbers are also a subset of real numbers. It is denoted by W.
3. Integers Z: The numbers …, -\(3,{\rm{ }} – 2,{\rm{ }} – 1,{\rm{ }}0,{\rm{ }}1,{\rm{ }}2,{\rm{ }}3,{\rm{ }}4,{\rm{ }}5,{\rm{ }} \ldots \) are called integers. Integers are also a subset of real numbers. It is denoted by Z.
4. Rational numbers Q: A rational number is defined as a number that can be expressed in the form of  \(\frac{p}{q}\), where \(p\)  and \(q\) are co-prime integers and \(q \ne 0\). Rational numbers are also a subset of real numbers. It is denoted by \([Q]\). Examples: \( – 23,{\rm{ }}0,{\rm{ }}5,310,{\rm{ }} \ldots ..\) etc.
5. Irrational numbers P: An irrational number is defined as the number that cannot be expressed in the form of  \(\frac{p}{q}\), where \(p\)  and \(q\) are co-prime integers and \(q \ne 0\). Irrational numbers are also a subset of real numbers. It is denoted by P. Examples: \(2,5,\pi \) , e, ….. etc.

How to Represent Zero in the Number line?

The representation of Zero in the number line is given below:

In the number line, positive numbers are placed at the right of zero. Negative numbers are placed at the left of zero \(\left( 0 \right).\) Zero is unbiased, which means neither positive nor negative. The arrows at either end of the number line indicate that the number line extends unlimited in both directions from zero.

Properties of Zero

1. Zero is considered the additive identity for all types of numbers. If zero is added to any number, the result remains the same. 
2. Zero is an even number.
3. Zero is neither positive nor negative.
4. Zero is a whole number.
5. Zero is an integer.

Interesting Facts About Zero

1. Brahmagupta invented Zero in \({\rm{628AD}}\) in India. Used later by the Persians and Arabs and then in Europe.
2. Zero is neither positive nor negative. In the number line, positive numbers start from \(0\) and extend towards the right of the number line. The negative number also start from \(0\) and extends towards the left side of the number line. It means zero comes in between negative and positive numbers in the number line.
3. Additive identity is a number, which when added to any number, gives the sum as the number itself. It means additive identity for any number is ‘\(0\)’, as by adding \(0\) to any number, we get the number itself as the sum.

Addition with Zero:

When \(0\) is added to any number, gives sum as the number itself.

That is, \({n + 0 = n}\)

Example: \(23 + 0 = 23\)

Subtraction with Zero:

\(0\) is subtracted from any number, gives the difference as the number itself.

That is, \({n – 0 = n}\)

Example: \(49 – 0 = 49\)

Multiplication with Zero:

When \(0\) is multiplied by any number, the product is \(0\).

That is, \(n \times 0 = 0\)

Example: \(27 \times 0 = 0\)

Dividing Zero:

When \(0\) is divided by any number, the result is \(0\).

That is, \({0 \div n = 0}\)

Example: \(0 \div 7 = 0\)

Square root of Zero:

The square root of \(0\) is \(0\).

Example: \(\sqrt 0  = 0\)

Power with Zero: 

The power of any number raised by zero is one.

That is, \({{n}^{\rm{0}}}{\rm{ = 1}}\)

Example: \({{\rm{5}}^{\rm{0}}}{\rm{ = 1}}\)

Is Zero Even or Odd Number?

An even number is a multiple of \(2\).

We see that zero can be represented as \(0 = 2 \times 0\).

Hence, zero is an even number, not an odd number.

What Type of Number is Zero?

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

The set of integer numbers:

\({\rm{Z  =  }}\left\{ {{\rm{0, 1, 2, 3, 4, 5 6, \ldots }}} \right\}\)

Zero is an element of the set of integer numbers:

\(0{\rm{ }} \in {\rm{ }}Z\)

Hence, zero is an integer.

A rational number is a number that can be expressed as the quotient of \(2\) co-prime integers.

\(Q = \left\{ {\frac{n}{m};n,m \in Z} \right\}\)

Zero can be written as a quotient of \(2\;\) integers.

Example: \(0 = \frac{0}{5}\)

Hence, zero is a rational number.

Zero is unbiased, which means there is no such things like \(+ 0\) or \(- 0\). 

Exponent:

\({0^n} = 0\)

Example: \({0^7} = 0\)

Logarithm: \({\log _b}(0)\) is undefined.

Factorial: \(0!{\rm{ }} = {\rm{ }}1\)

Sine: \(\sin {0^{\rm{o}}} = 0\)

Cosine: \(\cos {0^{\rm{o}}} = 1\)

Tangent: \(\tan {0^{\rm{o}}} = 0\) 

Solved Examples – Zero

Question-1: Add \(3509 + 0.\)
Ans: As we know, the addition of \(0\) to any number, the result is the number itself.
So, \(3509 + 0 = 3509\)
Hence, the sum is \(3509\).

Question-2: What will be the result, when you will subtract \(0\) from \(16998\)?
Ans : As we know, \(0\) is subtracted from any number gives the number itself as the difference.
So, \(16998 – 0 = 16998\)
Hence, the difference is \(16998\).

Question-3: Find the quotient after dividing \(0\) by \(521\).
Ans: We know, quotient is the result of division.
Again, we know, when \(0\) is divided by any number, the result is \(0\).
So, \(0 \div 57 = 0\)
Hence, the quotient is \(0\).

Question-4: What will be the product of \(987\) and \(0\).
Ans: As we know,
The product of \(0\) and any number is always \(0\).
Thus, \(987 \times 0 = 0\)
Hence, the product is \(0\).

Question-5: Find \({0^{10}}\) and \({10^0}\).
Ans: As we know, \({0^n} = 0,n\) is any number
So, \({0^{10}} = 0\)
Again, we know \({n^0} = 1,\,n\) is any number
So, \({10^0} = 1\)
Hence, the value of \({0^{10}} = 0\) and \({10^0} = 1\).

Summary

In this article, we learned many things about the number ‘0’ (Zero) which covered the history of zero like who invented it, from where the number zero came into existence, properties of zero, use of zero in Mathematics etc. And we hope the above details provided in the articled helped you to enhance your knowledge understand the basics of mathematics. 

Frequently Asked Questions About Zero

Let’s look at some of the commonly asked questions about zero:

Q1: Who found zero in India?
Ans: The first person to document 0 (zero) as a number was the astronomer and mathematician Brahmagupta in \(628\)CE. 

Q2: Who invented zero?
Ans: Zero appeared in Mesopotamia around \(3\) B.C (it was first recorded). The Mayans invented it independently in \(4\) A.D. It was later devised in India in the mid-fifth century, spread to Cambodia near the end of the seventh century, and into China and the Islamic countries at the end of the eighth. 
Aryabhata is the first of the great astronomers of the classical age of India. He gave the world the digit ‘0’ (zero) for which he became immortal.

Q3: What is the importance of zero?
Ans : Zero is a number, and the numerical digit used to represent the number in numerals. It has a great role in mathematics as the additive identity of the integers, real numbers and many other algebraic structures. As a digit, \(0\;\) is used as a placeholder in place values.

Q4: How do you write a number zero?
Ans: We write zero in number as ‘\(0\;\)’.

Q5: What is the concept of zero?
Ans: Absent of anything or nothing is equivalent to zero.

Q6: Who is the father of mathematics?
Ans: The Father of Mathematics is Archimedes. He was born in Syracuse, the Greek island of Sicily. Archimedes was serving the King Hiero II of Syracuse by solving mathematical problems and by creating interesting innovations for the king and their army.

Now that you have all the necessary information about the number Zero, we hope this detailed article is helpful to you. If you have any queries about this article or in general about Zero, let us know in the comment box below and we will get back to you as soon as possible.