Furthermore, it will take two steps to solve an inequality. The first one is to simplify using the reciprocal of addition or subtraction. The second one is to simplify more by using the reciprocal of multiplication or division. Note that when multiplying or dividing an inequality by a negative number, the inequality symbol must be reversed.
Inequality is a mathematical statement that explains that the two values are not equal and are different. The equation a≠b means that "a" is not equal to "b". Inequality is the same with any equation, with the (only) difference that inequality does not use an equal sign but inequality symbols. The inequality b>a means that b is greater than a. Speed limits, mark, and others use inequality to express them.
When solving an inequality, a person need to have a true statement. When you divide or multiply an inequality with a negative number on both sides, the statement is false. In order to make the statement correct with a negative number, one must reverse the direction of the inequality symbol. When a number is a positive number, one does not need to reverse the symbol. Inequality is about making a true statement.
For example, start with a true statement -6"y"<-12. When both sides are divided by -6, the result will become y<2. In this statement, the symbol need to be reversed in order to have a true statement, which produces y>2 as the correct answer. In the number line (see figure Inequality 2), a closed shaded circle points out that it is included in the solution set. An open circle points out that it is not included in the solution set.
Chechnya
Chechnya (; ), officially the Chechen Republic (; ), is a federal subject in Russia. It is located in the Caucasus region. The capital is Grozny.
Most people in Chechnya are Muslims and are of ancient Hurrian roots who spoke a Caucasian language. Most Chechens belong to the Shafi`i school of Sunni Islam.
Chechens speak their own language, not related to the Russian language. Chechen is part of the Northeast Caucasian, or "Vainakh", family, while Russian is a Slavic language.
During Soviet control, Chechnya was unified with Ingushetia. After the fall of the Soviet Union, Chechnya broke away from Ingushetia to form its own republic. The Chechens wanted independence. After the First Chechen War, Chechnya was de facto independent as the Chechen Republic of Ichkeria. After the Second Chechen War, Russia regained control of Chechnya.
Russia has claimed Chechnya as part of its country since the Russians invaded the Caucasus in the 18th century.
The current leader of the Chechen Republic is Ramzan Kadyrov. He is also the son of the 1st Chechen President, Akhmad Kadyrov.
Hilbert's paradox of the Grand Hotel
Hilbert's paradox of the Grand Hotel is a mathematical paradox named after the German mathematician David Hilbert. Hilbert used it as an example to show how infinity does not act in the same way as regular numbers do.
Normal hotels have a set number of rooms. This number is finite. Once every room has been assigned to a guest, any new guest that wants a room and does not have one yet cannot be served - in other words, the hotel is fully booked.
Now suppose that there is a hotel that has an infinite number of rooms. As a convenience, the rooms have numbers, the first room has the number 1, the second has number 2, and so on. If all the rooms are filled, it might appear that no more guests can be taken in, as in a hotel with a finite number of rooms. This is wrong, though. A room can be provided for another guest. This can be done by moving the guest in room 1 to room 2, the guest in room 2 to room 3, and so on. In the general case, the guest in room "n" will be moved to room "n+1". After all guests have moved, room 1 is empty, and the new guest now has a room to occupy. This shows how we can find a room for a new guest even if the hotel is already full, something that could not happen in any hotel with a finite number of rooms.
Another thing that can be done with this imaginary hotel is to double the number of people inside, again when all the rooms are already full. This is done by asking each guest to multiply their room number by two and move to that room. (If their previous room number was "n", this time they would move to room number "2n".) This would send the guest in room 1 to room 2, the guest in room 2 to room 4, the guest in room 3 to room 6, the guest in room 4 to room 8, and so on. After finishing, we find that all the rooms with the odd numbers are empty. Then we can put an infinite number of guests into these empty rooms. Now the number of guests in the hotel has been doubled without making the hotel bigger.
The guest at room 11 would move to room 101.
The second person of group 5 (Address 5-2) would go to room 25.
This is not really a paradox, it is only counterintuitive. In a normal hotel, with a finite number of rooms, the number of "odd-numbered" rooms, is smaller than the total number of rooms. In Hilbert's Hotel this does not seem to be the case.
The guest 1 of group 2 of vehicle 1 (1-2-1) goes to room 121.
Address 4-7-7-4 goes to room 4774.
Address 0-1 (hotel dweller) stays because 1-0-0-0-1 moves to room 10,001.
and so on.
Each pod holds 10 people.
Each megapod holds 10 pods. (100 people)
Each supermegapod holds 10 megapods. (1,000 people)
Each superdupermegapod holds 10 supermegapods.
Each ultrasuperdupermegapod holds 10 superdupermegapod.
Each ultrasuperduperubermegapod holds 10 ultrasuperdupermegapods. (1,000,000 people)
And so on.
This assumes that there is never an infinitieth layer. (The main ship)
Hilbert's paradox is a veridical paradox: it leads to a counter-intuitive result that is provably true. The statements "there is a guest to every room" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms. An analogous situation is presented in Cantor's diagonal proof.
At first, this state of affairs might seem to be counter-intuitive. The properties of "infinite collections of things" are quite different from those of "finite collections of things". The paradox of Hilbert's Grand Hotel can be understood by using Cantor's theory of transfinite numbers. In an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel, the quantity of odd-numbered rooms is not smaller than the total "number" of rooms. In mathematical terms, the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. Indeed, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets (sets with the same cardinality as the natural numbers) this cardinality is formula_1.
Put differently, for any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers. For example, the set of rational numbers—those numbers which can be written as a quotient of integers—contains the natural numbers as a subset, but is no bigger than the set of natural numbers since the rationals are countable: there is a bijection from the naturals to the rationals.
Opposite number
In mathematics, opposite number or additive inverse of any number is a number "n" which, if added to "n", results in 0 (the identity element of addition). The opposite number for "n" is written as "−n". For example, −7 is the opposite number of 7, because formula_1.
For positive numbers, the opposite number is simply the negative version of the number. For negative numbers, the opposite number is the positive version of the number.
Nth root
An "n"-th root of a number "r" is a number which, if "n" copies are multiplied together, makes "r". It is also called a radical or a radical expression. It is a number "k" for which the following equation is true: