You can also look at ratios of perfect squares, which is what 2.1^2 is. That’s 441/100.
Consider when you divide a perfect square by a perfect square and get an integer. Can you say anything about the result?
Look at the square root of 9 and 10 and compare them. Look at the square root of 441/100 and 442/100 and compare them. Can you see some ways perfect squares matter in relation to roots?
When you’re done you can ask AI what perfect squares and ratios of perfect squares are useful for.
? You divide a perfect square, such as 36, by another perfect square, 25, and get an integer, such as 1. Ok.
So it doesn’t work with the numbers I shared above. Lets use 36 and 4 that gives you 9. 9 is another perfect square.
Hmm. 36 is composed of 32 and 22 . Hmm. I think the integer would have to be another perfect square. The perfect square dividing into the first perfect square would also have to be one of the perfect squares that the numerator consists of. I think thats true for division in general. Like if you divide a non-perfect square like 18 by 2 and get 9, you get an integer because 9 “makes up” 18.
Anyways back to dividing a perfect square by a perfect square, the integer has to be a perfect square since a perfect square can only be composed of perfect squares.
sqrt(9) = 3
sqrt(10) = 3.16227766017
uhh the sqrt of 10 is just a bit bigger than the sqrt of 3. i got nothing else
sqrt(441/100) = 2.1
sqrt(442/100) = 2.10237960416
perfect squares give finite roots?
im psure the roots of 442/100 and 10 are infinite and are approximate. you can get close enough that it doesn’t even matter, but I don’t think their ever exact.
I asked Gemini “What are perfect squares used for?” :
Perfect squares are fundamental mathematical concepts with wide-ranging applications in various fields.
Here are the main areas where perfect squares are used:
Solving Quadratic Equations: They are essential in the method of completing the square, which is used to solve quadratic equations.
Factoring Expressions: Perfect squares are used in factoring algebraic expressions, particularly for the difference of squares (a2−b2=(a−b)(a+b)) and perfect square trinomials (a2+2ab+b2=(a+b)2).
Simplifying Square Roots: Understanding perfect squares is key to simplifying radicals, as the square root of a perfect square is an integer (e.g., 36!(data:image/svg+xml;utf8,)=6).
Area Calculation: The area of a square with side length ‘s’ is calculated as A=s2, which is a perfect square .
Pythagorean Theorem: They are integral to the Pythagorean theorem (a2+b2=c2), which is used to find the sides of a right-angled triangle. Perfect squares are used to calculate the square of the side length s .
Distance Formulas: Perfect squares are used in the calculation of the Euclidean distance between two points in 2-D or 3-D systems, which is an extension of the Pythagorean theor e m.
Properties and Puzzles: Perfect squares are a central topic in number theory, appearing in theorems and puzzles related to integer properties, prime factorization, and modular arithme t ic.
Factorization: Squarity testing can be used in the factorization of large numbers by applying the difference of two squares for m ula.
Finding Patterns: Perfect squares follow predictable patterns (e.g., the sum of the first n odd integers is n2), which aids in problem-so l ving.
Cryptography: Perfect squares and modular arithmetic play a role in various cryptographic algorithms used for data encryption and se c urity.
Algorithms and Optimization: They are relevant in the analysis of algorithm efficiency, particularly those with a time complexity proportional to the square of the input size (e.g., O(n2)).
Image Processing: Digital images are often represented using square matrices, which favors operations involving square-shaped portions of a picture.
Game Development: Perfect squares are often used in grid-based games for calculating distances and movements on the g a me board.
In short, perfect squares serve as a fundamental building block for solving numerous problems across mathematics, science, and t echnology.
Its answer for the above seemed kinda broad and unhelpful. Idk.
I did find its response for “What are ratios of perfect squares useful for?”:
- Defining Rational Squares: A key mathematical concept is that a rational number is a square if and only if it can be expressed as the ratio of two perfect squares (e.g., 169=4232=(43)2). This property is fundamental in number theory.
Try to find a counter-example.
So I said that if you take the square root of a perfect square you get a finite answer.
Hmm.
Hmm. Maybe the finiteness(?) here has to do with what we’re dividing by (in the case of 441/100 vs 442/100)?
So if the denominator is something that is not a multiple of 2 or 5, it will give a infinite root.
So sqrt (36/49) = 0.85714285714
Well its infinite because its a repeating decimal. Is that a counter-example?
Yes that’s infinite.
Another example is 1/9.
However, 1/9 has a finite decimal in base 3 or base 9 number systems instead of base 10.
Whether a decimal is infinite can depend on the number system base, which makes it not necessarily important.
So instead of talking about infinite decimals you can talk about other categories like whole, rational or irrational numbers.
So some positive integers have an irrational square root. Some, the perfect squares, have an integer square root.
Do any positive integers have a rational square root like 7/4 or 1/3?
In general, what causes finite or infinitely repeating decimals for rational numbers in base 10? And with other bases?
BTW are you clear on what rational numbers are? Look them up if not.
You should know what natural, whole, integer, rational, irrational, and real numbers are.
You should also know the very basics of complex numbers: they define i = the square root of negative one and they add that into the number system. You generally don’t need to know what that’s for or use complex numbers, but just knowing what the term means is worthwhile. You should know that with the other types of numbers, negative numbers don’t have square roots because if you square a negative or a positive you get a positive either way. Adding i allows square roots of all negatives. It also allows the other roots of negatives that weren’t already available with real numbers. Try figuring out which roots of negatives can be solved with real numbers and which can’t. like -5^(1/3), -5^(1/4), -5^(1/5) etc. If you already know you can just say the answer.
Number theory is the study of integers.
Really? Why not fractions? I mean, I can see the point of studying one specific thing, its just with a name like number theory I thought it would cover a lot of stuff. But then the term could be pointless? Idk.
10 is a multiple of 5, because 5 times 2 equals 10. 10 is also a multiple of, because 2 times 5 equals 10.
Oh yeah the integer part is important because otherwise all numbers could be “multiples” of each other because of fractions. Saying 10 is a “multiple” of 3, because 3 times 10/3 equals 10 is not helpful.
So 0 would be a multiple of everything? Also is everything the multiple of 1?
I did some math problems earlier today and from a few days. Some I got wrong. Will share soon. Just figured I’d post these notes to get them out of my forum draft to respond to your posts.
Is there a scenario that would make it important? With my current understanding I can see how an infinite/repeating decimal can give you information about a number (such as it not having a factor of 2 or 5), but outside of that it infinite decimals just seem like a quirk of the fact we do math in base 10 (though I don’t really understand other bases). Nothing else.
Can a positive integer - such as 1,2,3,4,5 - have a rational square root? We’ve shown that some positive integers have an irrational square root, others an integer square root. What about a rational square root?
Hmm. I don’t think they could. Rational means can be represented as a fraction. Right? I’m having trouble verbalizing that rn.
Idk. I just had an intuition which I then later confirmed with reddit (as shared way earlier). The post didn’t explain it though, it just confirmed it. I don’t really get why. I don’t even know how to count in other bases.
natural - counting numbers, numbers that appear naturally, whole positive numbers, 1,2,3,4
whole - numbers with no fractional part
integer - whats the difference between this and a whole number?
rational - can be represented as a fraction
irrational - can’t be represented as a fraction
real numbers - i think all of the above, i think the term has value because of imaginary numbers and stuff
Clear? No. I will do that soon. 
i know i is square of -1. i didn’t understand much of the math taught behind it in school. for two reasons: crappy teachers and hostility(?) to imaginary numbers. I think when I learned of them I was is in a headspace of “the world is irrational” and stuff. All the intellectuals are bad and say dumb stuff like we don’t exist. So when I heard “imaginary” numbers and how this number contradicted the math I understood (or seemingly contradicted) I was like this is all bs and didn’t care to learn it. I changed my mind on i years later because I saw a video explaining how i is an actual proper concept/thing. I still don’t get what it does though.
1/3,1/5, or just odd fraction powers. Actually is 1/3 considered odd like 3? If a power is “odd” it can give real number answers with negatives.
Can you set up the problem mathematically? Step 1 of trying to prove what the answer is.
Rational Numbers - Is a number that can be expressed as a fraction of two integers. 5 can be expressed as the fraction 25/5, for example. Rational has “ratio” in it. Do ratios have something to do with etymology here?
Integers - Numbers without a decimal part. Whole numbers. Integers include zero, positive numbers, and negative numbers.
Whole Numbers - Counting numbers, such as (positive) 1, 2, 3, 4, etc., along with 0. However google defines it as:
Is it synonymous with integer or a different thing entirely?
Irrational Numbers - Numbers that cannot be expressed as a fraction. Irrational numbers like pi, e, or sqrt(2), are all numbers that cannot be accurately expressed as a fraction. Fractions can approximate the numbers, such as 22/7 having the same first three digits as pi (3.14). Hmm. I wonder: how far does the precision of an irrational number matter? Could we not substitute rational numbers, i.e. integer fractions, in place of the irrational numbers in certain equations and stuff? Also from the linked wikipedia article:
In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers.
So ratios do play a part here. Hmm. I don’t really understand ratios much.
Natural - Positive integers. Sometimes includes 0 (which makes it synonymous with whole numbers) but sometimes does not (which would make it unique then).
Real Numbers - Rational and irrational numbers, but wikipedia exapnds on it.
In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a length, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences.
What is a continuous one dimensional quantity? (idk if this is important)
Ratios -
In mathematics, a ratio (/ˈreɪ.ʃ(i.)oʊ/) shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3).
One number contains another? Eight oranges, six lemons, with a ratio of eight to six. Is it representing that for every eight oranges we have six lemons? Every eight of something “contains” six of something?
Also what would this have to do with fractions? In the above we have fourteen pieces of fruit total. Would a ratio be the number representing how much there is of X fruit? 8/14 is the ratio of oranges?
A ratio may be specified either by giving both constituting numbers, written as "a to b " or "a :b ", or by giving just the value of their quotient
Hmm. Whats the problem I’m trying to solve mathematically? Do any positive integers (1,2,3,4, 10, 120) have a rational square root? Can they have a rational square root like 7/4 or 1/3 (as opposed to 25/5 or 36/6, that might as well be an integer square root).
A rational square root means that the square root of that positive integer can be represented as a fraction.
Setting up the problem mathematically (something like this?):
sqrt(x) = 1/3
or
sqrt(x) = y/z
Yeah that’s a good start. Are there any conditions on what numbers x, y and z can be?
I think they’re saying that because ratios are related to division. 24/6 tells you how many times 6 is contained in 24. And with 24:6 if you simplify the ratio you get 4:1, the same answer, 4.
They have to be integers. We’re looking for whether the sqrt of an integer can be a rational square root. A rational square root would consist of integers.
Oh yeah they have to be positive too.
Can z be 1?
No z can’t be 1. Otherwise y/z would be an integer.
It also can’t be 0, since you can’t divide by 0.
OK so write out what you know all together, then look at it and see if you can do any math steps to try to solve it or figure anything out about it.
