ZERO
People can find something interesting in everything. However, what is there interesting in nothing?
Number zero is one of the most interesting, most mysterious and most beautiful numbers in mathematics. Maybe it sounds surprising, but it’s one of the most important discoveries in the early history of mathematics.
Today, scientists agree that it ranks between the five most famous numbers: one, zero, Euler's number, Ludolph's number and number i. These numbers are connected in the relation:
eiπ + 1 = 0
In 1988 this relation won an inquiry in the newspaper The Mathematical Intellingencer as the most beautiful mathematical allegation in history.
0 AS A NUMBER
0 is the number preceding number 1.
0 is neither a positive nor negative number.
0 is neither a prime number nor a composite number, nor is it a unit.
Zero is a number which quantifies a count or an amount of null size.
The number 0 is a whole, rational and real number (as well as an algebraic number and a complex number). In the most cases zero isn’t considered a natural number. However, in the 19th century the first set theory in which zero belongs to natural numbers was developed. Nowadays, zero is a natural number also in logic, discreet mathematics and in IT.
0 AS A DIGIT
The modern numerical digit 0 is usually written as a circle, an ellipse, or a rounded rectangle. The height of the 0 is the same as of other digits..
On calculators, watches, etc., 0 is usually written with six line segments. However, on some historical calculator models it was written with four line segments.
The number zero is not the same as the digit zero, used in numeral systems using positional notation. The digit zero is used to skip a position and also in big numbers. A zero digit is not always necessary in a positional number system, for example, in the number 02.
A comparison of the zero and the letter O
HISTORY OF NUMBER ZERO
In past people did not use big numbers, so for a long time they did not need numeral zero. For example, ancient civilisations, such as Romans used numbers in a way that it did not permit them to enter really big numbers (if you wanted to write down the number 12 million in Roman numeral system, it would be very difficult) because in real life they did not need these numbers. When they wanted to write down a bigger number, e.g. 87 they had some problems. Greeks needed only 2 signs to express this number. However, Romans needed seven signs (LXXXVII) and Egyptians 12 signs (they wrote 8 horseshoes and 7 lines). And if they wanted to express a really big number, they could not do it. Fortunately, medieval penmen and clerks did not use a written form of numbers very often, because they used a special counting table, called an abacus, with gutters that expressed tens, thousands and hundreds.
The numeral zero did not appear until emergence of the most advanced type of a number system, which is called "positional" because the value of a character depends on its position. However, not all civilizations which used a positional system also used the numeral zero. For example, Babylonians used a positional system, but they did not write anything at places where we write numeral zero (that means that number 102 looked similar like number 12). And how did they know which number were they speaking about? From the context.
Later, when it was not clear from the context, they started to use a space instead of numeral zero (that means that number 102 looked like this: 1 2). And after 1500 years, they started to use a special symbol to express number zero.
Zero was used in the 3rd century BC for the first time. It is proved by a table that was found in Uruk.
A similar system of expressing zero was used by the Mayas, who wrote zero as a shell, and the Greeks (e.g. Greek astronomer Ptolemaios). However, zero as a number was discovered in India. They were the first who counted with number zero (it was around the year 458). In the 7th century the Indian mathematician and astronomer Brahmagupta introduced several rules about the number zero:
- 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 they are equal, 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 numerator and the finite quantity as denominator.
- Zero divided by zero is zero.
It’s clear that the last Brahmagupta’s rule is wrong because zero divided by zero is not zero.
In fact, a division by number zero has always been very disputable.
In 830, around 200 years after Brahmagupta wrote his rules, the Indian Mahavira wrote Ganita Sara Samgraha as an update of Brahmagupta's book. He said that:
A number remains unchanged when divided by zero.
Another Indian who was also interested in the number zero was Bhaskara. He said:
When we divide some number with zero, we get an infinity value.
So Bhaskara tried to solve the problem by writing n/0 = ∞. At first sight we might think that Bhaskara was correct, but of course he was not. If it was true then 0 times ∞ must be equal to every number n, so all numbers are equal.
However, Bhaskara correctly wrote other properties of zero: e.g. 02 = 0 and √0 = 0.
The journey of zero to Europe was not easy. The first person who described a new numeral system that used the number zero was a Persian mathematician Al-Khwarizmi. In his book he recommended an oval shape for the number zero.
In 967, a Christian monk Gerbert from Aurillac detected a mystery surrounding the number zero. He went to Spain and, masked as a Muslim, he went to an Arabian library in Cordoba and learnt everything about zero there. However, when he came back to France where he wanted to propagate his knowledge, he was accused of heresy and of conjunction with devil.
In 1202 an Italian mathematician Leonardo Fibonacci from Pisa wrote a book Liber Albaci where he underlined practical uses of zero. After that, Italian businessmen started to use number zero and they became an example for all of Europe.
However, for the Christian world, zero remained a big mystery, nay something elusive.
In 1828, the problem of division by zero was solved by a German, Martin Ohm, who solved it simply: by outlawing division by zero.
ETYMOLOGY OF THE WORD ZERO
The word zero comes from French zéro, from Italian (Venetian) zero, from Mediaeval Latin zephirum, from Arabic صفر (ṣifr, “nothing, cipher”) which was used to translate Sanskrit śūnya meaning void or empty.
RULES OF COUNTING WITH ZERO
Addition: x + 0 = 0 + x = x.
Subtraction: x − 0 = x and 0 − x = − x.
Multiplication: x · 0 = 0 · x = 0.
Division: 0/x = 0, x/0 is undefined
Exponentiation: x0 = 1, except that the case x = 0 (it may be undefined in some contexts), 0x = 0.
USES OF ZERO
Nowadays we cannot imagine our lives without the numeral 0. Number zero expresses nothing and it is also used as an empty place indicator in our numeral system. For example, if there was not the numeral zero, we could not recognise a difference between number 26 and number 2006, which means something quite different.
People use number zero every day. We often talk about temperatures (if it is higher or lower than zero), about altitude (if it is higher or lower than the sea level), etc. Every day, we use words as minus and plus which express if something is more or less than zero.
On the one hand, the number zero expresses nothing. On the other hand, it is also very important when we want to express big numbers, such as millions and billions.
Mathematics
- In set theory, 0 is the size (cardinality) of the empty set (if you do not have any apples, you have 0 apples). The empty set is a set with no elements.
- In set theory, 0 is also the least ordinal number (a number that expresses position in a series, such as the 1st, 2nd, or 3rd).
- In propositional logic, 0 may be used to express a false mathematical statement (the truth of the mathematical statement is expressed by number 1)
- In abstract algebra, 0 is commonly used to denote a zero element, which is a neutral element for addition and an absorbing element for multiplication.
- The zero function (or zero map) on a domain D is the constant function with 0 as its only possible value, i.e., the function f defined by f(x) = 0 for all x in D.
- Zero is the division between positive and negative numbers
Physics
The value of zero plays a special role for many physical quantities.
For some quantities, the zero level is naturally distinguished from all other levels. However, for some quantities the zero level has been chosen by physics. For example, on the Kelvin temperature scale, zero is the coldest possible temperature (negative temperatures exist but are not actually colder), but on the Celsius scale, zero is defined to be at the freezing point of water.
Chemistry
Zero has been proposed as the atomic number of the theoretical element tetraneutron. (an element with no protons and no charge on its nucleus).
Computer science
Zero is very useful also in computer science. All computers, notebooks, laptops, etc. work on the principle of a binary system which is founded on a long sequence of zeros and ones.
Computers` logic is very difficult. It can only differ if there is the current in a cable or if there is not. However, using the binary system computers can get information about all numbers. And with this information they can recognize also colours, sounds, words, etc. which are transcribed into these numbers.
The number zero expresses that there is not a current in a cable and number one expresses that there is a current.
How to convert numbers in the binary numeral system to the decimal numeral system? The binary (base two) numeral system has two possible values, often represented as 0 or 1, for each place-value. In contrast, the decimal (base ten) numeral system has ten possible values (0, 1, 2, 3, 4, 5, 6, 7, 8, or 9) for each place-value.
For example, let's convert the binary number 10011011 to a decimal number. At first, list the powers of two from right to left. Start at 20, evaluating it as "1". Increment the exponent by one for each power. Stop when the amount of elements in the list is equal to the amount of digits in the binary number. The example number 10011011 has eight digits, so the list, to eight elements, would look like this: 128, 64, 32, 16, 8, 4, 2, 1. The resultant number will we get after adding these elements.
Other fields
- In some countries, 0 on a telephone places a call for operator assistance.
- In Braille, the numeral 0 has the same dot configuration as the letter J.
- In classical music, 0 is very rarely used as a number for a composition. Famous symphonies No.0 are: Symphony No. 0 in D minor and Symphony No. 00 by Anton Bruckner and Symphony No. 0 by Alfred Schnittke.
INTERESTING INFORMATION ABOUT ZERO
- There is no year zero in the BC / AD year numbering system. When the numbering goes from BC (before Christ) to AD (anno domini) the year numbering goes: 3BC, 2BC, 1BC, 1AD, 2AD, 3AD. Therefore, the third millennium and the 21st century begin on 1 January 2001. However, many people throughout the world celebrated the new millennium on 1 January 2000. Of course they made an error and celebrated the passing of only 1999 years.
- Division by zero is undefined. Why? Take for example the fact that the result of division is bigger when the number by which you divide is bigger. We can see this on an easy example: if we divide the same number by 5 and then by 3, the answer you get as a result of the division by three would be bigger. So it would seem that if we divided a number by a fraction so small as to be almost zero the resulting answer would be infinitely large. So if this is true why division by zero cannot be defined? We know that the answer should be infinity. However, that is not correct. Let’s look at this simple equation:
2X + 6 = X + 3
When we solve this equation for X we get
X = -3
But we can also use the rules of algebra and then the result is:
2(X + 3) = X + 3
Divide both sides by X + 3 and the result is:
2 = 1
And it is not correct.
If you wanted to define division by zero you would find many problems like this that prove why it cannot be done.
ZERO IN FOREIGN LANGUAGES
English: nought, naught, ought, nil, null, zero
German: Null, Nullpunkt
Spanish: cero
Italian: zero
Holland: nul
Finish: nolla
Norwegian: null
Swedish: noll
Poland: zero
Hungarian: nulla, zéró
Czech: nula
Slovak: nula
Latin: nihildum, nusquam
Hebraic: efes [ אפס ]
Chinese: líng
Russian: нуль, ничто
Croatian: ništica, nula