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Number

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Numbers are used for counting, for measurement, for placing things in order, and for identification. For example: there are 3 apples in a bowl; a mile is about 1.609 km; she was third in the race; part number 378691236.

Most modern civilisations use the decimal number system (sometimes called base-10 or denary), with positional notation to represent numbers greater than 9, and to represent numbers which are not whole numbers (i.e. with a decimal point).

Thus a number written as 123 is interpreted as 1×100 + 2×10 + 3, with each position getting more weight as we move from right to left. This example can also be written as: 1×102 + 2×101 + 3×100, which shows more clearly how the powers of 10 change from one position to the next.
In a similar fashion a number with a decimal point 45.67 is interpreted as: 4×10 + 5 + 6/10 + 7/100 with numbers to the right of the decimal point being divided by successively greater powers of 10 as we move to the right. This example can also be written as: 4×101 + 5×100 + 6×10-1 + 7×10-2, which shows more clearly how the powers of 10 change from one position to the next.

It is widely believed that a decimal number system is used because humans have 10 fingers (or digits).

Numbers used for counting are sometimes called cardinal numbers; counting can sensibly start at zero. Numbers used for placing things in order are sometimes called ordinals; ordering normally starts at the first item.

Contents

Examples

cardinal
numbers
ordinal
numbers
Roman
numerals
binary
numbers
symbol name symbol name symbol symbol
0 zero 0
1 one 1st first I 1
2 two 2nd second II 10
3 three 3rd third III 11
4 four 4th fourth IV 100
5 five 5th fifth V 101
6 six 6th sixth VI 110
7 seven 7th seventh VII 111
8 eight 8th eighth VIII 1000
9 nine 9th ninth IX 1001
10 ten 10th tenth X 1010
16 sixteen 16th sixteenth XVI 10000
100 hundred 100th hundredth C 1100100
1000 thousand 1000th thousandth M 1111101000

For Roman numerals:

there is no symbol for zero,
it is not a positional notation; 2938 is written as MMCMXXXVIII. In this number, the combination 'CM' indicates '100 less than 1000'. Arithmetic is rather difficult in such a system.

History

In the beginning

Ordinal numbers have been used from the beginning. The creation account in Genesis 1:5-2:2 lists God's creation activities on the first day, second day, etc. e.g. "And there was evening, and there was morning — the first day."

Cardinal numbers also appear early. In Genesis 2:10 there is a report of four rivers flowing from the Garden of Eden.

Cardinal numbers are also used from the beginning to record how long various people lived or at what age they had children, e.g. Genesis 5:3-4

When Adam had lived 130 years, he had a son in his own likeness, in his own image; and he named him Seth. After Seth was born, Adam lived 800 years and had other sons and daughters. Altogether, Adam lived a total of 930 years, and then he died.

Supposed secular dating

Evolutionary speculation, on the other hand, presumes that numbering developed gradually as human intelligence developed, and assumes that numbering came about as the need arose to count things.

The first motivation for people to create number was the human desire to the manyness of a set of objects. In other words, to know how many duck’s eggs are to be divided amongst family members or even how many days until the tribe reaches the next watering hole, how many days wills it be until the days grow longer and the nights shorter, how many arrow heads do one trade for canoe? Knowing how to determine the manyness of a collection of objects must surely have been a great aid in all areas of human endeavour.[1]

Circa 43,000 BC, Lebombo Bone with tally marks.[1]

Circa 19,000 BC, Ishango bone with tally marks.

Counting didn't become significant until the rise of cities.

Circa 4000 BC, Sumerians used various marks on clay tablets to count possessions.

Circa 3000 BC, Egyptians used different symbols for different numbers.

Note that the existence of tally marks, marks on clay tablets, etc. is not generally disputed; it is the associated supposed dates which are in question.

Commonly accepted history

Circa 1750 BC, Babylonians used a base-60 number system. Traces of this still exist today: timekeeping has 60 seconds in a minute, and 60 minutes in an hour; angles have 60 minutes in a degree.

Circa 500 BC, Pythagoras distinguished between odd and even numbers (the start of number theory).

Circa 300 BC, Roman numerals develop from Etruscan numerals.

Circa 200 BC, Archimedes devised ways to write down very large numbers.

The decimal number system originated in India.

In the 5th century, Aryabhata developed the idea of positional notation (using 1..9).[2]
In the 6th century, Brahmagupta introduced a symbol for zero.[3]

These ideas spread to nearby countries, including Arabs, who translated the Hindu texts into Arabic.

Circa 760, Arabs introduced something close to the modern way of writing fractions (initially to deal with sharing out inheritances).

Circa 1200, these ideas spread to Europe. In 1202 Fibonacci published a book "Liber Abaci" with details of this different way of writing numbers.[4]

Because the Western world obtained these ideas from the Arabs, they are known to most Western countries as Arabic numerals.

Circa 1450 AD, negative numbers started to become grudgingly accepted.

1579, Bombelli introduces a notation for √(−1), to provide solutions to certain cubic equations.[5]

1637. Descartes refers to numbers involving √(−1) as imaginary numbers.

Number classification

The natural numbers are a subset of the the integers, which are a subset of the real numbers, which are a subset of the complex numbers. Rational and irrational numbers together make up the real numbers.

Natural numbers

Natural numbers, usually denoted by a double-struck , include the positive numbers 1, 2, 3, etc. Some authorities also include 0 as a natural number. These are often referred to as whole numbers.

"Number theory is a vast and fascinating field of mathematics, sometimes called "higher arithmetic," consisting of the study of the properties of whole numbers."[6]

Integers

The integers, usually denoted by a double-struck , include positive and negative whole numbers, along with zero:

. . . -3 -2 -1 0 1 2 3 . . .

Real numbers

The real numbers, usually denoted by a double-struck , are all rational and irrational numbers. In real life, all numbers with a decimal point are real numbers.

Rational numbers

The rational numbers, usually denoted by a double-struck , are all numbers which can be expressed as a fraction a/b, where a and b are integers and b is nonzero.

Irrational numbers

The irrational numbers are the decimal numbers which cannot be represented by the ratio of two integers. They include e, pi and the square roots of many integers. Unlike rational numbers, they have non-repeating digits when expressed as a decimal.

Complex numbers

The complex numbers, usually denoted by a double-struck , are all numbers of the form a + bi, where a and b are reals and i2 = −1 (or i is the square root of minus one). They can be used to create a two-dimensional version of the number line called the Argand plane.

Other classifications

Algebraic number

The algebraic numbers, usually denoted by a double-struck ₳, are all numbers which can be described as a root of a nonzero polynomial equation.

Pythagoras's constant, the square root of two, is an algebraic number. This is the first number known to have been proved to be irrational. It is the positive solution to the nonzero polynomial equation

x2 - 2 = 0.

The golden ratio, often abbreviated by the letter φ ("phi"), after the Greek sculpter Phidias, is the positive solution to the nonzero polynomial equation

x2 - x - 1 = 0.

Geometric construction, using the ancient Greek rules, can only produce algebraic numbers.

Transcendental numbers

The remaining real numbers, which are not algebraic, are all transcendental numbers.

Because π ("pi") is transcendental (proven in 1882), it is not possible to square the circle, at least not using the ancient Greek rules of geometric construction.

References

  1. 1.0 1.1 Parama Dutta, The History of Counting, Mon. 25th April, 2011Mon. April 25th, 2011.
  2. Aryabhata. Indian astronomer and mathematician
  3. Brahmagupta. Indian astronomer
  4. Fibonacci's Liber Abaci (Book of Calculation)
  5. A Short History of Complex Numbers
  6. What is number theory
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