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Need expert advice - Is 0 a perfect square on the GMAT? 11 Oct 2018, 13:57
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Apologies for the weird question. I searched and saw other threads, but didn't see any response from experts etc.

Does the GMAT consider 0 to be a perfect square, cube etc? - Mainly concerned in case a DS question states "x is a perfect square" - Then I need to know if 0 can be considered or not.
Moreover, would -27 be a perfect cube? - And more generally, for "perfect cubes", can I consider negative numbers since sign is kept?

Thank you so much.
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Need expert advice - Is 0 a perfect square on the GMAT? 13 Oct 2018, 20:52
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Apologies for the weird question. I searched and saw other threads, but didn't see any response from experts etc.

Does the GMAT consider 0 to be a perfect square, cube etc? - Mainly concerned in case a DS question states "x is a perfect square" - Then I need to know if 0 can be considered or not.
Moreover, would -27 be a perfect cube? - And more generally, for "perfect cubes", can I consider negative numbers since sign is kept?

Thank you so much.
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1) Zero is both a perfect square and a perfect cube.
2) -27 can be written as \((-1)*27=(-1)^3*3^3\); so cube root of -27=(-1)*3=-3. Hence , negative numbers could be perfect cubes.

Hope it helps.
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Need expert advice - Is 0 a perfect square on the GMAT? 07 Jan 2022, 00:37
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I stumbled upon this post made sometime back ... Had the same doubt.

I don't suppose '0' is either a perfect square or a perfect cube.

I admit that I could be wrong here.

My reasoning - A perfect square is a number which can be expressed as even powers of prime. However, 0 cannot be expressed as product of even powers of prime. Also, there are other properties of perfect square that 0 violates such as "The number of distinct factors is in a perfect square is ODD".

Similarly, I don't think 0 is a perfect cube either.

IanStewart
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Would love to hear your thoughts on this.
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Need expert advice - Is 0 a perfect square on the GMAT? 07 Jan 2022, 05:08
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AbhiroopGhosh


I don't suppose '0' is either a perfect square or a perfect cube.

I admit that I could be wrong here.
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It's just a question of how you choose to define those phrases, but the GMAT will never use the phrases "perfect square" or "perfect cube" anyway, so it doesn't actually matter for the test. GMAT questions always say "square of an integer" or "cube of an integer" (which both encompass zero, because 0^2 = 0 and 0^3 = 0), and it's only some prep company questions that use the phrase "perfect square".

But if you want an actual answer ( :) ) then yes, zero is usually considered a perfect square in math (though you'll probably find some definitions somewhere that seem to exclude zero). This paragraph will be completely irrelevant to GMAT test takers, so for interest only: technically, you need to specify what collection of numbers or things you're working with before you can even talk about "perfect squares" in math (the fraction 4/9 is a 'perfect square' if your collection of numbers is the rational numbers, but not if your collection is the integers, while even something like x^2 + 6x + 9 = (x + 3)^2 can be considered a "perfect square" if you're working with polynomials with integer coefficients, so that's why the GMAT just avoids the phrase altogether, and uses the much more precise "square of an integer"). And when we talk about any collections of numbers or other mathematical things where we might want to talk about "perfect squares", mathematicians want to include zero, because then we're working with something called a "field" in the terminology of abstract algebra (algebra that is miles beyond the scope of the GMAT), and in a field you can be sure you can have all kinds of nice properties that you can't be sure you have otherwise.
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Need expert advice - Is 0 a perfect square on the GMAT? 07 Jan 2022, 06:30
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AbhiroopGhosh


I don't suppose '0' is either a perfect square or a perfect cube.

I admit that I could be wrong here.

It's just a question of how you choose to define those phrases, but the GMAT will never use the phrases "perfect square" or "perfect cube" anyway, so it doesn't actually matter for the test. GMAT questions always say "square of an integer" or "cube of an integer" (which both encompass zero, because 0^2 = 0 and 0^3 = 0), and it's only some prep company questions that use the phrase "perfect square".

But if you want an actual answer ( :) ) then yes, zero is usually considered a perfect square in math (though you'll probably find some definitions somewhere that seem to exclude zero). This paragraph will be completely irrelevant to GMAT test takers, so for interest only: technically, you need to specify what collection of numbers or things you're working with before you can even talk about "perfect squares" in math (the fraction 4/9 is a 'perfect square' if your collection of numbers is the rational numbers, but not if your collection is the integers, while even something like x^2 + 6x + 9 = (x + 3)^2 can be considered a "perfect square" if you're working with polynomials with integer coefficients, so that's why the GMAT just avoids the phrase altogether, and uses the much more precise "square of an integer"). And when we talk about any collections of numbers or other mathematical things where we might want to talk about "perfect squares", mathematicians want to include zero, because then we're working with something called a "field" in the terminology of abstract algebra (algebra that is miles beyond the scope of the GMAT), and in a field you can be sure you can have all kinds of nice properties that you can't be sure you have otherwise.
Show more

Thank you so much for your reply IanStewart. As always ... I appreciate your wisdom and your help in resolving my queries!

I had a quick follow-up question -

In Mathematics (and in GMAT) the way we define squares (or perfect squares) i.e. they are number having even power of prime factors, is the definition - if I understand correctly - restricted to only integers.

To be honest, I always thought the term perfect squares to be associated only with integers (though logically square can be any number multiplied twice). I think that the assumption is flawed. Please let me know if you think otherwise.

Fun Fact :

I had discussion with a friend of mine sometime back and we argued on the topic - "Are perfect square only integers". While I was supporting the statement, he said it can be associated with any number. We had a bet, and thanks to Wikipedia that I won :lol:

An excerpt from Wiki (that I used in my defense :) ) -

Quote:
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself.
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Source : https://en.wikipedia.org/wiki/Square_number

I will have to apologies to him after this :D
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Need expert advice - Is 0 a perfect square on the GMAT? 07 Jan 2022, 09:24
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AbhiroopGhosh

In Mathematics (and in GMAT) the way we define squares (or perfect squares) i.e. they are number having even power of prime factors, is the definition - if I understand correctly - restricted to only integers.
Show more

You could define perfect squares (of positive integers) that way, but that would be an unconventional definition. Usually an integer is called a perfect square if it is equal to the square of an integer, i.e. if it is equal to some integer times itself. That definition has a few advantages, when you compare it with your definition; it is simpler (there's no need to know about prime factorizations to understand it) and there's no need to declare separately what is true about zero, or about one: zero and one are clearly perfect squares if we decide that a "perfect square" is the square of an integer. Then what you say, about even powers in the prime factorization of a perfect square when that square is a positive integer, becomes a consequence of the definition of "perfect square", rather than the definition itself.

This will get a bit philosophical, and is really only for interest (anyone who only cares about GMAT prep won't gain anything by reading any more of this post). In GMAT-level math, or in high school level math (which is the level of that wikipedia article you quote), when people talk about "perfect squares" they're essentially always talking about integers, and are using the definition I mention above. When you get beyond high school math, the definitions of all kinds of mathematical things are extended -- many test takers will be familiar with complex or imaginary numbers, which are defined by extending the definition of a square root so we can take roots of negative numbers, but that's just one of dozens of examples. In abstract algebra, which covers areas of math known as "group theory", "field theory", "ring theory", and more, you start to extend the meaning of mathematical operations like "multiplication" and "squaring", and the definition of "perfect square" can then extend to cover other collections of numbers, algebraic expressions, even geometric operations like rotation and reflection. You'd never even run into any abstract algebra unless you got to the second year of an undergrad math degree (or perhaps if you were learning quantum physics or some advanced computer science), so it's all way beyond GMAT scope, but in your bet, you and your friend were both right, depending on what level of math you're discussing. And if you search for "perfect square" on the internet, and look at definitions, you'll find some that support your point of view, and some that support your friend's.
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Need expert advice - Is 0 a perfect square on the GMAT? 10 Jan 2022, 11:39
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In Mathematics (and in GMAT) the way we define squares (or perfect squares) i.e. they are number having even power of prime factors, is the definition - if I understand correctly - restricted to only integers.

You could define perfect squares (of positive integers) that way, but that would be an unconventional definition. Usually an integer is called a perfect square if it is equal to the square of an integer, i.e. if it is equal to some integer times itself. That definition has a few advantages, when you compare it with your definition; it is simpler (there's no need to know about prime factorizations to understand it) and there's no need to declare separately what is true about zero, or about one: zero and one are clearly perfect squares if we decide that a "perfect square" is the square of an integer. Then what you say, about even powers in the prime factorization of a perfect square when that square is a positive integer, becomes a consequence of the definition of "perfect square", rather than the definition itself.

This will get a bit philosophical, and is really only for interest (anyone who only cares about GMAT prep won't gain anything by reading any more of this post). In GMAT-level math, or in high school level math (which is the level of that wikipedia article you quote), when people talk about "perfect squares" they're essentially always talking about integers, and are using the definition I mention above. When you get beyond high school math, the definitions of all kinds of mathematical things are extended -- many test takers will be familiar with complex or imaginary numbers, which are defined by extending the definition of a square root so we can take roots of negative numbers, but that's just one of dozens of examples. In abstract algebra, which covers areas of math known as "group theory", "field theory", "ring theory", and more, you start to extend the meaning of mathematical operations like "multiplication" and "squaring", and the definition of "perfect square" can then extend to cover other collections of numbers, algebraic expressions, even geometric operations like rotation and reflection. You'd never even run into any abstract algebra unless you got to the second year of an undergrad math degree (or perhaps if you were learning quantum physics or some advanced computer science), so it's all way beyond GMAT scope, but in your bet, you and your friend were both right, depending on what level of math you're discussing. And if you search for "perfect square" on the internet, and look at definitions, you'll find some that support your point of view, and some that support your friend's.
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IanStewart - Thanks a lot for your response. As always appreciate your response !
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