 |
| sin 0 |
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| sin 30 |
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| sin 45 |
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| sin 60 |
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| sin 90 |
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| sin 120 |
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| sin 135 |
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| sin 150 |
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| sin 180 |
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| sin 210 |
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| sin 225 |
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| sin 240 |
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| sin 270 |
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| sin 300 |
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| sin 315 |
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| sin 330 |
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| sin 360 |
The unit circle is like a movie that shows us mathematics. Here you see some pictures of the cult TV series.
Thursday, November 12, 2015
The sine values
The unit circle can be considered not only for individual trigonometric functions, but also under the microscope each function separately. Here are the sine values for different angles.
The unit circle is like a movie that shows us mathematics. Here you see some pictures of the cult TV series.
The unit circle is like a movie that shows us mathematics. Here you see some pictures of the cult TV series.
Wednesday, November 11, 2015
Unit circle
Mathematicians consider themselves clever and all the rest. But not all of us are as smart as mathematics. The unit circle mathematics invented for themselves. For those who are just beginning to study mathematics, I suggest a simpler version - separately cosines and sines separately.
If we remove from the unit circle all that relates to the sine, we get the unit circle cosines.
If we remove from the unit circle everything about cosines, we get a unit circle sinuses.
Now, you will not confuse the values of sines and cosines.
If we remove from the unit circle all that relates to the sine, we get the unit circle cosines.
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| Unit circle cosines |
If we remove from the unit circle everything about cosines, we get a unit circle sinuses.
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| Unit circle sinuses |
Now, you will not confuse the values of sines and cosines.
Tuesday, November 10, 2015
Degrees, minutes, seconds
I was write: "Help me to understand. It is necessary to determine how many degrees, minutes, and seconds contains 4563 seconds corner. Only by actions, and I completely confused."
No harly-barly! We sit quietly and remember that one minute contains 60 seconds, in one degree contains 60 minutes. Now we pimenyaem our knowledge into practice. Transform seconds in a minutes.
4563 : 60 = 76.05 minutes
Determine how many seconds we leave intact:
4563 - 60 * 76 = 4563 - 4560 = 3 seconds
Now 76 minutes translate into degrees, still sadly dividing by 60:
76 : 60 = 1.26667 degrees
We calculate how many minutes do not fit into the clothes degrees:
76 - 60 * 1 = 76 - 60 = 16 minutes
Now our suffering ended and we in good conscience can write the final result:
4563 seconds = 1 degree 16 minutes 3 seconds
Checking? We translate everything back to the second:
1 * 60 * 60 + 16 * 60 + 3 = 3600 + 960 + 3 = 4563
As the math, the conversion is correct.
No harly-barly! We sit quietly and remember that one minute contains 60 seconds, in one degree contains 60 minutes. Now we pimenyaem our knowledge into practice. Transform seconds in a minutes.
4563 : 60 = 76.05 minutes
Determine how many seconds we leave intact:
4563 - 60 * 76 = 4563 - 4560 = 3 seconds
Now 76 minutes translate into degrees, still sadly dividing by 60:
76 : 60 = 1.26667 degrees
We calculate how many minutes do not fit into the clothes degrees:
76 - 60 * 1 = 76 - 60 = 16 minutes
Now our suffering ended and we in good conscience can write the final result:
4563 seconds = 1 degree 16 minutes 3 seconds
Checking? We translate everything back to the second:
1 * 60 * 60 + 16 * 60 + 3 = 3600 + 960 + 3 = 4563
As the math, the conversion is correct.
Sunday, September 13, 2015
How to find the area of a rectangle?
Here's a problem about the area of a rectangle of a textbook on algebra in 7 class (Russian school program):
If the width of the rectangle by 2 dm enlarge and reduce a length of 0.5 m, then we obtain a square, an area 50 dm² less than the area of the rectangle. Find the area of a rectangle.
Interestingly, in 7 class school studying systems of linear equations with two unknowns? Judging by the fact that the task of the textbook on algebra, so it is necessary to solve this problem. Stupidly make up the system of equations, bluntly we decide - anguish. If I'm here just write a solution, and you simply write off, the smarter you from it will not. I propose first to solve this problem, and the system of equations with the decision at the end we write.
What is square? It's a rectangle whose sides are equal. What is rectangle? It is a square, whose side are different. You are their teachers of mathematics do not govoriteovetuyu - for them it sounds like a desecration. I like "definitely" use constantly. After all, the mathematical properties of geometric objects they pass very accurately.
To solve the problem, we denote the side of the rectangle: a - the length, b - the width. Now we start again to read the statement of the problem.
"If the width of the rectangle to increase by 2 dm ...". In the language of mathematics it can be written as: b + 2.
"... And to reduce the length of 0.5 m ..." Here please pay attention - only completely illiterate people in a problem using different units of length. For example, meters and decimeters. We are educated people, as opposed to the author of the textbook, and translate everything in decimeters. Why decimeters? Because our area is measured in square decimetres. How many decimeters in one meter? That's right, ten. A 0.5 meter - decimeter is how many? Yes, 0.5 * 10 = 5 decimeters. Now we can take our phrase in the language of mathematics: a-5.
a-5=b+2
What does this leave us? Juggle this expression, we can express the length of one side through the length of the other side. In the future, it is useful to us. Personally, I do not like the "minus" sign. Now we get rid of him.
a-5=b+2
a=b+2+5
a=b+7
Something we digress from the conditions of the problem. Turn rewind and read the phrase: "... then we get a square with an area of 50 dm² is less than the area of a rectangle".
a*b-(a-5)*(b+2)=50
a*b-(a*b-5b+2a-10)=50
a*b-a*b+5b-2a+10=50
5b-2a+10=50
5b-2a=50-10
5b-2a=40
What's next? Now we can party instead of the original a substitute the result of juggling a=b+7.
5b-2a=40
5b-2(b+7)=40
5b-2b-14=40
3b=40+14
3b=54
b=18
The width of the rectangle we already know - 18 decimeters. We are looking for long.
a=b+7
а=18+7
а=25
Now we can easily determine the area of a rectangle 25 * 18 = 450 dm². The notebook can write all this as a system of two equations with two unknowns. I will give just two equations, choose any.
On the left side, we expressed a square area over the length of the rectangle on the right side - in width. In the course of solving the problem, we have considered a third option - the area of the square is represented as the product of the length by the width. All three options provide the same result. Here on the use of mathematics equations to solve problems.
If the width of the rectangle by 2 dm enlarge and reduce a length of 0.5 m, then we obtain a square, an area 50 dm² less than the area of the rectangle. Find the area of a rectangle.
Interestingly, in 7 class school studying systems of linear equations with two unknowns? Judging by the fact that the task of the textbook on algebra, so it is necessary to solve this problem. Stupidly make up the system of equations, bluntly we decide - anguish. If I'm here just write a solution, and you simply write off, the smarter you from it will not. I propose first to solve this problem, and the system of equations with the decision at the end we write.
What is square? It's a rectangle whose sides are equal. What is rectangle? It is a square, whose side are different. You are their teachers of mathematics do not govoriteovetuyu - for them it sounds like a desecration. I like "definitely" use constantly. After all, the mathematical properties of geometric objects they pass very accurately.
To solve the problem, we denote the side of the rectangle: a - the length, b - the width. Now we start again to read the statement of the problem.
"If the width of the rectangle to increase by 2 dm ...". In the language of mathematics it can be written as: b + 2.
"... And to reduce the length of 0.5 m ..." Here please pay attention - only completely illiterate people in a problem using different units of length. For example, meters and decimeters. We are educated people, as opposed to the author of the textbook, and translate everything in decimeters. Why decimeters? Because our area is measured in square decimetres. How many decimeters in one meter? That's right, ten. A 0.5 meter - decimeter is how many? Yes, 0.5 * 10 = 5 decimeters. Now we can take our phrase in the language of mathematics: a-5.
a-5=b+2
What does this leave us? Juggle this expression, we can express the length of one side through the length of the other side. In the future, it is useful to us. Personally, I do not like the "minus" sign. Now we get rid of him.
a-5=b+2
a=b+2+5
a=b+7
Something we digress from the conditions of the problem. Turn rewind and read the phrase: "... then we get a square with an area of 50 dm² is less than the area of a rectangle".
 |
| How to find the area of a rectangle? |
a*b-(a-5)*(b+2)=50
a*b-(a*b-5b+2a-10)=50
a*b-a*b+5b-2a+10=50
5b-2a+10=50
5b-2a=50-10
5b-2a=40
What's next? Now we can party instead of the original a substitute the result of juggling a=b+7.
5b-2a=40
5b-2(b+7)=40
5b-2b-14=40
3b=40+14
3b=54
b=18
The width of the rectangle we already know - 18 decimeters. We are looking for long.
a=b+7
а=18+7
а=25
Now we can easily determine the area of a rectangle 25 * 18 = 450 dm². The notebook can write all this as a system of two equations with two unknowns. I will give just two equations, choose any.
 |
| How to find the area of a rectangle Decision. |
On the left side, we expressed a square area over the length of the rectangle on the right side - in width. In the course of solving the problem, we have considered a third option - the area of the square is represented as the product of the length by the width. All three options provide the same result. Here on the use of mathematics equations to solve problems.
Friday, May 15, 2015
The problem about the juice
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| The problem about the juice |
A liter of grape juice is worth 6 manat. It was mixed with a liter of mulberry juice manat. A liter of juice sold for 10 manats. What benefits can be obtained from the sale of 10 liters of the mixed juice?
For those other than dollars knows nothing, to announce that manat - is the monetary unit of Azerbaijan (there is such a country). By the way, Muslims do not eat pork, mathematicians so do not use math-degree measure of angles in Calculus. As they say, find ten differences. It was the information for the overall development, but will return to the problem about the juice.
The number in the value of mulberry juice available. The man forgot to write. But mulberry juice can be stolen. Then it really does not cost anything. Such "schemes" are thriving in our lives. The lack of numbers in mathematics to denote the numeral zero. If we substitute the value of zero in mulberry juice, then the problem is easily solved.
For a start, we determine the number of cocktail, which is obtained by mixing two different juices. Each juice we take in the amount of one liter. If you do not like liters, take one gallon.
1 + 1 = 2 liters
Now we consider the cost of the resulting cocktail
6 + 0 = 6 manats
Calculate the cost per liter of cocktail
6: 2 = 3 manats
Who is the most interesting point - we determine the profit from the sale of a liter of cocktail
10 - 3 = 7 manats
At the end of the general view of the profits from this scam
10 * 7 = 70 manats
Conclusions:
1. With revenues of these scams is not to compare, but enough to start.
2. The juice can be diluted with water, then stealing nothing.
Disclaimer:
1. For the theft could be imprisoned.
2. For juice dilution with water can beat face.
If in the problem still listed price for mulberry juice, instead of the zero substitute that number. The solution of the problem will not change. Incidentally, the legal business from the business of the criminal also a little different.
Saturday, May 09, 2015
The problem about the garden area
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| The problem about the garden area |
Why such a strange image to task about the area of the garden? Because the problem itself, to put it mildly, very strange. Here's how it sounds:
The garden occupies 80 hectares. Apple trees occupy 5/8 of the area, and 31% of cherry trees. How many hectares of area under apple trees larger than the area under the cherry trees?
Let's first solve this problem for those students who want to write off the stupid decision, and only then talk about the strangeness of this problem.
The first action we define the area that is occupied by apple trees in the garden. To do this, the total area of the garden should be multiplied by the fractional expression of the area under apple trees.
80 * 5/8 = 50 ha
The second area of action is determined that the garden occupied cherry trees. Take the common garden area, multiplied by the number of cherry trees per cent and 100 per cent share. Interest on interest reduced and as a result we get an area of hectares.
80 * 31%: 100% = 24.8 ha
The area under apple trees we really get more than the area under the cherry trees. Takes away from smaller and larger area of the results.
50 - 24.8 = 25.2 ha
Answer: The area under apple trees on 25.2 hectares more than the area under the cherry trees.
Without checking any decision can be considered incorrect. How to check the solution to this problem? It is necessary to put together the area under apple trees and the area under the cherry trees. This result should be compared with the total area of the garden. If the sum is greater than the total area, so we decided not to challenge properly. If the sum is equal to or less than the total area, so our decision is correct.
50 + 24.8 = 74.8 hectares of less than 80 hectares
In most curious students immediately arises a natural question: what else is growing in this delicious daze, about what we were afraid to tell?
It was a children's nursery problem solution. Now the conversation for adults. This is the task of the textbook, which approved the Ministry of Education as an educational tool. The condition of this problem and at the same time used the percentage of fractional parts of a whole. Normal literate people'd never allow it. They used a ratio or percentage. Only an idiot is able to dump everything into one pile. The author of this task is illiterate idiot who either do not understand what he was doing, or for approval of a textbook command is ready to do anything. Stati, the quality of textbooks is very well characterizes the quality of all education. Our education is built on the principle of "one fools compose tutorials, others say they are fools".
For example, I'll write down the number on the same principle that is used by the author about the problem area of the garden. I simultaneously use two forms of records: number and letter. That's what I got:
2 thousand three hundred 45
As you can see, only idiots can do so. Competent people will write down this number as follows:
2345 or two thousand three hundred forty-five
Why am I so vehemently opposed to such problems? Children - they are like a sponge that absorbs everything. If it is written in the book, then you can do so. As a result, we get the next portion of the idiots who write stupid textbooks that stupid leadership approves a flock of idiots. Just because they are so used to be taught.
What would you say there is not mathematics, but knowledge of mathematics lies not in the ability to accurately repeat all the things the teacher taught. Knowledge of mathematics - is the ability to competently and simply express their thoughts in the language of mathematics.
Wednesday, May 06, 2015
The six faces of a parallelepiped
Here is such a question: "Parallelepiped. Explain where he has six faces?". If mathematics is not able to clearly explain to you the parallelepiped design, then I'll try to do it. Will speak only about the faces of the parallelepiped, without delving into other structural details of the mathematical model, we are not in the showroom, and I'm not a manager trying to sell you an outdated model of the parallelepiped.
So, imagine that you, as if nothing had happened, went to sleep in his rectangular (this clarification is important) room. And at night you wake up in the current model of the box in full size! No need to panic. Easy start counting the brink of mathematical miracle. The wall with the window - this is the first bound. The wall opposite the window - this is the second face. The walls of the left and right of the window - this is the third and fourth edges. The floor - this is the fifth face. Ceiling - this is the sixth and final face. Great mathematical revelation: the number of faces does not depend on the order of their conversion, the main thing - do not miss it.
If you are up to this point has not yet fallen asleep, the next question is what to do next? Mentally, we develop a mathematical papyrus entitled "Set Theory", looking for the head of "Infinite math scores sheep" and start counting. People say this is very good mathematical procedure helps insomnia.
Just I want to honestly admit that I have lied a little. Not a rectangular room is a working model of a rectangular parallelepiped, but quite the contrary - a cuboid is a mathematical model of the room. This is particularly evident during the repair. The area of the walls will be the surface area of the side faces of the cuboid. The area of floor or ceiling is determined as well as in the area of the base of the box. Of course, the builders made their nuances in the rules for determining the areas of math, but we will not specify now.
Incidentally, the rectangular room is entirely dependent on the quality of construction. It is only in ancient Greece mathematics was so developed that famous building of the Parthenon in Athens was built with almost no right angles and straight lines. There's a base architecture of the building was laid not a mathematical perfection, and optical illusions. I'm afraid of modern mathematics such a task is not on the shoulder - too high in the clouds, they soar. But we digress from faces of a parallelepiped.
If you count the face of the parallelepiped you wanted during the day and not at night, then pull out from the wardrobe of a rectangular box with shoes. The bottom of the box - it is one face, it is also the lower base of the box. The lid of the box - a second side, she's upper base. The four walls of a shoebox - it faces the third to sixth.
We have considered six faces of a rectangular parallelepiped. If angles are not straight and curves? In this case we are dealing with a conventional parallelepiped, not rectangular. On the number of faces is not affected. Well, think, a little crushed a parallelepiped. By the way, mathematics bend rectangular parallelepipeds or align usual? I'm on the algebra of interesting to see. However, the mathematician's simple: uttered the sacred mantra "Let there be given a parallelepiped" and that he had gleaming white chalk on a blackboard. Life is complicated. There are many ways to straighten the curvature and parallelepipeds - by a heavy sledgehammer, to flirty "Oh, please!". About algebra of these methods can not even ask.
Seriously speaking, the algebra and rectangular, and the usual parallelepipeds is exactly the same. Bends and leveled a parallelepiped using sines of the angles between the ribs. In all corners of the rectangular parallelepipeds straight sinuses and their unity. Lazy math just does not write these sinuses in formulas. In conventional Parallelepipeds sines of the angles is less than one, so willy-nilly have to mathematicians in their formulas to write.
In conclusion, as they say teachers, fix the passed material. As a fixative use a simple children's coloring book, which paint the six faces of a parallelepiped.
I recall that the parallelepiped, unlike the quadrangular prism opposing faces are parallel, and the base is a quadrilateral with parallel sides: square, rectangle, diamond or parallelogram. However, mathematicians believe that a parallelepiped is a kind of prism. So they have written in the definition. With the same success it can be argued that the prism is a kind of parallelepiped. Simply rewrite the definition, as a textbook on mathematics - it is not the Bible.
So, imagine that you, as if nothing had happened, went to sleep in his rectangular (this clarification is important) room. And at night you wake up in the current model of the box in full size! No need to panic. Easy start counting the brink of mathematical miracle. The wall with the window - this is the first bound. The wall opposite the window - this is the second face. The walls of the left and right of the window - this is the third and fourth edges. The floor - this is the fifth face. Ceiling - this is the sixth and final face. Great mathematical revelation: the number of faces does not depend on the order of their conversion, the main thing - do not miss it.
If you are up to this point has not yet fallen asleep, the next question is what to do next? Mentally, we develop a mathematical papyrus entitled "Set Theory", looking for the head of "Infinite math scores sheep" and start counting. People say this is very good mathematical procedure helps insomnia.
Just I want to honestly admit that I have lied a little. Not a rectangular room is a working model of a rectangular parallelepiped, but quite the contrary - a cuboid is a mathematical model of the room. This is particularly evident during the repair. The area of the walls will be the surface area of the side faces of the cuboid. The area of floor or ceiling is determined as well as in the area of the base of the box. Of course, the builders made their nuances in the rules for determining the areas of math, but we will not specify now.
Incidentally, the rectangular room is entirely dependent on the quality of construction. It is only in ancient Greece mathematics was so developed that famous building of the Parthenon in Athens was built with almost no right angles and straight lines. There's a base architecture of the building was laid not a mathematical perfection, and optical illusions. I'm afraid of modern mathematics such a task is not on the shoulder - too high in the clouds, they soar. But we digress from faces of a parallelepiped.
If you count the face of the parallelepiped you wanted during the day and not at night, then pull out from the wardrobe of a rectangular box with shoes. The bottom of the box - it is one face, it is also the lower base of the box. The lid of the box - a second side, she's upper base. The four walls of a shoebox - it faces the third to sixth.
We have considered six faces of a rectangular parallelepiped. If angles are not straight and curves? In this case we are dealing with a conventional parallelepiped, not rectangular. On the number of faces is not affected. Well, think, a little crushed a parallelepiped. By the way, mathematics bend rectangular parallelepipeds or align usual? I'm on the algebra of interesting to see. However, the mathematician's simple: uttered the sacred mantra "Let there be given a parallelepiped" and that he had gleaming white chalk on a blackboard. Life is complicated. There are many ways to straighten the curvature and parallelepipeds - by a heavy sledgehammer, to flirty "Oh, please!". About algebra of these methods can not even ask.
Seriously speaking, the algebra and rectangular, and the usual parallelepipeds is exactly the same. Bends and leveled a parallelepiped using sines of the angles between the ribs. In all corners of the rectangular parallelepipeds straight sinuses and their unity. Lazy math just does not write these sinuses in formulas. In conventional Parallelepipeds sines of the angles is less than one, so willy-nilly have to mathematicians in their formulas to write.
In conclusion, as they say teachers, fix the passed material. As a fixative use a simple children's coloring book, which paint the six faces of a parallelepiped.
 |
| The six faces of a parallelepiped |
I recall that the parallelepiped, unlike the quadrangular prism opposing faces are parallel, and the base is a quadrilateral with parallel sides: square, rectangle, diamond or parallelogram. However, mathematicians believe that a parallelepiped is a kind of prism. So they have written in the definition. With the same success it can be argued that the prism is a kind of parallelepiped. Simply rewrite the definition, as a textbook on mathematics - it is not the Bible.
Monday, April 27, 2015
Increased - that's a plus or multiply?
If you have a question: "Increased - is that a plus or multiply?", then definitely no answer to it. You need to read or listen to what's next. If "increased by as much as something something", then this addition and the plus sign. For example, the number 7 is increased by 2 units. What are we doing? To previous value we added increase and get new value:
7 + 2 = 9
If "increased in so many times," it is multiplication. For example, the number 7 is increased in 2 times. It is necessary to multiply the previous value of the specified number of times, and we get a new meaning:
7 * 2 = 14
As you can see, the numbers in the above examples are the same, and the result is different. It all depends on how to formulate a thought.
7 + 2 = 9
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| Increased |
If "increased in so many times," it is multiplication. For example, the number 7 is increased in 2 times. It is necessary to multiply the previous value of the specified number of times, and we get a new meaning:
7 * 2 = 14
As you can see, the numbers in the above examples are the same, and the result is different. It all depends on how to formulate a thought.
Tuesday, April 21, 2015
Snail on a pole
Objective: Snail crawling on a pole height of 10 meters. During the day, she rises to 4 meters per night descends to 3 meters. For some time the snail reaches the top of the column?
To solve this problem about a snail on a post need to revise some of its condition. The day the snail takes off the pole to a height of 4 meters. The post has a height of 10 meters. Reformulate the question: at what height should be a snail to a running start to fly to the top of the column? The answer to this question is easy to calculate:
10 - 4 = 6 meters
Now the second question: how many days should snail fidgeting up and down the pole to reach a height of 6 meters. Again, take the hands of the math and think:
6 / (4 - 3) = 6 days
As a result, we have six days cochlea will need to enter the starting position for a breakthrough victory, and one day for the race victory. The result is that the triumph of a snail on a post we will see after 7 days.
The same mathematical problem can be formulated with a modern twist. Look how it will sound in the Russian mathematical textbooks soon: From the border to Russia at the borders of NATO distance of 10 thousand kilometers. Green men move per day from the Russian border to NATO at the border at a distance of 4000 kilometers. During the night, NATO troops cast of unknown green men on the 3,000 kilometers back to Russia. How many days disappear NATO? The answer to this problem is the same: after 7 days of the great light of the Russian world illuminate the whole of Europe. By the way, the light of the Christian faith had once lit up Europe funeral pyres of the Inquisition. And the Third Reich sprawling across Europe recently. We must not forget history.
 |
| Snail on a pole |
To solve this problem about a snail on a post need to revise some of its condition. The day the snail takes off the pole to a height of 4 meters. The post has a height of 10 meters. Reformulate the question: at what height should be a snail to a running start to fly to the top of the column? The answer to this question is easy to calculate:
10 - 4 = 6 meters
Now the second question: how many days should snail fidgeting up and down the pole to reach a height of 6 meters. Again, take the hands of the math and think:
6 / (4 - 3) = 6 days
As a result, we have six days cochlea will need to enter the starting position for a breakthrough victory, and one day for the race victory. The result is that the triumph of a snail on a post we will see after 7 days.
The same mathematical problem can be formulated with a modern twist. Look how it will sound in the Russian mathematical textbooks soon: From the border to Russia at the borders of NATO distance of 10 thousand kilometers. Green men move per day from the Russian border to NATO at the border at a distance of 4000 kilometers. During the night, NATO troops cast of unknown green men on the 3,000 kilometers back to Russia. How many days disappear NATO? The answer to this problem is the same: after 7 days of the great light of the Russian world illuminate the whole of Europe. By the way, the light of the Christian faith had once lit up Europe funeral pyres of the Inquisition. And the Third Reich sprawling across Europe recently. We must not forget history.
Monday, April 20, 2015
9 multiply 7 equals
Immediately write a ready answer: 9 multiply 7 equals 63. So we have the multiplication table says that we all have been taught, but to the end, many of us have not learned. I'm not an exception. Natural laziness and chronic multiple sclerosis did not allow me to remember this vital piece of mathematics. But it is knowledge of mathematics allows me to easily cope with this column multiplication tables. I just calculate the result I needed.
We all know that when multiplied by a permutation of the factors does not change the result. If somewhere sometime mathematics will tell you otherwise, do not believe them - they are wrong. Simple math multiplication like to call it, that in fact is not a multiplication. For example, if we call a cow "Multiplication", it then becomes to turn milk and poop in the hay. After all, the cow to the mathematical action has nothing to do.
So, I always use the following equation:
9*7 = 7*9
Next is quite simple. That number, which must be multiplied by 9, I multiply by 10 and take away his once:
7*9 = 7*10-7 = 70-7 = 63
Multiplied by 10 is very simple - have added to the number of the toe and the result is ready. Subtract this number from a small number of possible even without a calculator. Now you can easily cope with any stitch multiplication table 9.
For example, if we need to multiply 9 by 9, proceed as follows:
9*9 = 90-9 = 81
If needed multiply 9 by 3:
9*3 = 3*9 = 30-3 = 27
In all these transformations have no miracles, we just use the normal math.
My personal opinion is: better to use the thinking abilities of their brain than clog the brain remembering any nonsense. Even if this nonsense forced to learn mathematics.
We all know that when multiplied by a permutation of the factors does not change the result. If somewhere sometime mathematics will tell you otherwise, do not believe them - they are wrong. Simple math multiplication like to call it, that in fact is not a multiplication. For example, if we call a cow "Multiplication", it then becomes to turn milk and poop in the hay. After all, the cow to the mathematical action has nothing to do.
So, I always use the following equation:
9*7 = 7*9
Next is quite simple. That number, which must be multiplied by 9, I multiply by 10 and take away his once:
7*9 = 7*10-7 = 70-7 = 63
Multiplied by 10 is very simple - have added to the number of the toe and the result is ready. Subtract this number from a small number of possible even without a calculator. Now you can easily cope with any stitch multiplication table 9.
For example, if we need to multiply 9 by 9, proceed as follows:
9*9 = 90-9 = 81
If needed multiply 9 by 3:
9*3 = 3*9 = 30-3 = 27
In all these transformations have no miracles, we just use the normal math.
 |
| 9 multiply |
My personal opinion is: better to use the thinking abilities of their brain than clog the brain remembering any nonsense. Even if this nonsense forced to learn mathematics.
Stupid triangle top view
Maybe it's someone not very pleasant, but stupid triangles does not happen, there are only stupid people. But wits on the Internet are often looking for a stupid triangle, top view which they would very much like to see. Show.
First, let's talk about the name of this type of triangles. Call similar triangles stupid - a sign of illiteracy. As they say now, "politically correct" will call such triangles "obtuse". All the triangles which have one angle greater than 90 degrees, are obtuse triangle. It's not a shame, is not a defect, just that physique is obtuse triangles. However, each obtuse triangle always has a couple of sharp corners. So, just in case.
Now let's talk about the kinds. No triangles on the kinds and types of top side of the window. Obtuse triangle top looks exactly the same as the bottom. But the side view is not pleasing to the eye - is just a regular interval. From the window you are unlikely to consider anything, but in a notebook at a neighbor's party this triangle you can see from many angles. In this case the obtuse triangle will look quite different, what it looks like from above. Describe such a wonderful transformation is possible by means of projective geometry, descriptive geometry, trigonometry, or poems. Who like more.
 |
| Stupid triangle top view |
First, let's talk about the name of this type of triangles. Call similar triangles stupid - a sign of illiteracy. As they say now, "politically correct" will call such triangles "obtuse". All the triangles which have one angle greater than 90 degrees, are obtuse triangle. It's not a shame, is not a defect, just that physique is obtuse triangles. However, each obtuse triangle always has a couple of sharp corners. So, just in case.
Now let's talk about the kinds. No triangles on the kinds and types of top side of the window. Obtuse triangle top looks exactly the same as the bottom. But the side view is not pleasing to the eye - is just a regular interval. From the window you are unlikely to consider anything, but in a notebook at a neighbor's party this triangle you can see from many angles. In this case the obtuse triangle will look quite different, what it looks like from above. Describe such a wonderful transformation is possible by means of projective geometry, descriptive geometry, trigonometry, or poems. Who like more.
Wednesday, April 08, 2015
How not to change the signs inside the brackets?
All of us are taught to change signs when opening or closing brackets of the expression in brackets, if before the brackets minus sign. Let's look at this tedious process for half-dead examples.
11-(2+5-4) = 11-3 = 8
Before the expression in brackets is a minus sign, it means that the disclosure of the brackets need to change all the signs to the contrary all the numbers that are inside the parentheses. Consider the same example, but without the brackets.
11-(2+5-4) = 11-2-5+4 = 9-5+4 = 4+4 = 8
Now let's try to take a part of the expression in parentheses.
1+2+3+4 = 10
Naturally, you ask: "Where is the minus sign?!" Do not worry, now appear.
1+2-(-3-4) = 3-(-7) = 3+7 = 10
I put in front of the brackets and the minus sign to change the sign before the numbers inside the parentheses. In disclosing the brackets again I changed the sign, because before my braces a minus sign. In the end, the result remained unchanged.
Now a more complex example.
17-6+9 = 20
17-(6-9) = 17-(-3) = 17+3 = 20
As you can see, continuous headache turns out, when suddenly appears before the brackets the minus sign. How not to change the signs inside the brackets? Very simply - no need to put a minus in front of the brackets. Look how it's done.
17-6+9 = 17+(-6+9) = 17+(3) = 17+3 = 20
Now consider the last two examples under the microscope. In the first case I put the first parenthesis after the minus sign. I like a knife cut a negative number into two parts - a minus sign and a positive number. The minus sign was in front of the bracket, and a positive number - within the brackets. Look.
17-(6........
In fact, we are in parentheses conclude a positive number, which until then was negative. Changing the sign of the first number in parentheses passed on complete autopilot without our intervention. A sort of a machine circumcision minus sign in numbers. But with the other numbers that fall into these brackets are already having problems. Signs they need to be changed manually.
In the second case I put the opening bracket before the minus sign. In fact, I put in brackets a negative number with a minus sign. Here's how it looks initially.
17(-6.........
Now, between numbers 17 and bracket there is no sign that mathematics implies multiplication. But I do not have anything to multiply. To answer in the solution remained the same example, I set the bracket additional "plus" sign.
17+(-6.........
Now everything is correctly written. Before the parentheses appears the plus sign and signs before the numbers inside the brackets intact. No mathematical crime I did not commit, just competently get rid of extra action on the replacement characters inside the brackets. Why mathematics always do? They had not taught this. If it is not in the curriculum, and then teach you that no one will. Math enough to know, you need to also know how to use her.
11-(2+5-4) = 11-3 = 8
Before the expression in brackets is a minus sign, it means that the disclosure of the brackets need to change all the signs to the contrary all the numbers that are inside the parentheses. Consider the same example, but without the brackets.
11-(2+5-4) = 11-2-5+4 = 9-5+4 = 4+4 = 8
Now let's try to take a part of the expression in parentheses.
1+2+3+4 = 10
Naturally, you ask: "Where is the minus sign?!" Do not worry, now appear.
1+2-(-3-4) = 3-(-7) = 3+7 = 10
I put in front of the brackets and the minus sign to change the sign before the numbers inside the parentheses. In disclosing the brackets again I changed the sign, because before my braces a minus sign. In the end, the result remained unchanged.
Now a more complex example.
17-6+9 = 20
17-(6-9) = 17-(-3) = 17+3 = 20
As you can see, continuous headache turns out, when suddenly appears before the brackets the minus sign. How not to change the signs inside the brackets? Very simply - no need to put a minus in front of the brackets. Look how it's done.
17-6+9 = 17+(-6+9) = 17+(3) = 17+3 = 20
Now consider the last two examples under the microscope. In the first case I put the first parenthesis after the minus sign. I like a knife cut a negative number into two parts - a minus sign and a positive number. The minus sign was in front of the bracket, and a positive number - within the brackets. Look.
17-(6........
In fact, we are in parentheses conclude a positive number, which until then was negative. Changing the sign of the first number in parentheses passed on complete autopilot without our intervention. A sort of a machine circumcision minus sign in numbers. But with the other numbers that fall into these brackets are already having problems. Signs they need to be changed manually.
In the second case I put the opening bracket before the minus sign. In fact, I put in brackets a negative number with a minus sign. Here's how it looks initially.
17(-6.........
Now, between numbers 17 and bracket there is no sign that mathematics implies multiplication. But I do not have anything to multiply. To answer in the solution remained the same example, I set the bracket additional "plus" sign.
17+(-6.........
Now everything is correctly written. Before the parentheses appears the plus sign and signs before the numbers inside the brackets intact. No mathematical crime I did not commit, just competently get rid of extra action on the replacement characters inside the brackets. Why mathematics always do? They had not taught this. If it is not in the curriculum, and then teach you that no one will. Math enough to know, you need to also know how to use her.
Saturday, January 31, 2015
Two angles of a triangle
Consider a very basic problem about two angles of a triangle are known. This problem sounds like this:
Two angles of a triangle are 53 degrees and 57 degrees. Find it the third corner of the triangle.
In any triangle all three angles. That is why the triangle is called. The value of the two angles of the three we know. Now I ask you a couple questions that will help solve this problem.
The first question. What is the sum of the angles of a triangle? This sacred knowledge of mathematics tease "A theorem on the sum of the angles of a triangle." As if they did not call it a law of nature, its essence does not change. Incidentally, the sum of the angles of a triangle belongs to the category of the mathematical knowledge that is easily stored for a long time, but that you never use not awake in their daily lives. Useless knowledge? No, but people use this knowledge is very limited range of professions, such as surveyors.
The second question. If you know that the sum of all the angles of a triangle is 180 degrees, with the arithmetic yourself cope? Here everything is simple. From the sum of the angles of a triangle 180 degrees subtract two prominent corner and get the value of the third angle of the triangle.
180 - 53 - 57 = 70 degrees
I do not want to show here ready-made solution, but ... First, the calculator have a lot of different buttons and accidentally be confused. In such cases, the scientists disappear satellites of Mars. So a complete solution for monitoring, can not hurt. Just check yourself.
Secondly, this is a very good opportunity to do what we do mathematics is strongly not recommended. We are taught to perform tasks with minimal downtime, and possibly without saving intermediate results. Actually, I did. On the one hand, it is correct. On the other hand, it does not give us the opportunity to understand, but what do we actually do?
Personally, I like to consider solving mathematical problems under the microscope in slow motion. Sometimes the impression is that we observe the focus by illusionist and all the secrets of the focus immediately crawl out. Let's look at the detailed solution of this problem on two well-known corners of the triangle and one unknown. Here's how it looks.
And so. Someone measured the angles in a triangle is real. The measurements were performed only for the two corners. Man in high school and knows that the third angle can be simply calculated. This is the condition of the problem. Now, a detailed description of the meaning of the decision and the action carried out by us.
1. Write a law that establishes a relationship between the angles of a triangle, in algebraic form. I have already said that in mathematics it is called "A theorem on the sum of the angles of a triangle." The geometric shape of this law is shown in the first picture.
2. Transform the algebraic form of the law on the corners of a triangle to solve our specific problem.
3. Enter this formula in the data from the task ahead of us. Pass from the algebraic form to the physical.
4.Analiziruem physical model for solving the problem. Mathematical apparatus introduced the decimal system of numbers, other notations are absent. The physical device is represented by a measure of the degree angles, other angle units available. Only under these conditions we can perform addition and subtraction.
5. Go to the mathematical model of the physical problem and perform mathematical operations with numbers using a calculator, a sheet of paper or in your mind.
6. Get ready solution to the problem in physical form.
Here's a novel in verse about I turned to a very simple task. The accuracy of the description of this literary opus does not claim because the school did not teach me this, had to invent on the fly. All the described actions we perform automatically, without going into detailed explanations. I agree with mathematicians that stupid every solution of the problem in as much detail paint. But even more stupid stupid to perform the actions that you teach. In this case, the formation is converted into a conventional animal training.
Two angles of a triangle are 53 degrees and 57 degrees. Find it the third corner of the triangle.
In any triangle all three angles. That is why the triangle is called. The value of the two angles of the three we know. Now I ask you a couple questions that will help solve this problem.
The first question. What is the sum of the angles of a triangle? This sacred knowledge of mathematics tease "A theorem on the sum of the angles of a triangle." As if they did not call it a law of nature, its essence does not change. Incidentally, the sum of the angles of a triangle belongs to the category of the mathematical knowledge that is easily stored for a long time, but that you never use not awake in their daily lives. Useless knowledge? No, but people use this knowledge is very limited range of professions, such as surveyors.
 |
| Sum of the angles of a triangle |
The second question. If you know that the sum of all the angles of a triangle is 180 degrees, with the arithmetic yourself cope? Here everything is simple. From the sum of the angles of a triangle 180 degrees subtract two prominent corner and get the value of the third angle of the triangle.
180 - 53 - 57 = 70 degrees
I do not want to show here ready-made solution, but ... First, the calculator have a lot of different buttons and accidentally be confused. In such cases, the scientists disappear satellites of Mars. So a complete solution for monitoring, can not hurt. Just check yourself.
Secondly, this is a very good opportunity to do what we do mathematics is strongly not recommended. We are taught to perform tasks with minimal downtime, and possibly without saving intermediate results. Actually, I did. On the one hand, it is correct. On the other hand, it does not give us the opportunity to understand, but what do we actually do?
Personally, I like to consider solving mathematical problems under the microscope in slow motion. Sometimes the impression is that we observe the focus by illusionist and all the secrets of the focus immediately crawl out. Let's look at the detailed solution of this problem on two well-known corners of the triangle and one unknown. Here's how it looks.
 |
| Two angles of a triangle |
And so. Someone measured the angles in a triangle is real. The measurements were performed only for the two corners. Man in high school and knows that the third angle can be simply calculated. This is the condition of the problem. Now, a detailed description of the meaning of the decision and the action carried out by us.
1. Write a law that establishes a relationship between the angles of a triangle, in algebraic form. I have already said that in mathematics it is called "A theorem on the sum of the angles of a triangle." The geometric shape of this law is shown in the first picture.
2. Transform the algebraic form of the law on the corners of a triangle to solve our specific problem.
3. Enter this formula in the data from the task ahead of us. Pass from the algebraic form to the physical.
4.Analiziruem physical model for solving the problem. Mathematical apparatus introduced the decimal system of numbers, other notations are absent. The physical device is represented by a measure of the degree angles, other angle units available. Only under these conditions we can perform addition and subtraction.
5. Go to the mathematical model of the physical problem and perform mathematical operations with numbers using a calculator, a sheet of paper or in your mind.
6. Get ready solution to the problem in physical form.
Here's a novel in verse about I turned to a very simple task. The accuracy of the description of this literary opus does not claim because the school did not teach me this, had to invent on the fly. All the described actions we perform automatically, without going into detailed explanations. I agree with mathematicians that stupid every solution of the problem in as much detail paint. But even more stupid stupid to perform the actions that you teach. In this case, the formation is converted into a conventional animal training.
Tuesday, January 20, 2015
Right triangle trigonometry
I will not tell you the rules and definitions of trigonometric functions on a right triangle. Mathematics is happy to do without me. I'll just show you a picture, which shows a right triangle trigonometry.
The top row shows who is who in the trigonometric zoo. Sine and cosine of the angle alpha - a relationship of legs to the hypotenuse. Tangent and cotangent - a relationship of legs. With hypotenuse usually no problem, and it is located opposite one of the right angle. But the legs as much as two and they are different. One is located in front of the angle alpha is called opposed (the picture side a). Another gently pressed against the corner called alpha and an adjacent (on the picture side b). Now, looking at the picture, you can easily formulate the definition of trigonometric functions on a right triangle.
The bottom row of images shows how to find the sides of a right triangle, if we know one side and the angle alpha. Known side is green. Using this side and trigonometric functions, you can easily find the other two sides of a right triangle.
Spin this picture as you like, turn face down and look up to the light - trigonometric dependence in a right triangle from it are not changed.
This picture you can come in handy in the future, the study of physics, theoretical mechanics, in engineering calculations. By the time you've forgotten how to define and use trigonometric functions in a right triangle.
And the saddest thing in the end. If you teach mathematics to use mathematics, these pictures you drew to themselves for a few minutes, without any textbooks. After all this is done simply elementary.
 |
| Right triangle trigonometry |
The top row shows who is who in the trigonometric zoo. Sine and cosine of the angle alpha - a relationship of legs to the hypotenuse. Tangent and cotangent - a relationship of legs. With hypotenuse usually no problem, and it is located opposite one of the right angle. But the legs as much as two and they are different. One is located in front of the angle alpha is called opposed (the picture side a). Another gently pressed against the corner called alpha and an adjacent (on the picture side b). Now, looking at the picture, you can easily formulate the definition of trigonometric functions on a right triangle.
The bottom row of images shows how to find the sides of a right triangle, if we know one side and the angle alpha. Known side is green. Using this side and trigonometric functions, you can easily find the other two sides of a right triangle.
Spin this picture as you like, turn face down and look up to the light - trigonometric dependence in a right triangle from it are not changed.
 |
| Spin this picture |
This picture you can come in handy in the future, the study of physics, theoretical mechanics, in engineering calculations. By the time you've forgotten how to define and use trigonometric functions in a right triangle.
And the saddest thing in the end. If you teach mathematics to use mathematics, these pictures you drew to themselves for a few minutes, without any textbooks. After all this is done simply elementary.
Saturday, January 17, 2015
Hilbert's sixth problem
Here I have in the comments appeared one modest request:
Help solve the problem associated with the extend of Kronecker–Weber theorem on abelian extensions of the rational numbers to any base number field.
Such a number of clever words in one sentence, I have not met. I immediately suspected that not all of the school textbook is taken, and from some other place. Type in the search phrase Wikipedia "Kronecker–Weber theorem" and look at the result. That's right, I slipped twelfth problem of Hilbert, has not been solved so far. Suddenly her uncle decides - all would be funny!
You should not expect. For me, the phrase "abelian extensions", "the rational numbers to any base number field" is no more than a collection of letters. In higher mathematics, I did not go through dog training and have no idea what to do, when he heard these phrases. Moreover, I do not even have a clue what the root of the unit n-th degree is different from the unit itself. In mathematics, the same thing can have different names, the same name can mean a variety of things. Hence, problems arise.
More than a hundred years ago David Hilbert formulated twenty-three problems of mathematics. Some problems have been resolved, some partially solved two problems remain unsolved to this day. There are also a number of problems that are simply hushed for clarity: "too vague" or "requires clarification of the phrase". Among these "jammed" caught my attention problems Hilbert's sixth problem, which is: Mathematical treatment of the axioms of physics. Mathematics jumped from the solution of this problem with the phrase "too vague". This is where I disagree with them.
Formulation of Hilbert very clearly, that's just to express nothing - not taken root in physics ideas about mathematics axioms, as his time in the chemistry did not take the idea of negative numbers. This mathematics all their theories verify with the axioms of physics as his theories verify with the experimental results. There physics postulates, but it's just a temporary patch on the white spots of our knowledge. Sooner or later, the postulates are replaced by physical laws. Temporary postulates physicists do not go to any comparison with the monumental firmness axioms of mathematics.
Here and there is a very sudden decision Hilbert's sixth problem - the language of mathematics is much easier to state "axioms" of religion than the axioms of physics. Looks mathematical presentation of the fundamentals of religion something like this:
- Sacred texts in religion - axioms and definitions in mathematics;
- All that is in this world created by God - in mathematics, the phrase "Let us given ..." by default assumes that everything God gives us;
- The story of Noah's ark and the "pair of every creature" - set theory;
- Man is composed of body and soul - complex numbers consist of real and imaginary parts;
- Kingdom of God - a complex space;
- God and the devil, good and evil - the positive and negative numbers;
- Holy Cross - Cartesian coordinate system.
If desired, you can thoroughly examine the sacred texts of religion and the sacred texts of Mathematicians (axioms and definitions) in search of other matches. In the language of religion, math is pretty good presents.
In my opinion, the problems in modern mathematics and believers alike - autism. They live in an imaginary world and not pay attention to their surroundings. Teaching of mathematics is very similar to the missionary preaching - we need to learn and do what they say preachers. All attempts to draw attention to the preachers of the surrounding reality ends are being sent to the sacred texts: "Read the Bible", "Read the definition."
Help solve the problem associated with the extend of Kronecker–Weber theorem on abelian extensions of the rational numbers to any base number field.
Such a number of clever words in one sentence, I have not met. I immediately suspected that not all of the school textbook is taken, and from some other place. Type in the search phrase Wikipedia "Kronecker–Weber theorem" and look at the result. That's right, I slipped twelfth problem of Hilbert, has not been solved so far. Suddenly her uncle decides - all would be funny!
You should not expect. For me, the phrase "abelian extensions", "the rational numbers to any base number field" is no more than a collection of letters. In higher mathematics, I did not go through dog training and have no idea what to do, when he heard these phrases. Moreover, I do not even have a clue what the root of the unit n-th degree is different from the unit itself. In mathematics, the same thing can have different names, the same name can mean a variety of things. Hence, problems arise.
More than a hundred years ago David Hilbert formulated twenty-three problems of mathematics. Some problems have been resolved, some partially solved two problems remain unsolved to this day. There are also a number of problems that are simply hushed for clarity: "too vague" or "requires clarification of the phrase". Among these "jammed" caught my attention problems Hilbert's sixth problem, which is: Mathematical treatment of the axioms of physics. Mathematics jumped from the solution of this problem with the phrase "too vague". This is where I disagree with them.
Formulation of Hilbert very clearly, that's just to express nothing - not taken root in physics ideas about mathematics axioms, as his time in the chemistry did not take the idea of negative numbers. This mathematics all their theories verify with the axioms of physics as his theories verify with the experimental results. There physics postulates, but it's just a temporary patch on the white spots of our knowledge. Sooner or later, the postulates are replaced by physical laws. Temporary postulates physicists do not go to any comparison with the monumental firmness axioms of mathematics.
Here and there is a very sudden decision Hilbert's sixth problem - the language of mathematics is much easier to state "axioms" of religion than the axioms of physics. Looks mathematical presentation of the fundamentals of religion something like this:
- Sacred texts in religion - axioms and definitions in mathematics;
- All that is in this world created by God - in mathematics, the phrase "Let us given ..." by default assumes that everything God gives us;
- The story of Noah's ark and the "pair of every creature" - set theory;
- Man is composed of body and soul - complex numbers consist of real and imaginary parts;
- Kingdom of God - a complex space;
- God and the devil, good and evil - the positive and negative numbers;
- Holy Cross - Cartesian coordinate system.
If desired, you can thoroughly examine the sacred texts of religion and the sacred texts of Mathematicians (axioms and definitions) in search of other matches. In the language of religion, math is pretty good presents.
In my opinion, the problems in modern mathematics and believers alike - autism. They live in an imaginary world and not pay attention to their surroundings. Teaching of mathematics is very similar to the missionary preaching - we need to learn and do what they say preachers. All attempts to draw attention to the preachers of the surrounding reality ends are being sent to the sacred texts: "Read the Bible", "Read the definition."
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