Let me tell you first, I am not stopping you from using √(-1), it is just a symbol, here is just to discuss the properties of some numbers
We first need to discuss a property of the next root form. The following will tell a story about Xiao Ming:
His teacher wrote such a question on the blackboard
a very normal question
His classmate Xiaohong is very smart and got the correct answer right away
A very standard problem-solving procedure
But looking at Xiao Ming, he seems a little weird
He misunderstood the title first, and finally misunderstood the result. The steps are as follows
Misunderstood the question first
then count
Finally got the result wrong again
Eh? Actually right!
But in fact, this is not an unreasonable accident.
We first set a^2=5, and then calculate (8+a)^2
step
And because a^2=5, so there is
I admit that I am very busy, otherwise I would not post this article
At this time, it is obvious that there is no difference whether a=√5 or a=-√5
This is why Xiao Ming was actually right in the end.
But we can't stop, because there is a problem before us:
If they are said to have the same nature in the calculation process, how do we distinguish them?
For ±√5, this problem is particularly easy to solve, because they are real numbers and can be compared
We know that there is such a number 2.23, its square is 4.9729
We also know that there is such a number 2.24, its square is 5.0176
So we get an interval, 2.23<√5<2.24
This interval can be kept small, so we know that √5 is a real number, but it is not easy to express
it is approximately equal to
an approximate value
By analogy, -√5 can also be expressed in this way, after all, it is also a real number
another approximation
We found that we can derive an approximation of ±√5, so we can at least distinguish between the two by approximation
At this point we can finally face our problem:
√(-1) Why is this mark unreasonable?
The proof process we have just done is universal for all numbers that open the square root sign, so we can know that ±√(-1) has the same property in the operation process
Here comes the question, how do we distinguish between ±√(-1)?
The method just now cannot be used, because we know that for any real number, its square must be a non-negative number
We can't find such real numbers, and the difference between them and ±√(-1) can be kept small, so we can't find the so-called interval just now, and we can't find the real number approximation
an "approximate value"
In other words, for ±√(-1), we cannot distinguish without a new definition
We only know that they are opposite to each other, but because of their special nature, we cannot make a simple distinction between them
Just ±√5 in terms of calculation, even if the two can be exchanged at the same time and the calculation result does not change much, but at least the approximate value has changed
Yes, they are at least slightly different in their approximations to real numbers
But in front of ±√(-1), you can't calculate the approximate value. At this time, if you exchange the two at the same time, there is really no difference
At this time, the ± outside the root sign loses its meaning in real numbers
So defining i=√(-1) is not very accurate, because you can define i=-√(-1), and you will find that such a change will not make any difference in the operation
So generally define i like this, i^2=-1
At this time, (-i)^2=-1 is also defaulted
So if we use i, we will be less stingy like the author on the way to learn mathematics
That's why I think √(-1) is not a good notation
At this time, everyone should be able to understand why Descartes deliberately set the imaginary number to i, in order to prevent the occurrence of double precision.
Of course, at this time, you will definitely understand why Descartes set the imaginary number as the letter i
imaginary => imaginary
imaginary number=>imaginary number