Impact of Zero

The invention of Zero had several impacts on the development of math.

Numbers as Abstract Concepts

Until this time, in the Greek tradition, numbers were associated only with sets of things.

After 0 was accepted as a number, even the numbers 1 to 9 had to be recognised as entities independent of "set of objects".

This was the first step of numbers becoming "abstract ideas" rather than "concrete quantities."

That is to say a set of “human eyes”, “bird’s eyes”, “human legs”, “human hands” all possess the quality of “twoness”.

The word “two” used as an adjective in the phrase “two eyes”, became an abstract noun in the phrase “the set of human eyes has a property of “twoness”.

Natural Numbers to Whole Numbers

Numbers 1 to 9 & so on are called Counting or Natural numbers.

Along with 0, numbers 0 to 9 & so on are called Whole numbers.

Cardinality of a set

The idea of whole numbers gave rise to the property of “cardinality” of sets with which several sets can be compared on the basis of their cardinality.

We can say that the set of “human fingers in one hand” is more than the “set of eyes.” Or the cardinality of a set of “human fingers in one hand” is more than the cardinality of a “set of eyes”.

Zero as a Point on the Number Line

Numbers could be “plotted” on a number line.

Zero was the natural “starting point” of the number line.

Zero and Integers

The placement of zero at the beginning of the number line enabled negative numbers to be conceptually marked on the opposite direction to the positive whole numbers.

So zero gave a mathematical base for understanding negative numbers.

They were defined as "additive inverses" of whole numbers. The sum of a number and its "additive inverse" was 0.

Zero as a point on the Cartesian Plane

When we study the idea of a 2 D plane in Coordinate Geometry, 0 can be arbitrarily placed anywhere and becomes the reference point around which the line or the plane are constructed.

Every point is referred to in relation to the position of 0.

Zero and Vectors

While dealing with integers or vectors, a value of 0 for an operation is interpreted as "returning back to the starting point or position".

Moving from a particular position and coming back to it after certain moves defines the sum total of these moves as 0 is A+(-A) = 0

Zero as a Measure/Score

Having a numeral for zero helps us to indicate that the score at half time in a match is 3:0.

It also helps a teacher to indicate that a student had not scored any marks in an examination.

Zeros in Algebra

In the 18th century, a new method was found for solving algebraic equations by shifting all the terms to one side of an equation leaving the other side equal to 0.

An expression like x2 + 4x = 3 was re-written as x2 + 4x - 3 = 0.

This method had far-reaching consequences in the solution of algebraic equations of various types. The solutions of an equation also were referred to as "zeros" of the equation!

This method also provides a visual idea of the solutions to the polynomial. When the polynomial is plotted on a XY plane, the solutions are values of x where the curve intersects the X axis (where y=0)

Position of Zero changes value of a number

With the invention of the place value system, the value of the numeral 0 started depending on its position.

If 0 is added to the right end of a number it will be the effect of increasing the value tenfold, i.e multiplying it by 10! 45 becomes 450 which is ten times 45.

If 0 is added to the left end of a number, its value does not change at all. This is because the place value of the rest of the numerals remains unchanged and the value of 0 still remains 0. 45 is same as 045. 0 is in the hundred's place and its value is 0.

In whole numbers "leading zeros" have no value and hence can be omitted.

But while learning decimal numbers, we will see that these rules need to be modified as the conceptual understanding behind decimal representation is slightly different. In decimal numbers "trailing zeros" have no value and hence can be omitted.

Zero & Data Entry

Modern data collection necessitates the use of 0 in certain places for purpose of clarity.

Many times, in government forms, 8 spaces are given to write your date of birth and in any government form every space is to be filled.

If your date of birth is 6th April 1960, you have to fill the form as 06 04 1960, writing 6 as "06" and 4 as "04"

A Typical Birthday Confusion

The subtle fact that 0, though included in the number system, is not a counting number can cause some confusions in calculating some national days.

Consider that a school is celebrating the Independence Day on 15th August 2015. What is the count of the Independence Day in 2015?

If a child was born on 15th August 1948, its age on 15th August 2015 is calculated as 2015-1948 = 67. It is considered that on the day of birth, the child is 0 years old.

With the same logic, many a time, the Independence Day in 2015 is sometimes called the 67th Independence Day. This is wrong because for a country, 15th August 1948 was the 1st Independence Day and that on 2015 is actually the 68th.

Absolute Zero

We have been talking till now of zero as a mathematical concept. But advancements in science have pointed to the existence of a zero in the physical world - the Absolute Zero temperature.

Temperature is defined as the average kinetic energy of particles. Hotter objects have particles with higher speeds, while colder objects have particles with lower speeds. In other words, temperature and the average speed of particles are directly proportional.

Lord Kelvin showed that as we cool the material, the motion of the molecules and hence their energy reduces. There is a point at which all motion ceases.

Absolute Zero temperature is the "hypothetical" temperature at which all motion ceases. Hypothetical because such a temperature is physically unattainable.

It is the starting point of the Kelvin Scale. Absolute zero or 0K corresponds to -273.15° in the Centigrade scale.