Understanding Zero 1

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In the chapter on Number Systems, we mentioned that the Roman system did not require a Zero and that the Indian Place Value System uses 0 in a special way.

What is Zero? The answer is given by most teachers is that &ldquo;Zero is Nothing&rdquo;. This is not only incorrect but also incomplete.

Zero plays multiple roles in Mathematics. Let us see all of them.

Zero cannot be represented with physical materials as the numbers 1 to 9 can be. Zero is just a mental concept.


 * 1) Zero is not &ldquo;Nothing&rdquo;. 				If we think for a little while we will realise that there is nothing in this world which can be called Nothing, in an absolute sense. Even if we go into remote parts of the Universe, there is space and energy.The idea of Nothing can only be contextual. Mathematically Zero indicates an absence of a particular object, idea or event in a specified context. In the context of a classroom, we say there are 23 boys when asked for the number of boys, though there are other things like girls, chairs &amp; benches also in that room. If it is a boys-only school, we could say that there are &ldquo;zero&rdquo; girls in that classroom.The idea of &ldquo;nothing&rdquo; (as in what is left when 5 pebbles are removed from a collection of 5 pebbles) evolved in all civilisations. However, most civilizations did not attach a mathematical sense or shape to &ldquo;nothing&rdquo;.
 * 2) Zero as a Place Holder 				As number systems evolved, various cultures realised that there has to be a symbol or a place holder to differentiate between &ldquo;thirty-six&rdquo; and &ldquo;three hundred and six&rdquo;, both of which use only the numerals for three &amp; six.Sumerians used to just leave a blank space where today we write 0.There is evidence that Hindus were using a dot &ldquo;.&rdquo; in place of zero, as early as the 3rdcentury AD. Later this evolved into the familiar shape &ldquo;0&rdquo;.  This is intimately related to the place value system which was invented by Hindus</li></ol>
 * 3) Zero is a number 				It was Hindus who gave a mathematical perspective to the idea of &ldquo;nothing&rdquo;. They also gave it a numeral shape. It has gone through several shapes and finally emerged as &ldquo;0&rdquo;.</li>Hindu &amp; Buddhist philosophies give a lot of importance to the ideas of &ldquo;emptiness&rdquo; which they called &ldquo;sunya&rdquo;. This cultural mooring possibly helped then to conceive of &ldquo;nothing&rdquo; as something!</li>The Hindus also realised that the symbol &ldquo;0&rdquo; can be treated as a numeral &ldquo;0&rdquo; which has no countable value. They treated number 0 as saying that there is nothing to count of a particular thing in a particular context. 0 was the numeral which represented the number Zero.</li>Hence,  we can add, subtract and multiply with zero. We can say 3 + 0 = 3, 3 - 0 = 3, 3 X 0 = 0 and 3 &divide; 0 cannot be defined mathematically. We will see about these in later chapters. The Indian mathematician Brahmagupta (7thcentury) was the first to give these rules.</li>It took many centuries before human civilizations accepted 0 as a number on par with 1 to 9. Numbers 1 to 9 are called counting numbers. Along with 0, numbers 0 to 9 are called whole numbers indicating quantity.</li>Until this time, in the Greek tradition, numbers were associated only with sets of things. Hence, before 0 was accepted as a number, even the numbers 1 to 9 had to be recognised as entities independent of &ldquo;set of objects&rdquo;. This was the first step of numbers becoming &ldquo;abstract ideas&rdquo; rather than &ldquo;concrete quantities&rdquo;.</li>Use of 0 also lets us inject some humour into the classroom and train children to say that &ldquo;I have 0 Geometry Box&rdquo; instead of &ldquo;I have not brought my Geometry Box&rdquo;!</li></ol>
 * 4) Division by zero 				The first attempt to define the division by zero was done by Brahmagupta around 628 CE. After that, Bhaskaracarya (c. 1150), while discussing the mathematics of zero in Bijaganita, explains that infinity (ananta-rasi) which results when some number is divided by zero is called khahara.</li></ol>
 * 5) Zero as the Origin OR Starting Point on the Number Line 				After the invention of the number line and the Cartesian plane, 0 has also acquired a meaning of an &lsquo;origin&rsquo;.</li>When we study the idea of a number line or the 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 reference to the position of 0.</li></ol>
 * 6) While dealing with integers or vectors, a value of 0 for an operation is interpreted as &ldquo;returning back to the starting point or position&rdquo;. 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

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