Categories
Themes in Mathematics

Why is the number zero most important in mathematics?

Zero is the smallest number non-negative integer the immediately precedes 1. It is an even number as it as it is divisible by 2 with the remainder itself 0 { 0 ≡ 0 (mod 2)} i.e. no remainder . It cannot be termed as a positive or a negative number. The correct way of describing zero would be a number which equals to cardinality or an amount of null size.

1 is the natural number that follows 0 and there is no natural number that precedes 0. 0 is usually not considered as a natural number but it is definitely an integer and therefore a rational and real number. It also falls under the category of complex and algebraic numbers.

0 is usually presented as the central number in a number line. 0 can definitely not be termed as a prime number as it has a number of factors and cannot be composite as well. The reason behind 0 not been termed as a composite number is the inability to express the digit as a product of prime numbers as 0 is itself a factor.

In the field of Mathematics, there are some basic rules for working with the number 0.

Let x be any real or complex number

Subtraction can be done in 2 ways

x-0=x (positive number)

0-x= -x (negative number)

In addition 0 is the identity element i.e.

x+0=x

on adding any number with o the result is the number itself.

Division again yields different results

\(\frac{0}{x}\)=0

but \(\frac{x}{0}\)= undefined, as no real number multiplied by 0 produces 1 thus 0 does not contain any multiplicative inverse.

Multiplication of any number with 0 yields 0

x.0=0

and,

\(\frac{0}{0}\)=0 ,this expression is expressed in order to find the limit of the indeterminate form \(\frac{f(x)}{g(x)}\).This is called the indeterminate form.This implies that if the limit of \(\frac{f(x)}{g(x)}\) exists then it can be solved using L’Hospital’s rule.

Another very interesting fact about 0 is that 0! yields 1. It is an exceptional case of empty product.

0 is also used in propositional statements where it usually represents true or false depending upon a specific condition. It is also denoted as a zero element for addition and if defined, then zero is denoted an absorbing element for multiplication in the filed of abstract algebra.

It has several other applications in set theory ( where it is represented as the lowest ordinal number), lattice theory where zero is represented as the bottom element of a lattice(bounded), category theory and recursive theory as well. In category theory zero represents a initial value or object of a category.

Categories
Themes in Mathematics

How did Aryabhatta invent zero? How did he get this idea? Why did he give zero an oval shape?

Aryabhata was one of the major Mathematician-Astronomers belonging to the classical age of Indian Astronomy and Mathematics. Born in Pataliputra,Magadha, he is regarded as one of the greatest Mathematician of all time. His famous works include the ‘Aryabhatiya’ whose Mathematical parts consists of topics on algebra, trigonometry and arithmetic, continued fractions, sum of power series, quadratic equations and sine tables.

One of his discoveries is the approximation of pi which is given by him in Aryabhatia,

“Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached.”

The calculation is obtained as 3.1416 which is close to the actual value of \(\pi\)(3.14159).

Before going to Aryabhata’s invention of zero lets know a little bit about the Indian History of number zero.

Acharya Pingala, a Sanskrit scholar and an Indian Mathematician first used the Sanskrit word ‘Sunya’, referred to as Zero.The word ‘Sunya’ means void or empty. It is believed that the first text to use the decimal place value system(includes zero) was first used in Jain text or Cosmology named ‘Lokavibhaga’ . This is where the term ‘Sunya’ was used.

‘Bakshali Manuscript’, an Arithmetic manual on merchants records the symbol of zero which is a dot like structure having a hollow structure signifying void or nothing..These manuscripts were brought up by Radiocarbon dating ( which is a method of determining the age of an object using radiocarbon) in 2017. The ages were recorded to come from 224-383 AD, 680-779 AD, and 885-993 AD. This marks the world’s oldest record of the application of the symbol of Zero.

In Mathematics there is a term called the Decimal place Value System also called Positional Notation. This means that the value of a number is determined by the position of the digit that is the value of a number is actually the product of the digit by a factor which is determined by the position of the digit.

For example lets take three identical digits 999. Here the interesting part is in words the number is written as nine hundred and ninety nine . The hundreds tens and the units here are being determined by the position of the digits that is digit at the first place represents the units, second place represents the tens and the third place represents hundreds. Similarly any digit at the fourth place shall reprimand thousands.

This concept of the place value system, although was first used in ‘Bakshali Manuscript’ held a very important place in Aryabhata’s work. But the symbol for Zero was not used by Aryabhata. The use of Zero as a ‘digit’ was first used in India during the Gupta Period.

George Ifrah, a French Mathematician stated that the concept and understanding of zero as a ‘digit’ was first given by Aryabhata in his place value system because the counting system of digits is not possible without the place value system or zero. Also calculation performed by Aryabhata on square and cubic roots cannot be done if the numbers are not arranged in accordance with the place value system or zero. This concept of Zero is considered to be one of the best and greatest achievements of Indian Mathematics.

Now the rules for using Zero as a digit was first introduced in Brahmasputha Siddhanta, by Bramhagupta whereas in some stances his rules differ from the modern rules, one being on dividing zero by zero the result yields zero.