Proving a^2 is Always Positive

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In summary, the conversation is about proving that a^2 is positive and the different approaches and assumptions used. The first approach involves considering two cases - a≥0 and a≤0 - and using basic algebra to show that a^2 is indeed positive. The second approach involves multiplying both sides of the inequality by a, but it is pointed out that this assumes the knowledge of what a*a means. The conversation also touches on the topic of building algebra from real number axioms.
  • #1
jaqueh
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0

Homework Statement


prove a^2 is positive

I have tried this, but i don't know if it's conclusive:

i) Suppose a≥0
then a+1≥1
then a(a+1)≥a
then a^2+a≥a
then a^2≥0
ii) Suppose a≤0
then 1+a≤1
then a(a+1)≥a since a is negative, by a lemma
then a^2+1≥a
then a^2≥0
 
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  • #2
Seems pretty ok. You aren't stating all of your assumptions and lemmas here, though. And there is a typo in your second argument.
 
  • #3
ok great, yeah i'll be more specific and i do see my typo: a+1 instead of 1+a
 
  • #4
jaqueh said:
ok great, yeah i'll be more specific and i do see my typo: a+1 instead of 1+a

I actually meant turning a(a+1) into a^2+1, but I'm sure that's not news.
 
  • #5
oh gosh i didn't even realize i had typed that, haha well thanks anyways.

btw you are always incredibly helpful, and for that i am really grateful!
 
  • #6
Why don't you just multiply both sides immediately by a?

i.e.

Case 1: a≥0
a2≥0

Case 2: a≤0
a2≥0

Seems a lot more direct and it's the same idea.
 
  • #7
because you don't know what a*a means yet
 
  • #8
jaqueh said:
because you don't know what a*a means yet

What do you mean, exactly?
 
  • #9
well I am taking a beginning real analysis course and basically we're building the basic structure of algebra from scratch from a couple of real number axioms.
 
  • #10
jaqueh said:
well I am taking a beginning real analysis course and basically we're building the basic structure of algebra from scratch from a couple of real number axioms.

OK. But saying you don't know the implication of a*a means you don't know the implication of a*(a+1) either.

You're already making an implicit assumption when you reverse the sign of the inequality upon multiplying by negative a. Multiplying a≤0 by a assumes no more than multiplying a+1≤1 by a.
 
  • #11
the multiplying done by the negative constant was proved in a lemma but you're right, it wasn't known before. and when i did a(a+1) i guess i knew that it was a*a+a, but i didn't know what a*a was still so i had to describe it using the inequalities
 

1. What is the concept of proving a^2 is always positive?

The concept of proving a^2 is always positive is based on the fact that the square of any real number (a) will always result in a positive number, regardless of the value of a. This can be proven mathematically using algebraic manipulation and the properties of real numbers.

2. Why is it important to prove a^2 is always positive?

Proving a^2 is always positive is important because it is a fundamental concept in mathematics and has many applications in various fields such as physics, engineering, and economics. It also helps us understand the behavior of quadratic functions and their graphs.

3. How can I prove a^2 is always positive?

To prove a^2 is always positive, you can use the definition of squaring a number, which is multiplying it by itself. Then, you can use algebraic manipulation and the properties of real numbers to show that the result will always be a positive number.

4. What are some examples of proving a^2 is always positive?

One example is proving that (2)^2 is always positive. This can be done by multiplying 2 by itself, which results in 4, a positive number. Another example is proving that (-3)^2 is always positive. This can be done by using the property that a negative number squared is equal to a positive number, so (-3)^2 is equal to (3)^2, which is a positive number.

5. Can a^2 be negative?

No, a^2 cannot be negative. As mentioned earlier, the square of any real number will always result in a positive number. Therefore, a^2 will always be positive, regardless of the value of a.

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