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2.08.2011

How many will be metres and decimeters?

"How many there will be 22 metres of 2 decimeters a minus of 1 metre of 8 decimeters?" Difficulty of this problem consists that for a designation of one number is used at once two units of measurement: metres and decimeters. If to translate told on normal language of blondes it almost the same that "I have bought yesterday a phrase two jackets and one has presented to the girl-friend" to replace with a phrase "I yesterday has bought two jackets and four sleeves, and then one jacket and two sleeves has presented to the girl-friend". It is natural that last phrase will confuse you at once: "And sleeves that, separately from jackets are on sale?". Certainly, no. Sleeves are sewn to jackets, here we only consider them separately. What for? What you to confuse. Problems in the mathematician always get confused, as a ball of threads. You should untangle and receive them the right answer. It develops ingenuity, here. And simple writing off from the answer book develops dullness. Now we will start the decision.

Means, a problem at us such: how many will be, if from 22 metres and 2 decimeters to take away 1 metre and 8 decimeters? The main thing what it is necessary to know at the decision of this problem - how many decimeters in one metre? Or how many metres in one decimeter.

1 metre = 10 decimeters

1 decimeter = 0.1 metres

One metre is equal to ten decimeters. One decimeter is equal the one tenth metre. Here now we have everything what to solve a problem. Is the whole three ways of the decision of this problem - in metres, in decimeters, in two units of measurement at once.

We solve a problem in metres. For this purpose decimeters reduced (this that number from which subtract) and subtracted (this that number which subtract) it is translated in metres.

2 decimeters = 0.2 metres

8 decimeters = 0.8 metres

We add these tails to our numbers in metres:

22 metres + 0.2 metres = 22.2 metres

1 metre + 0.8 metres = 1.8 metres

Now it is possible to execute subtraction and to receive a difference of these metres. It is possible to translate the answer at once in metres and decimeters:

22.2 metres - 1.8 metres = 20.4 metres = 20 metres of 4 decimeters

The second way is to solve the same problem in decimeters. For this purpose we will translate metres in decimeters:

22 metres = 220 decimeters

1 metre = 10 decimeters

To the received decimeters it is added those decimeters which at us are on a statement of the problem:

220 decimeters + 2 decimeters = 222 decimeters

10 decimeters + 8 decimeters = 18 decimeters

We carry out subtraction and we translate decimeters again in metres and decimeters:

222 decimeters - 18 decimeters = 204 decimeters = 20 metres of 4 decimeters

It was the mathematics. Now for blondes. What to understand that we did, let's replace metres with bows, and decimeters on ruches. Thus we will agree that it is possible to make ten ruches of one bow, and it is possible to make one bow of ten ruches – well to sew. We copy a statement of the problem: «you have 22 bows and 2 ruches. And 8 ruches you should give 1 bow to the girl-friend». With bows all is simple – we take one and we give. And how to be with ruches? They obviously do not suffice us. We will look, as we solved this problem math in two ways. From outside it looks so.

The first way. You take all bows and untie them, smooth all ruches. Take a needle with a thread and sew all it in one long tape. After you have stopped to sew, measure the necessary quantity from a tape and cut off scissors. Solemnly hand over to the gone nuts girlfriend a tape piece. Then you with the girlfriend cut the tapes, do of them bows and ruches. As a result you with the girlfriend, all in bows and ruches, stand, as two marriageable silly women.

The second way. You untie all bows and cut them on ruches. Do a heap of ruches and from this heap select 18 pieces for the girl-friend. After that solemnly hand over to the gone nuts girl-friend a small group of ruches. Then together begin to smooth ruches and to sew from them bows. Thus your girl-friend thinks: «In the silly woman! Because of this ugly creature I should sew now a bow». You think: «In the silly woman! Because of this ugly creature I at first cut, and now should sew already twenty bows». To tell the truth, I do not envy you.

As you can see, math correct ways from outside look is not absolutely practical. How any blonde will arrive in this situation? As the two-nuclear processor. Takes one bow from a small group of bows and will make of it ten ruches. Now you have one small group from twenty one bows and the second small group from twelve ruches. From the first small group you take one bow, eight ruches a beret from the second small group. Hand over to the happy girl-friend one bow and eight ruches. Then you admire the treasures and is simple blah-blah-blah instead of puffing with scissors or needles. It also is the third way of the decision of a problem. Coming back to our metres and decimeters, math it can be written down so:

22 metres and 2 decimeters = 21 metre and 12 decimeters

(21 metre and 12 decimeters) – (1 metre and 8 decimeters) = (21 metre – 1 metre) and (12 decimeters – 8 decimeters) = 20 metres and 4 decimeters

We with two different units of measure carry out two mathematical actions of subtraction simultaneously, in the same way, as two-nuclear processors Intel for computers work. I think, after such decision in your writing-book mathematicians will start to go nuts. Though so all calculations practically become. In the mathematician it is accepted to say that it is necessary to execute at first subtraction of decimeters, and already then subtraction of metres. Basically, mathematicians are right. So it is heavier to get confused. At subtraction of decimeters you can always steal one metre from metres if decimeters do not suffice you. Then from the remained metres subtract metres. Only do not forget about one metre if you have already dissipated it in decimeters.

The table of factorials to 255

Exact values of a factorial of natural numbers to 50 we have already considered. In practice, at mathematical calculations, use approximate values of a factorial which are collected in the table of factorials to 255 is more often.

A table of factorials  is a picture. Approximate  values of factorial of  numbers to 255. What a  factorial is equal to. Mathematics for blondes.

Factorial

Factorial in the mathematician name product of all natural numbers, including the specified number. The factorial by an exclamation mark after number, for example 4 is designated!. So, if you have met an exclamation mark in the mathematician, it at all does not mean "Vau! Number!". It simply factorial. From sacred mathematical texts it is necessary to learn one phrase "a zero Factorial it is equal to unit". Why the zero factorial is equal to unit? Read under the reference. Exact values of factorials of numbers to 50 are resulted in drawing.

Factorial. Exact meaning of factorials of numbers to 50. Mathematics for blondes.
Factorial
On a picture it is shown how to consider a factorial for natural number 7. Calculation of a factorial of other numbers is made in the same way: all numbers from one to specified before an exclamation mark are multiplied among themselves.

The factorial of 1 (unit) is equal to unit.
1! = 1
The factorial of 2 (two) is equal to two.
2! = 1 · 2 = 2
The factorial of 3 (three) is equal six.
3! = 1 · 2 · 3 = 6
The factorial of 4 (four) equals to twenty four.
4! = 1 · 2 · 3 · 4 = 24
The factorial 5 is equal hundred twenty.
5! = 1 · 2 · 3 · 4 · 5 = 120
Well and so on.

In a general view the formula for a factorial finding can be written down so:

n! = 1 · 2 · 3 · 4 · ... · (n - 2) · (n - 1) · n

The table of factorials to 255 is presented on separate page.

By the way, if you go at the wheel the car and will see an exclamation mark in a triangle on a white or yellow background is not a lesson of mathematics with factorials, it is a traffic sign "Attention!".

Factorial. Mathematics for blondes.
Factorial )))
Here it is not necessary anything the friend on the friend to multiply. It is necessary to postpone away the beautician, to cease to stir by a mobile phone and more strong to keep for a car wheel. Attentively look not on the parties, and at road. Ahead there can be unpleasant surprises. That unpleasant surprises on road did not turn to unpleasant situations, them designate this traffic sign.

The multiplication table to 20

The multiplication table to 20 square still is called the table of Pythagoras. To download the multiplication table it is possible by means of the right button of the mouse, having chosen in the menu "to Keep the image as...". After preservation, this multiplication table can be unpacked. To learn or not to learn this multiplication table - business voluntary. What for to load the memory the multiplication table to twenty when there is a picture and calculators? In your operative memory, that is in a head, there is enough multiplication table to ten. It three times is less numbers.

The multiplication table to 20, multiplication chart. Mathematics for  blondes.
The multiplication table to 20

table multiplication 11 à 20 - a picture can be looked here.

We do repair

Blondes sometimes too do repair. Here again without mathematics application it is impossible to manage in any way. It is necessary to pay For repair money. Thus inevitably there is a question: how many and for what from me take money? Whether correctly builders have counted repair cost? How to count amounts of works at apartment repair? Here, as it is impossible by the way, knowledge of mathematics will be useful to you. And so, if we do repair it is necessary to know volumes of forthcoming works approximately at least. Calculation of amounts of works - business tiresome enough, but necessary. How many it is necessary to buy wall-paper? How many it is necessary to buy a tile? These figures as undertake from amounts of works.

Amounts of works are considered very simply, basically, on the rectangle area - the length is multiplied for the width and the area (in full conformity with the multiplication table) turns out. If it is a floor - that the length and width of a room undertake. Measurements are spent by a roulette between opposite walls over a plinth. As the measuring tool it is possible to use metre by which you measure volume of hips, waists, breasts. For example, if at us a room in the size 5,0 on 3,0 metres, in this room of 15 metres of a square floor.

5 · 3 = 15

It is amount of works on packing of a tile, a laminate, linoleum, the device of a coupler, a floor first coat and so forth in this room. In addition the floor area in a doorway and радиаторной to a niche can be considered. This area increases to area already received by us. By builders at estimate calculation factors of complexity of works can be applied, but masters should prove that work is executed not the standard. For example, in a bathroom the tile of two colours was killed on a curve that, naturally, more difficultly, than simply to put a tile. Even for selection of drawing of a tile cost of works can be increased.

If the room has the difficult form then the floor area is calculated on the areas of simple geometrical figures into which it is necessary to break mentally a room and for which you know formulas of a finding of the areas. In the practice I very often used the formula of Gerona for a finding of the area of a triangle on length of three parties.

If any works are regarded in running metres (for example, plinth installation) then it is simply measured by metre or a roulette on room perimetre. In a room in the size 5 on 3 metres the perimetre makes 16 metres running.

(5 + 3) · 2 = 16

For calculation of quantity of a plinth it is necessary to take away width of a doorway, usually 0,8 - 0,9 metres, to add a plinth on door slopes (if in a room it is). Let in our case the door will be 0,9 metres, and the door slope will be in the width 0,3 metres. Total we will receive 15,7 metres running plinth installations.

16 - 0,9 + 0,3 · 2 = 15,7

If any works are regarded for a piece (installation of corners on a plinth, installation of sockets, etc.) It is counted up at random by a finger in a product. Two times and too the product is not considered one. It is necessary to mean that installation box for setting of electric wall outlet and socket installation are different works which are paid for different quotations though are carried out consistently in one place. By the way, boxes for setting of electric wall outlets are established and for switches. Check is carried out simply: quantity of switches plus quantity of sockets to equally quantity box for setting of electric wall outlet. If figures do not converge, then is considered at random by a finger of the builder (already the builder should count all at your presence) and listened its arguments, after all situations happen different (the telephone socket, the television socket, the computer socket - all of them are put in box for setting of electric wall outlet). box for setting of electric wall outlet is cheaper than an assembly box and it can be used in junctions of wires - such beggar an European-quality repair variant.

The ceiling area is considered in the same way, as well as the floor area - the sizes from a wall to a wall are measured. The area of niches increases, and the area of columns is subtracted from the ceiling area. Usually the ceiling area is equal or less areas of a floor (remember, in the floor area the area in doorways, radiator niches can be added).

The area of walls is considered multiplication walls on height of a room minus the area of door and window apertures are long. Let at us height of a room of 2,5 metres, a window in the size 1,4 on 1,5 (height) of metres, a door 0,9 on 2,1 (height) of metres. Then the area of walls is equal to room perimetre (we already so considered a plinth) increased by height of a room minus the area of apertures about makes 36,0 metres square:

((5 + 3) · 2) · 2,5 - 1,4 · 1,5 - 0,9 · 2,1 = 16 · 2,5 - 2,1 - 1,9 = 40 - 4,0 = 36

Furnish of slopes is considered, usually, in running metres. The window slope in our case makes 4,4 metres of the running:

1,5 · 2 + 1,4 = 4,4

The door slope equals 5,1 metres of the running:

2,1 · 2 + 0,9 = 5,1

If the quotation for slopes in square metres then the received running metres it is multiplied separately: length of a window slope running metres for width of a window slope, length of a door slope for width of a door slope. If in a room the oil panel in height of 1,8 m then the panel area is calculated separately is executed and it equals of 25,6 metres of the square:

((5 + 3) · 2) · 1,8 - 1,4 · 1,0 - 0,9 · 1,8 = 28,8 - 1,4 - 1,6 = 25,6

Here for a window of 1,0 metres is a distance from a window sill to top of the oil panel. The area of furnish of walls over the oil panel makes remained from a total area of walls of the square of 10,4 metres:

36 - 25,6 = 10,4

The areas in other premises are considered precisely also. More simple method still nobody has thought up. Write down the calculations in a writing-book on each room separately and summary calculation on apartment - separately: циферка in циферку with all signs on mathematical actions. The error in calculations can be the most banal - on the calculator not that key have pressed at calculation of addition, multiplication, subtraction. From builders you in the right to demand explanations concerning the overestimated amounts of works. In disputable cases in common make the necessary gaugings and together do calculations - here you quickly find out, who and where exactly was mistaken.

You can compare quotations of builders to the quotations presented in my building catalogue. Usually the highest quotations for civil work - in Moscow.

Division by zero - the question formulation

Division by zero is maybe - to such conclusion we have come. But to solve a problem about division of number into a zero we and could not. Then let's solve not a specific target on division of number into a zero, and a zero problem as a whole. We begin all from the very beginning.

What is the zero? All-knowing Wikipedia says that the zero is a number. This number designates a point on a numerical straight line which separates positive numbers from the negative. Give also we will look at this well-known numerical straight line in which the dog on a nickname "Zero" is buried.

Number line. Mathematics for blondes.
Number line
And now we will look, as the zero in the basic mathematical operations behaves agrees the standard mathematical rules:

a + 0 = a

0 + a = a

a - 0 = a

0 - a = -a

a - a = 0

a · 0 = 0

a : 0 = ?

0 : a = 0

0 : 0 = ?

In that mathematicians are mistaken, including division into a zero impossible, we have already understood. Instead of whether and mathematicians are mistaken in other places at the formulation of results of mathematical actions with zero? Quite probably that some from resulted above equalities are false statements.

The problem with zero in the mathematician dares simply and gracefully, in style of blondes. Therefore mathematicians to such never will guess. Here the sober and critical sight of the person from outside, without fanatical belief in the received mathematical knowledge is necessary. Blondes for a zero solution of a problem approach as well as possible. About the mathematician they have the most general representations. Their mentality differs from the standard.

Whether you can find a solution of a problem with zero? The variants of the decision leave in comments.

2.07.2011

Division of number into a zero

Last time we have considered possibility of division into a zero, and have come to conclusion that Division by zero is maybe. But it was only the half of a problem of division into a zero which decision we undertook. There is one more set of the equations of division into a zero which we are simply obliged to consider.

We have very cheerfully laughed over Wikipedia, now has come turn of all of the others to laugh over us. We will try to answer on a question that will turn out if to try to divide any number into a zero. That the number as a result cannot turn out, we have neatly noticed. Then, what can turn out? Not clearly that. We will designate it "not clearly that" which turns out as a result of division of number into a zero, a question mark. At us such small set of the mathematical equations will turn out:

a : 0 = ?


0 · ? = a


a : ? = 0

Now the equations received from the equation of division of number on a zero, we will try to sound and compare to the rules accepted in the mathematician. If a zero to divide on not clear that, any number as a result will turn out. As we know from the previous message, it is possible to assume only that any number turns out as a result of division of zero into a zero.

If any number to divide on not clear that, the zero as a result will turn out. As we know, in the mathematician all occurs just what isn't needed: the zero turns out as a result of division of zero into any number.

We include logic of blondes and we start to think, how to us with this most "to be not clear that"? How, how? Yes in any way! We will substitute instead of a question mark a zero - and there are not problems. Then at us rather nice equations with zeroes will turn out:

a : 0 = 0


0 · 0 = a


a : 0 = 0

Here! The first and last equations coincide now and it is not necessary nothing to think out! Well, and that the zero increased by a zero at us equals to any number, means nothing. There should be whence that any numbers? It is such sleeve of the mathematician-conjurer from which it always gets them. "Let any number is given us..." All around sit, mouths поразевали, have listened openmouthed, eyes around ransack in search of any number, and mathematics in the meantime, imperceptibly, from the sleeve, gets this most any number and shows to spectators. All spectators in delight clap in palms. But we that know that in a sleeve at the mathematician division of zero into a zero is hidden. Or multiplication of a zero to a zero? Oh, with this zero absolutely it is possible to get confused.

Here we, as real mathematicians, have come to the equation:

0 : 0 = 0 · 0 = a

And after all all know that any number increased by a zero, equals to zero, instead of any number. Again at all of us will laugh. Mathematicians this problem cannot already solve some hundreds years, that already only did not think out. Now we stand near to them in a deaf corner in which ourselves have tired out ourselves and from which there is no exit, we look against each other and we wipe the snotty noses.

It is possible to put, of course, instead of a question mark an infinity badge. But what such infinity? Basically, this same any number, only very much the big. Means, this variant is not necessary.

As we see, the problem with division into a zero does not dare. Though we have come to a conclusion that the decision should be.

Let's next time try begin all with the beginning. Only not from that beginning from which, and division into a zero begins with that beginning with which the zero begins.