Equal sizes
The meaning is different
As a recognized labor model, in addition to working every day, the supermodel Jun also cultivates his cousin's scientific research ability and spirit from an early age.
Today, the supermodel Jun supervises the homework of her 8-year-old cousin as usual, and in a question where 0.1 is not equal to 0.10, the cousin does not hesitate to write an equal sign.
The supermodel told his cousin that you can write an equal sign for this question, but they are not exactly the same.
The cousin was in a hurry, and the teacher clearly said that no matter how many zeros there are in the back of 0.1, it is the same!
The supermodel Jun did not hold back, so he taught her a lesson in advance!
Is 0.1 the same as 0.10?
If we only learned exact decimals, the problem would seem redundant.
Because in the exact decimal:
0.1 = 1/10, 0.10 = 10/100, but the fraction of 10/100 is reduced to 1/10. So the values of the two are exactly the same. Generally speaking, 0.10 is not written in the simplest decimal way, so it is considered that the last zero is not necessary.
But in approximate decimals, this problem becomes very important.
When rounded to approximate, the decimal 0.1 may be obtained from 0.05 with "five in" or from 0.14 with "rounded". Thus, an approximate decimal place of 0.1 indicates that its exact value is between greater than or equal to 0.05 and less than 0.15.
If x is used for its exact value, then, 0.05≤x<0.15.
What if it's written as 0.10? This approximate decimal may be obtained from 0.095 with "five in" or from 0.1049 with "rounded".
If x is used to represent its exact value, then 0.095≤ x< 0.105, its range is much smaller than 0.1.
So in the approximate decimals, the difference between 0.1 and 0.10 is large.
For example, in chemical research, there will be weighing, preparation of solutions and other operations, each number is followed by various units, this time accurate to which digit, the decimal point after the 0 also becomes very important, 0.1 and 0.10 here there is a difference, a little careless will get different results!
For example, in financial accounting bookkeeping, it is usually in yuan as a unit, and the angles and points are expressed in decimals, and the points are not deleted. For example, 10 yuan 1 corner is recorded as 10.10 yuan, and the last 0 should not be removed as 10.1 yuan. 0.10 is also not counted as 0.1.
Under the training of supermodel Jun, the 8-year-old cousin already has the signs of scientific research spirit, throwing out a question: the precise decimal is simple and simple, why should you propose an approximate decimal?
Why should there be approximate decimals?
In fact, in practical problems, many values cannot be completely accurate, and many values require that they do not have to be completely accurate, but only consider the approximate values of these values.
For example, how old are you when someone asks you? You say 8 years old, this is an approximation, if you want to be precise it becomes very troublesome, you have to talk about 8 years and months, 8 years old months and days, 8 years old months have a few days and a few hours and a few minutes...
There is no approximate decimal, it takes a few minutes to report the age, to think to calculate, not necessarily accurate!
Different things require different levels of precision! Like reporting age, we generally only need to approximate the year. But in atomic physics , " superons " have a lifespan of only 10^-10 to 10^-8 seconds , which is very short , and to find out their ages is at least approximately 10 ^ - 10 to 10 ^ - 8 seconds.
Back to the mathematical problem. Mathematically, there are numbers with countless numbers after the decimal point, and if there are no approximate decimals, these numbers are difficult to use.
Here, the magical pi π, a number with a history of thousands of years, must be nominated. π the numbers after the decimal point have not yet been calculated, or will never be calculated, and on March 14, 2019, Google announced that Pi has now reached 31.4 trillion decimal places.
The status of Pi does not need to be said, after all, every year on March 14th, this day belongs to it. This endless number is almost indispensable in our scientific research or life!
For example, when we calculate the area of a piece of land with a round shape, we can only approximate the π to 3.14 to get an exact number, in order to be clear;
Calculus, the higher trigonometric identity, was developed by the study of pi. π plays a very important role in promoting the development of mathematics.
By calculating π you can also test whether there are problems with the computer, including software and hardware problems.
The 8-year-old cousin asked again, but how could there be such an unreasonable number?
The mathematical crisis of irrational numbers
Heck, it's just irrational numbers!
An irrational number is a number full of blood. We all know that people who tell the truth are often targeted, especially when telling the truth offends the interests of others.
In 500 BC, the ancient Greek mathematician Pythagoras put forward the view that "all things are numbers": the element of numbers is the element of all things, the world is composed of numbers, everything in the world is not expressed by numbers, and numbers themselves are the order of the world.
In that feudal era, Pythagoras dominated academia. At this time, The Pythagorean disciple Ofsus discovered a striking fact:
The diagonal of a square and the length of one side of it are unjustifiable (if the side length of the square is 1, the length of the diagonal line is not a rational number), which is very different from the philosophy that "everything is number" (referring to the rational number).
This discovery triggered the first great crisis in the history of mathematics, and Pythagus, standing on the academic genius, panicked and worried about his position, tried to block the truth and exclude Hepzos. Hebzos was forced into exile, and eventually the Disciples of the Bi clan were brutally thrown into the water and killed.
Truth can never be erased. In 1872, the German mathematician Dedekin, proceeding from the requirement of continuity, used the "division" of rational numbers to define irrational numbers, ending the era when irrational numbers were considered "irrational" and ending the first great crisis in the history of mathematics that lasted for more than 2,000 years.
Irrational numbers finally fill the gap of "everything is number of digits".
Now there is a very bold idea for the most representative π of irrational numbers: all the information in the universe is stored in the numbers behind the π, and when needed, it is enough to search, and all the storage will be replaced!
This idea also appeared in the film and television drama "Suspect Tracking", can it be realized? We don't know.
Harold Finch said this:
"The ratio of the circumference of a circle to the diameter is endless and never repeated. In this string of numbers, every possible combination is included. Your birthday, locker password, social security number, all in one of them. If you convert these numbers to letters, you can get all the words, countless combinations. The first syllable you uttered as a baby, the name of your sweetheart, the story of your life from beginning to end, everything we have done or said, all the infinite possibilities in the universe, are in this simple circle. ”
This time, the supermodel Jun has expanded the program of her 8-year-old cousin a little, cultivating her to look at any problem and hold a rigorous spirit! 0.1 and 0.10 are numerically the same, but not identical.