Why is a equal to negative a in this scenario?

[noahsdev]

I've been messing around with numbers (as you do) and I'm wondering why this occurs..
lets let a = b-c.
√a
= √(b-c)
=√(-(c-b))
=i√(c-b)
=i√(-(b-c))
=i²√(b-c)
=-√(b-c)
=-√a
For example if you let a = 1, b = 2, and c = 1.

 

[Mark44]

I've been messing around with numbers (as you do) and I'm wondering why this occurs..
lets let a = b-c.
√a
= √(b-c)
=√(-(c-b))
=i√(c-b)

The step above is where the problem is. You're using the property that $\sqrt{ab} = \sqrt{a}\sqrt{b}$
There are restrictions on this and some of the other square root properties - both a and b have to be nonnegative.

=i√(-(b-c))
=i²√(b-c)
=-√(b-c)
=-√a
For example if you let a = 1, b = 2, and c = 1.

 

[noahsdev]

The step above is where the problem is. You're using the property that $\sqrt{ab} = \sqrt{a}\sqrt{b}$
There are restrictions on this and some of the other square root properties - both a and b have to be nonnegative.

That makes sense. Thanks.

 

[FactChecker]

$\sqrt[2]{-1} = \pm i.$ The choice of the '+' or '-' depends on the situation. So your result is '$a = \pm a$' where you must decide which sign is correct.

 

[CellCoree]

First of all, it is important to note that in mathematics, the equal sign (=) means that the two expressions on either side are equivalent or have the same value. It does not necessarily mean that they are identical or the same thing.

In this case, a is equal to negative a because of the properties of square roots and negative numbers. When we take the square root of a number, we are essentially finding the number that, when multiplied by itself, gives us the original number. So, when we take the square root of a negative number, we are essentially looking for a number that, when multiplied by itself, gives us a negative result.

Now, let's look at the expression √a. When we plug in a = b-c, we get the expression √(b-c). This means that we are looking for a number that, when multiplied by itself, gives us b-c. However, when we plug in a = -(c-b), we get the expression √(-(c-b)). This means that we are looking for a number that, when multiplied by itself, gives us -(c-b). But, as we know, a negative number multiplied by a negative number gives us a positive number. So, √(-(c-b)) is equivalent to √(c-b).

This is why we can simplify the expression to i√(c-b) and then further to -√(b-c). By simplifying, we are essentially making the expressions equivalent to each other. So, even though we started with two different expressions, they both simplify to -√a, showing that a is equal to negative a.

In your example, when a = 1, b = 2, and c = 1, we get √1 = 1 and -√1 = -1, which shows that 1 is equal to -1. This may seem counterintuitive, but it is important to remember that in mathematics, we are dealing with abstract concepts and not physical objects. So, we cannot always rely on our intuition to understand mathematical concepts.

I hope this helps to clarify why a is equal to negative a in this scenario. It is a result of the properties of square roots and negative numbers. Keep exploring and experimenting with numbers, and you will continue to discover interesting mathematical concepts and relationships.