We often say that 1 is 1 and 2 is 2, but in mathematics, 1=0.99999 can be proved. These two numbers are obviously different, but it is strange that they can be equal. Why? There are many similar controversies in the field of mathematics. The following exploration series will introduce the controversy in mathematics when 1=0.99999!
1=0.99999 Mathematical controversy.
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1, operation flow
2. The university teacher explained that
3. Mathematics and reality
4. Similar disputes in mathematics.
5. Strange math problems
Arithmetic process
a=0.99999…
10a=9.99999…
10a=9+0.99999…
10a=9+a
9a=9
a= 1
This is an example to prove 1=0.99999. According to this line of thinking, there seems to be no problem, but there always seems to be something wrong.
1=0.99999 Can you solve the controversial and strange mathematical problems in the field of mathematics?
The math teacher at Korea University explained
People who think that 0.99999 is equal to 1 are because1/3 = 0.333331/3x3 =1and 0.333X3=0.99999= 1. Ordinary people's thinking is a circular decimal followed by an infinite loop, which is difficult to understand. Now I tell you, in fact, there are many other ways of cyclic number, such as multi-bit cycle and so on. Now I'll tell you in a popular way.
1=0.99999 Can you solve the controversial and strange mathematical problems in the field of mathematics?
The cycle of 0.999999999999, 9 is the unit cycle. Now let's add a multi-bit cycle number, such as1/7 = 0.1428 571428 57142857. Let's calculate 1/X and 0.99999/X, and see if 1/X is equal to 0.999999/x. If 0.99999= 1, the calculation results are definitely equal. In the process of calculation, you will find a very amazing phenomenon, (calculate first, think analogously with other cycle numbers) whether you can calculate infinite cycles, which is very amazing. This is math. We can also set x to another acyclic number.
Mathematics and reality
Mathematics can have nothing to do with reality, and its key is definition. Different definitions can make him equal or unequal.
If you stay on the definition of rational number (i.e. fraction) and assume that 0.9999 ... is rational number, then 0.9999 ... is converted into component number, which is 11,which is undoubtedly1.
If you stay on the definition of real numbers, assuming that 0.9999 ... is a real number, then there are no other real numbers ... and 1 between 0.9999, and they are equivalent whether they are converted into sequence representation or de-de-de division, so they are also equal.
1=0.99999 Can you solve the controversial and strange mathematical problems in the field of mathematics?
If we go beyond the real number and define a new number system containing "numbers infinitely close to 1", then it is not equal to 1.
In fact, people who think it is equal to 1 have created 1 incomplete new number systems, which are beyond real numbers and contain "numbers infinitely close to a real number".
Of course, mathematics and reality are inseparable, and many things in life should be applied to the principles of mathematics.
A similar debate in the field of mathematics
1, Zeno paradox
This is also a controversy in the field of physics. Achilles and Zhi Nuo the tortoise ran together. Before Achilles started running, the tortoise was 65,438+000 meters ahead of Alisky.
Achilles runs 100 meter, the tortoise runs one meter more, Achilles runs one meter, and the tortoise runs one centimeter more. It can be inferred that Achilles will never outrun the tortoise. Although it runs very fast in reality, it seems that it will never catch up with it in mathematics.
1=0.99999 Can you solve the controversial and strange mathematical problems in the field of mathematics?
2. ants and rubber bands
An ant is crawling from one end of the rational elastic rope to the other at a speed of 1cm per second. The elastic rope is stretched uniformly at the speed of 1m per second. Can ants climb to the finish line?
It seems useless, but it is very useful in mathematics. Assuming that the speed of the elastic rope is 0.9 cm per second, ants can intuitively climb to the finish line. The uniform elongation of the elastic rope means that there is always a point on it with a speed of 0.9 cm per second, which means that ants can climb to this point. Next, just segment the whole elastic rope. There are some math problems that are very strange.
Strange math problems
One night, three people went to a hotel and stayed in 300 yuan for one night. Three people just each paid 100 to make up 300 yuan for the boss. When they returned to their room, the boss forgot to give them today's discount and asked the waiter to return it to them. The waiter felt that 50 yuan was not easy to divide, so he took 20 yuan. After each of the three people got 10 yuan, it should be that each person only spent 90 yuan's money for one night, 3*90=270, and the service fee was taken from 20 yuan, and 270+20=290 yuan. 1where did 0 yuan go? 300-290 = 65,438.