How to determine which one is greater between the two?

Indranil

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How to determine which one is greater between the two 30 and 30√2? Could you explain, please?
 
Let's begin by noting that mathematicians are often a lazy bunch and we tend to want to write as few symbols as possible. Hence, when two expressions are written side-by-side, it almost always means the two are meant to be multiplied and we've simply chosen not to include any multiplication symbols. In this case, that means that the expression \(\displaystyle 30 \sqrt{2}\) is read aloud as "thirty times the square root of 2." Accordingly, the most straightforward way to proceed would be to punch \(\displaystyle 30 \sqrt{2}\) into your calculator and see what you get. Is the result bigger or smaller than 30 (or perhaps the result is still exactly 30)? How do you know? (I only ask this since you seemed to be struggling with this concept in a previous thread)

However, my preferred method is a bit different and does not require the use of a calculator. Without even knowing what the exact value of \(\displaystyle \sqrt{2}\) is, we can use the definition of square root to learn some facts about it. Namely, we know we're looking for some number that, when multiplied by itself, equals 2. We also know that any number less than 1 gets smaller when multiplied by itself, and \(\displaystyle 1 \times 1 = 1\), we can conclude that \(\displaystyle \sqrt{2} > 1\). Based on this, what can you say about the value of \(\displaystyle 30 \sqrt{2}\)? Is it going to be smaller than 30? Exactly 30? Bigger than 30? How do you know?
 
Are you saying that you do not know whether \(\displaystyle \sqrt{2}\) is larger than 1 or not?
 
Are you saying that you do not know whether \(\displaystyle \sqrt{2}\) is larger than 1 or not?
No, I think it is far more fundamental than that. He has no clue about any type of number other than non-negative integers. He may also not know what "larger than" means in a mathematical sense.
 
Let's begin by noting that mathematicians are often a lazy bunch and we tend to want to write as few symbols as possible. Hence, when two expressions are written side-by-side, it almost always means the two are meant to be multiplied and we've simply chosen not to include any multiplication symbols. In this case, that means that the expression \(\displaystyle 30 \sqrt{2}\) is read aloud as "thirty times the square root of 2." Accordingly, the most straightforward way to proceed would be to punch \(\displaystyle 30 \sqrt{2}\) into your calculator and see what you get. Is the result bigger or smaller than 30 (or perhaps the result is still exactly 30)? How do you know? (I only ask this since you seemed to be struggling with this concept in a previous thread)

However, my preferred method is a bit different and does not require the use of a calculator. Without even knowing what the exact value of \(\displaystyle \sqrt{2}\) is, we can use the definition of square root to learn some facts about it. Namely, we know we're looking for some number that, when multiplied by itself, equals 2. We also know that any number less than 1 gets smaller when multiplied by itself, and \(\displaystyle 1 \times 1 = 1\), we can conclude that \(\displaystyle \sqrt{2} > 1\). Based on this, what can you say about the value of \(\displaystyle 30 \sqrt{2}\)? Is it going to be smaller than 30? Exactly 30? Bigger than 30? How do you know?
I don't understand the points you provided. Could you please simplify the portion so that I can grasp the concept easily? The portion is below:
'we can use the definition of square root to learn some facts about it. Namely, we know we're looking for some number that, when multiplied by itself, equals 2. We also know that any number less than 1 gets smaller when multiplied by itself, and \(\displaystyle 1 \times 1 = 1\), we can conclude that \(\displaystyle \sqrt{2} > 1\). Based on this, what can you say about the value of \(\displaystyle 30 \sqrt{2}\)? Is it going to be smaller than 30? Exactly 30? Bigger than 30? How do you know?'
 
It would be extremely helpful if you would tell us your own thinking on this. We're not sure what kind of help you need.

You want to compare the numbers 30 and 30√2; why is this difficult for you? Is there some particular issue you have in mind?

Also, what do you know about it?

Do you know what √2 is? Do you know that √2 is greater than 1, so that multiplying by √2 results in a larger number, just as multiplication by 5 increases a (positive) number because 5 > 1?
 
What in the world are you studying? It appears that you need a personal tutor for basic arithmetic. Or you are a troll.
 
What in the world are you studying? It appears that you need a personal tutor for basic arithmetic. Or you are a troll.
I am preparing for pre-medical Exam. It requires basic mathematics to higher mathematics for Physics and Chemistry.
I consider you all my personal teachers because the way you all teach me is amazing, just perfect for me. I can't learn anywhere else what I am learning here from you, from the basic to higher level, I learn a great deal here. I really appreciate the kind efforts you all doing for me. That's why I shall be very grateful to you all.
 
I am preparing for pre-medical Exam. It requires basic mathematics to higher mathematics for Physics and Chemistry.
I consider you all my personal teachers because the way you all teach me is amazing, just perfect for me. I can't learn anywhere else what I am learning here from you, from the basic to higher level, I learn a great deal here. I really appreciate the kind efforts you all doing for me. That's why I shall be very grateful to you all.

Thanks for telling us your goal. It would be better if you also explained your background -- what have you learned in mathematics, and why do you need help with such basic questions? Also, what are you using to study? Perhaps you need something better suited to your background.

I hope one thing you have learned here is that, in order to get helpful answers, you need to give us all relevant information so we can tell what help you need, and how to express it that you will understand.

In this case, you have not yet told us what makes this question difficult for you, so we still have no idea what to say.
 
I am preparing for pre-medical Exam. It requires basic mathematics to higher mathematics for Physics and Chemistry.
I consider you all my personal teachers because the way you all teach me is amazing, just perfect for me. I can't learn anywhere else what I am learning here from you, from the basic to higher level, I learn a great deal here. I really appreciate the kind efforts you all doing for me. That's why I shall be very grateful to you all.
I agreed with JeffM. I think that it just very hard to understand how someone preparing for that exam can be so seemingly unknowledgeable of such basic number facts.
If you have plans to do a medical degree you will find just how important mathematics will be.
My brother was a chief of a division at a major Medical School. He has told me that often he looks at the grades in calculus first when reviewing applications. His reason being that high grades in calculus show a commitment to hard work.

That said. Here is a question that I designed for a test-prep class.
Given \(\displaystyle 0 < \sqrt a < \dfrac{1}{{\sqrt b }} < 1\) then
arrange the following is ascending order.

\(\displaystyle \sqrt a ,\;\sqrt b .\;\dfrac{1}{a},\;\dfrac{1}{b},\;a,\;b,\;{a^2},\;{b^2}\)
 
I am preparing for pre-medical Exam. It requires basic mathematics to higher mathematics for Physics and Chemistry.
I consider you all my personal teachers because the way you all teach me is amazing, just perfect for me. I can't learn anywhere else what I am learning here from you, from the basic to higher level, I learn a great deal here. I really appreciate the kind efforts you all doing for me. That's why I shall be very grateful to you all.
There is a whole mathematical field called

https://en.m.wikipedia.org/wiki/Order_theory

I shall let you make what you can from that introductory article. Not attempting to be rigorous, we might say

\(\displaystyle a > b \iff (a - b) > 0.\)

That is, we say 3 is greater than 2 because we can take 2 objects from 3 objects and still have 1 or more objects left. That is a primitive intuition about what we mean by "greater than." We extend the concept from non-negative integers to real numbers a and b.

\(\displaystyle a > b \iff (a - b) > 0.\)

Now from this we can devise certain additional criteria, such as

\(\displaystyle a > b \text { and } a > 0 \implies 1 > \dfrac{a}{b}.\)

If we divide 3 into 1, we get a result less than 1.

\(\displaystyle c > 0 \text { and } a > b \implies ac > bc.\)

The product of 5 and 3 is 15, which is greater than 10, the product of 5 and 2.

Good to here?

At this point, say that we do not know what is the order relationship between

\(\displaystyle 30\) and \(\displaystyle 30\sqrt{2}.\)

Now a square root is non-negative by definition, but it can be zero. Let's assume

\(\displaystyle \sqrt{2} = 0 \implies \sqrt{2} * \sqrt{2} = \sqrt{2} * 0 \implies 2 = 0.\)

Well, 2 is not zero so that assumption was absurd.

\(\displaystyle \therefore \sqrt{2} > 0.\)

Actually this is a general theorem: \(\displaystyle x > 0 \iff \sqrt{x} > 0.\)

Maybe \(\displaystyle 1 > \sqrt{2} > 0 \implies \sqrt{2} * 1 > \sqrt{2} * \sqrt{2} \implies\)

\(\displaystyle \sqrt{2} > 2 \implies 1 > 2 \ \because \ 1 > \sqrt{2} \text { by assumption.}\)

And 1 is not greater than 2 so that assumption also was absurd.

But maybe the square root of 2 equals 1 so let's assume it to be so.

\(\displaystyle 1 = \sqrt{2} \implies 1 * 1 = \sqrt{2} * \sqrt{2} \implies 1 = 2.\)

More nonsense. So 1 is neither greater than nor equal to the square root of 2. Therefore, 1 is less than the square root of 2.

In fact, this is another general theorem \(\displaystyle x > 1 \iff \sqrt{x}> 1.\)

But 2 is greater than 1 so our conclusion is absurd, which means our assumption was false.

\(\displaystyle \therefore \sqrt{2} > 1 \implies 30\sqrt{2} > 30 * 1 = 30.\)

Most of the ideas about the real numbers are simple extensions of ideas that you already know about the natural numbers. There are rigorous developments of them studied in fields like foundations of mathematics, but that requires a course, not a tutoring site.
 
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There is a whole mathematical field called https://en.m.wikipedia.org/wiki/Order_theory
I shall let you make what you can from that introductory article. Not attempting to be rigorous, we might say
\(\displaystyle a > b \iff (a - b) > 0.\)
Actually the axioms that establish the real number field \(\displaystyle \mathbb{R}\) do not address order.
There is an order axiom.
That is a set \(\displaystyle \mathcal{P}\subset\mathbb{R}\) such that
1) \(\displaystyle \mathcal{P}\) is closed with respect to addition & multiplication.
2) \(\displaystyle (\forall x\in\mathbb{R})\) exactly one3if the following is true:
\(\displaystyle \left( {x = 0} \right)\;\,\underline \vee \;\,\left( {x \in P} \right)\;\,\underline \vee \;\,\left( { - x \in P} \right)\) that is the exclusive or.

 
Actually the axioms that establish the real number field \(\displaystyle \mathbb{R}\) do not address order.
There is an order axiom.
That is a set \(\displaystyle \mathcal{P}\subset\mathbb{R}\) such that
1) \(\displaystyle \mathcal{P}\) is closed with respect to addition & multiplication.
2) \(\displaystyle (\forall x\in\mathbb{R})\) exactly one3if the following is true:
\(\displaystyle \left( {x = 0} \right)\;\,\underline \vee \;\,\left( {x \in P} \right)\;\,\underline \vee \;\,\left( { - x \in P} \right)\) that is the exclusive or.

PKA

I am not being argumentative. I understand the axiom and its meaning and can perhaps see how we can rigourously deduce the normal properties of order from it. (It has been 50 years since I worked on foundations.) I am not sure if you are suggesting that I modify or delete my post or, alternatively, pointing out the basis for a more rigorous approach than I attempted. I appreciate either, but I would like to know which is the case. If I am misleading the OP, I want to correct or delete my post as soon as possible.

Thank you.
 
PKA

I am not being argumentative. I understand the axiom and its meaning and can perhaps see how we can rigourously deduce the normal properties of order from it. (It has been 50 years since I worked on foundations.) I am not sure if you are suggesting that I modify or delete my post or, alternatively, pointing out the basis for a more rigorous approach than I attempted. I appreciate either, but I would like to know which is the case. If I am misleading the OP, I want to correct or delete my post as soon as possible.
Sorry for any confusion. I was simply pointing out another approach. You are correct about my interest in rigor, That is all I meant. Again, sorry for not being clear on that point.
 
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