When a person was just learning to count, his fingers were enough to determine that two mammoths walking by the cave were smaller than that herd behind the mountain. But as soon as he realized what positional reckoning is (when a number has specific place in a long line), he began to think: what's next, what largest number?
Since the best minds began to look for how to calculate such quantities, and most importantly, what meaning to give them.
Dots at the end of a row
When students are introduced to original concept at the edges of a series of numbers, it is prudent to put ellipsis and explain that the largest and smallest number is a meaningless category. It is always possible to add one to the largest number, and it will no longer be the largest. But progress would not be possible if there were not those who wanted to find meaning where it should not be.
Infinity beyond the frightening and uncertain philosophical significance, created purely technical difficulties. I had to look for notation for very large numbers. At first, this was done separately for the main language groups, and with the development of globalization, words have appeared that name the largest number, generally accepted throughout the world.
ten, one hundred, one thousand
In every language, for numbers that have practical value, found its own name.
In Russian, first of all, this is a series from zero to ten. Up to a hundred further numbers named or based on them, with little change roots - “twenty” (two by ten), “thirty” (three by ten), etc., or are compound: “twenty-one”, “fifty-four”. Exception - instead of "four" we have a more convenient "forty".
The largest two-digit number - "ninety-nine" - has a compound name. Further from their own traditional names - "one hundred" and "thousand", the rest are formed from the necessary combinations. The situation is similar in other common languages. It is logical to think that established names were given to numbers and figures that most people dealt with. ordinary people. Even an ordinary peasant could imagine what a thousand head of cattle is. With a million it was more difficult, and confusion began.
Million, quintillion, decibillion
In the middle of the 15th century, the Frenchman Nicolas Chouquet, in order to designate the largest number, proposed a naming system based on numerals from Latin generally accepted among scientists. In Russian, they have undergone some modification for the convenience of pronunciation:
- 1 - Unus - un.
- 2 - Duo, Bi (double) - duo, bi.
- 3 - Tres - three.
- 4 - Quattuor - quadri.
- 5 - Quinque - quintes.
- 6 - Sex - sexty.
- 7 - Septem - septi.
- 8 - Octo - okti.
- 9 - Novem - noni.
- 10 - Decem - deci.
The basis of the names was to be -million, from "million" - " big thousand» - i.e. 1,000,000 - 1000^2 - a thousand squared. This word, to mention the largest number, was first used by the famous navigator and scientist Marco Polo. So, a thousand to the third power became a trillion, 1000 ^ 4 became a quadrillion. Another Frenchman, Peletier, suggested using the ending “-billion” for the numbers that Schuke called “thousand millions” (10 ^ 9), “thousand billions” (10 ^ 15), etc. It turned out that 1,000,000,000 is a billion, 10^15 is a billiard, one with 21 zeros is a trillion, and so on.
Terminology French mathematicians has been used in many countries. But it gradually became clear that 10 ^ 9 in some works began to be called not a billion, but a billion. And in the USA they adopted a system according to which the ending -million received degrees not of a million, like the French, but of thousands. As a result, there are two scales in the world today: “long” and “short”. To understand what number is meant by the name, for example, a quadrillion, it is better to clarify to what degree the number 10 is raised. including in Russia (however, we have 10 ^ 9 - not a billion, but a billion), if 24 is the "long" accepted in most regions of the world.
Tredecillion, vigintilliard and millillion
After the last numeral is used - deci, and a decillion is formed - the largest number without complex word formations - 10 ^ 33 on a short scale, combinations of the necessary prefixes are used for the next digits. It turns out complex compound names type tredecillion - 10 ^ 42, quindecillion - 10 ^ 48, etc. own names the Romans were awarded: twenty - viginti, one hundred - centum and a thousand - mille. Following the rules of Shuquet, one can form monster names for an infinitely long time. For example, the number 10 ^308760 is calledion.
But these constructions are of interest only to a limited number of people - they are not used in practice, and these quantities themselves are not even tied to theoretical tasks or theorems. It is for purely theoretical constructions that giant numbers are intended, sometimes given very sonorous names or called by the author's last name.
Darkness, Legion, Asankheya
The question of huge numbers also worried the “pre-computer” generations. The Slavs had several, in some they reached great heights: the largest number is 10 ^ 50. From the heights of our time, the names of numbers seem like poetry, and only historians and linguists know whether all of them had a practical meaning: 10 ^ 4 - "darkness", 10 ^ 5 - "legion", 10 ^ 6 - "leodr", 10 ^7 - crow, raven, 10^8 - "deck".
No less beautiful by name, the number asaṃkhyeya is mentioned in Buddhist texts, in ancient Chinese and ancient Indian collections of sutras.
The researchers give the quantitative value of the asankheyya number as 10^140. For those who understand, it is full of divine meaning: just so much space cycles the soul must pass in order to be cleansed of all the bodily accumulated for long haul rebirth, and reach the blissful state of nirvana.
Google, googolplex
A mathematician from Columbia University (USA) Edward Kasner from the early 1920s began to think about large numbers. In particular, he was interested in sonorous and expressive name for the beautiful number 10^100. One day he was walking with his nephews and told them about this number. Nine-year-old Milton Sirotta suggested the word googol - googol. The uncle also received a bonus from his nephews - a new number, which they explained as follows: one and as many zeros as you can write until you get completely tired. The name of this number was googolplex. On reflection, Kashner decided that it would be the number 10^googol.
Kashner saw the meaning in such numbers more pedagogically: at that time science did not know anything in such a quantity, and he explained to future mathematicians by their example what is the largest number that can keep the difference from infinity.
The chic idea of the little geniuses of naming was appreciated by the founders of the company to promote the new search engine. The googol domain was taken, and the letter o dropped out, but a name appeared for which an ephemeral number could someday become real - that's how much its shares would cost.
Shannon number, Skewes number, mezzon, megiston
Unlike physicists, who periodically stumble upon the limitations imposed by nature, mathematicians continue on their way towards infinity. Chess lover Claude Shannon (1916-2001) filled the number 10 ^ 118 with meaning - that is how many variants of positions can arise within 40 moves.
Stanley Skuse of South Africa was engaged in one of the seven tasks included in the list of "millennium problems" - It concerns the search for patterns in the distribution prime numbers. In the course of his reasoning, he first used the number 10^10^10^34, which he designated Sk 1 , and then 10^10^10^963, Skuse's second number, Sk 2 .
Even the usual notation system is not suitable for operating with such numbers. Hugo Steinhaus (1887-1972) suggested using geometric figures: n in a triangle is n to the power of n, n squared is n in n triangles, n in a circle is n in n squares. He explained this system using the example of numbers mega - 2 in a circle, mezzon - 3 in a circle, megiston - 10 in a circle. It is so difficult to designate, for example, the largest two-digit number, but it has become easier to operate with colossal values.
Professor Donald Knuth proposed arrow notation, in which repetition was denoted by an arrow, borrowed from the practice of programmers. The googol in this case looks like 10102, and the googolplex looks like 1010102.
Graham number
Ronald Graham (b. 1935) American mathematician, in the course of the study of the Ramsey theory associated with hypercubes - multidimensional geometric bodies- introduced special numbers G 1 - G 64 , with the help of which he marked the boundaries of the solution, where the upper limit was the largest multiple number that received his name. He even calculated the last 20 digits, and the following values served as the initial data:
G 1 \u003d 33 \u003d 8.7 x 10 ^ 115.
G 2 \u003d 3 ... 3 (the number of superdegree arrows \u003d G 1).
G 3 \u003d 3 ... 3 (the number of superdegree arrows \u003d G 2).
G 64 = 3…3 (number of super power arrows = G 63)
G 64, simply referred to as G, is the world's largest number used in mathematical calculations. It is included in the book of records.
It is almost impossible to imagine its scale, given that the entire volume known to man of the universe, expressed in the smallest unit of volume (a cube with a face of the Planck length (10 -35 m)), is expressed by the number 10 ^ 185.
On the this lesson you can find numbers that use two digits. Such numbers are called double digits. The following are examples of two-digit numbers, as well as a comparison of two-digit numbers. Then you can check out general rules number comparisons.
Lesson: Single and double figures
In this lesson, we will look at numbers that consist of tens and ones.
Consider the following numbers:
16, 61, 5, 10, 8, 99, 1
What groups can these numbers be divided into?
The first group - 5, 8, 1
The second group - 16, 61, 10, 99
In the first group, those numbers are written, in the record of which one character is one digit. Such numbers are called unambiguous.
The second group contains numbers with two digits. Such numbers are called two-digit.
The smallest two-digit number is the number 10 .
The largest two-digit number is the number 99 .
Consider more number 10. The number 10 is two-digit and round because it has the number 0 in the units place.
Now consider the number 99. The number 99 is two-digit and non-circular, since this number has the number 9 in the units place.
Try the description of the number, guess what number it is:
1. A two-digit number, when counting, it is called immediately after the number 16.
The correct answer is 17.
2. A two-digit number, it has 1 ten and 5 units.
2. Festival pedagogical ideas "Public lesson" ().
1. Divide the numbers 10, 13, 55, 60, 23, 32, 30 into two groups, round numbers and non-circular numbers.
2. Compare the numbers.
Open the page where our lesson is located. How can we find it? By content. Revisit the topic of the lesson.
Task number 1. Familiarize yourself with the task. What numbers need to be compared in the task? Name the largest two-digit number.
Compare with it any two-digit number. Write the result of the comparison in the form of inequalities with the > sign.
Why is the greater than sign chosen?
Task number 2. Work in progress already with what numbers? Name the smallest three-digit number.
Compare these three-digit numbers with it. Write the result of the comparison in the form of inequalities with the ">" sign.
What are your inequalities?
Task number 3. What numbers are you working with here?
Compare the smallest three-digit number with the largest two-digit number. Write the result of the comparison as an inequality with the > sign.
Why is this comparison sign offered to us by the author?
What conclusion can be drawn?
What problem did we start solving with you?
What is task 2?
Open notebooks on page 9. Let's complete task number 1. Let's apply our skill. Read the task.
What numbers do we meet here?
We concluded that three-digit numbers are larger than two-digit ones. Compare with two-digit and three-digit numbers single digits.
Pair check.
What did you get. Read.
I show two numbers. What number is greater in that direction and look. (22 and 90, 33 and 330, 456 and 7)
Let's remember what we are striving for. What is the purpose?
Fill out our article interesting facts. We work in pairs. Task on the desk. solve individually in notebooks.
The mass of an adult bear is 700 kg, the mass of a 6-month-old bear cub is 70 kg. Whose mass is greater? Write it down as an inequality.
Growth of the tall man is 2m 46 cm. short person- 74 cm. Write down the comparison of inequality as an inequality.
Take the smaller numbers in your right hand.
What numbers are in the right hand?
Take the larger numbers in your left hand.
What numbers are in the left hand?
What conclusion can you draw?
Start by saying: I know that
What problem did you solve?
Read the output in the textbook. page 21 on a blue background.
What is task 2? Let's read it.
Why, when comparing numbers, did you not find a card with a nominal number of 2 m 46 cm?
Let's apply this knowledge, as well as the ability to compare three-digit and two-digit numbers when solving task No. 3 in a notebook .. (On the board)
Read the task. Who understands the task?
We check in pairs. There is an example on the board.
What is the next task of the lesson?
To complete it, you need to answer the questions of task 7.
Formulate a rule for performing a difference comparison of numbers.
How to perform a difference comparison of a three-digit and two-digit number?
Why do we subtract from a three-digit number?
Read the output in the textbook.
