Imaginary Numbers and Properties: A Puzzling Case

In summary, the properties of the square root operation do not work with imaginary numbers. This is because in order to extend the operation to work with complex numbers, not just positive reals, it must be self-consistent and not self-contradictory. Therefore, not all properties of the square root operation can be retained.
  • #1
[C]alculus
2
0
-1/1=1/-1
sqrt(-1/1)=sqrt(1/-1)
i/1=1/i
i*(i/1)=i*1/i
i^2/1=i/i
-1/1=1
-1=1 <-- Well my conclusion is that properties don't work with imaginary numbers or did i do something wrong?
 
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  • #2
Lines 2-to-3 wrong.
Don't bother to post similar riddles at these forums.
 
  • #3
arildno said:
Lines 2-to-3 wrong.
Don't bother to post similar riddles at these forums.
Why[tex]\sqrt\frac x{y}\not =\frac{\sqrt x}{\sqrt y}[/tex] ??
 
  • #4
If x and y are both positive real numbers that is true, but not in general for x and y being complex numbers.
 
  • #5
kntsy said:
Why[tex]\sqrt\frac x{y}\not =\frac{\sqrt x}{\sqrt y}[/tex] ??
To expand on GibZ's comment.
In order to extend the square root operation to work on real&complex number, and not just on the positive reals, you will not be able to retain <i>all</i> properties of the square root operation if you wish it to be self-consistent, and not self-contradictory.
 

1. What are imaginary numbers?

Imaginary numbers are numbers that are expressed in the form of a real number multiplied by the square root of -1. They are used to represent complex numbers and cannot be represented on a traditional number line.

2. Why are they called "imaginary"?

Imaginary numbers got their name because they were initially thought to be "imaginary" or non-existent. However, they were later found to be useful in solving mathematical problems and are now an important part of complex numbers.

3. What is the difference between real and imaginary numbers?

Real numbers are numbers that can be represented on a traditional number line and include all rational and irrational numbers. Imaginary numbers, on the other hand, cannot be represented on a number line and are always expressed in terms of the square root of -1.

4. What are some properties of imaginary numbers?

Some properties of imaginary numbers include the fact that when squared, they result in a negative real number. Additionally, they follow the same rules of arithmetic as real numbers, such as addition, subtraction, multiplication, and division.

5. How are imaginary numbers used in the real world?

Imaginary numbers are used in various fields of science and engineering, such as electrical engineering, quantum mechanics, and signal processing. They are also used in financial modeling and computer graphics.

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