Several equal but different ones (1)?

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  • Thread starter Izy Amisheeva
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In summary: In other words, the square root of a number is the same as the number whose square root is that number. This is always true when a is positive. However, it is not always true when a is negative. For example, the square root of -1 is -1, but the square root of -2 is -0.5.
  • #1
Izy Amisheeva
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TL;DR Summary
I ran into a little problem trying to solve the (√x )+ 1 = 0 equation.
Summary: I ran into a little problem trying to solve the (√x )+ 1 = 0 equation.

Obviously there will be no solution in R, so I tried the following

(√x )+ 1 = 0

ei α/2 = ei π

i α / 2 = i pi

so: α = 2 pi

so the solution will be e2 i π

This solution actually works when replaced in the equation.

The question is:

e2 i π works, but 1 which is the same does not work, just as e4 i πdoes not work either.

It seems that of the infinite possibilities of complex representation of 1 only one of them is solution!

Anyone could explain this?

Tanks
 
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  • #2
There are many solutions: ##e^{(4n+2)\pi i}##, for all integer ##n##.
 
  • #3
√1 = ±1
 
  • #5
DaveE said:
√1 = ±1
This isn't true. ##\sqrt 1 = +1## only, if we're talking about the real square root function that maps nonnegative real numbers to the same set of numbers.

In any case, the original equation was ##\sqrt x + 1 = 0##, or ##\sqrt x = -1##. If we square both sides, we get ##x = 1,## but this is not a solution of the first equation. In short, the equations ##\sqrt x + 1 = 0## and ##x = 1## are not equivalent.
 
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  • #6
Izy Amisheeva said:
(√x )+ 1 = 0

ei α/2 = ei π

i α / 2 = i pi

This is not equivalent to the previous equation since ##e^A = e^B## does not imply ##A = B##

so the solution will be e2 i π
This solution actually works when replaced in the equation.

The solution for ##\alpha## works when replaced in the equation ##i \alpha / 2 = i \pi##, but that equation is not equivalent to the original equation.
 
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  • #7
DaveE said:
√1 = ±1
No, √1= 1. The square root of a positive real number, √a, is defined as the positive number, x, such that [itex]x^2= a[/itex].
 

Related to Several equal but different ones (1)?

1. What does it mean for something to be "equal but different"?

When something is described as "equal but different", it means that while two or more things may have some similarities or qualities that are the same, they also have distinct differences that set them apart.

2. How can something be both equal and different at the same time?

This concept is often seen in science and mathematics, where objects or phenomena may have equal values or properties, but differ in other aspects such as size, shape, or composition.

3. Can you provide an example of something that is "equal but different"?

One example could be two different species of birds that have the same number of feathers, but have different colors and beak shapes.

4. What is the significance of studying things that are "equal but different"?

Studying things that are "equal but different" can help us better understand the underlying principles and patterns that govern our world. It can also lead to new discoveries and advancements in various fields of science.

5. How can we apply the concept of "equal but different" in our daily lives?

Recognizing and understanding the concept of "equal but different" can help us appreciate and respect diversity and differences among individuals and cultures. It can also help us solve problems and make decisions by considering multiple perspectives and finding common ground.

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