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ZERO – The guy that never was and will never be ANYTHING!
Perhaps one of the most important discoveries in Math, the number ZERO revolutionized the entire branch of Math. And GMAT doesn’t shy away from according this status to ZERO, by testing it in many questions.
Importance of Zero in Arithmetic
Well, the easiest concept about Zero relates to our knowledge of the Number line. On the number line,
However, Zero itself does not belong to either of the categories. That leads us to the first property of Zero viz.,
Zero is neutral, neither negative nor positive.
Result:
This means that when Zero is combined with positive integers, we obtain the set of non-negative integers, which many students confuse with positive integers.
To set the record straight,
Well, what’s the big difference you ask? Not much, just the ZERO, perhaps!
This weakness (of not being able to distinguish between positive and non-negative) is heavily exploited by the GMAT in questions from topics like Number properties, Statistics, Inequalities and Absolute Values.
So, remember – Non-negative is NOT the same as positive.
This then leads us to the next point of confusion for many GMAT takers. We know that integers can be broadly classified into TWO categories based on their parity – Evens and Odds.
What about ZERO? Is it even or odd?
Well, since zero is neither negative nor positive, surely, zero must be neither even nor odd… and then everything went downhill from there!
(Sigh)
It can be seen clearly that Zero CANNOT be expressed as 2n + 1 / 2n – 1, but it can be expressed as 2n.
Zero is EVEN
This is a property that’s tested quite often in questions on Number properties – specifically, questions on Odds & Evens.
Result:
Always remember to include Zero in the set of even numbers. If the question mentions that a particular variable is even, never forget to consider Zero as part of your number list.
A corollary of the fact that we discussed above is the fact that zero is an integer, and hence a rational number too.
Divisibility & Remainders
Following this, are the two properties of Zero that show up quite often in questions on Divisibility and remainders.
When there is a question on Factors and Multiples, it is observed that, while students are very careful about not taking ZERO as a factor, most students forget the fact that zero is a multiple of all numbers. In some questions, not counting ZERO as a multiple of a certain number could just be the difference between the right and the wrong answer.
In questions on Remainders, the most common mistake observed is when students do not consider the quotient as ZERO. Again, GMAT has taken advantage of this weakness and created many questions in which this trap is set up; unfortunately, many test-takers end up falling for it.
On the GMAT, the remainder and the quotient are non-negative integers. Therefore, remember that the quotient CAN be Zero and this corresponds to the case where the dividend is smaller than the divisor.
Result:
In questions on remainders, remember that
Quotient could be ZERO, therefore the first number in your number list should correspond to this case
Remainder could be ZERO and this means that the dividend is a direct multiple of the divisor
Zero in Algebra
Moving on from the module of Arithmetic to Algebra, there are a couple of properties that get tested very often in topics from this module.
Equations, Exponents, Inequalities & Absolute Values
Zero is the smallest value for a perfect square
“\(x^2\) is always positive” is the bane of a lot of test-takers, in questions related to exponents, inequalities and absolute values.
Unless otherwise specified in the question data, one should not forget the possibility of x being Zero; in this case, the statements ‘\(x^2\) is always positive’ does not hold.
Result:
It is very important to remember that ‘\(x^2\) is always non-negative’ or in other words, the smallest value for a perfect square is ZERO.
Another common mistake that test-takers commit is in the topic of equations, by not considering the value of x as zero, when simplifying expressions/equations.
For example, when an equation like \(x^2\) = x is given, a lot of students end up cancelling x from both sides resulting in x = 1; this is incorrect. Because the equation is a Quadratic, there will be two values for the unknown viz x = 0 and x = 1.
This is, again, a result of not understanding the nuance that \(x^2\) can be zero.
Result:
Never cancel off variables if there is no data about the signs of the variables, the variable could just be 0.
Functions & Operators
Zero is the additive/subtractive identity
When zero is added to / subtracted from any number, the result will be the number itself. This property of Zero is tested in questions on Functions and Operators
In such cases, the test-taker must exercise caution and not assume that the function/operator refers to addition/subtraction alone. For example, if a * 0 = a, we cannot say that the operator * represents addition alone.
How Zero is tested in Statistics
Lastly, zero has a say in the topic of Statistics, specifically in questions on Standard Deviation.
Zero increases the SD when introduced in a set of values
A very common mistake that many test-takers make is to think that introducing Zero into a set of values will reduce the Standard Deviation. In actuality, introducing a Zero increases the SD.
Result:
Remember that introducing Zeroes into a set of numbers increases the deviation and hence the SD
Zero is the smallest value for SD and this happens only when all the values in the set are same.
Yet another property that has been tested historically in GMAT questions is the concept of SD being Zero. It’s an easy fact to remember that the SD for a set of numbers will be Zero when all the numbers in the set are same.
Result:
The SD of a set of numbers is Zero only when all the numbers in the set are equal / same.
With this, we come to the end of our article on the interesting and important properties of the number Zero which are relevant to GMAT aspirants. We hope that this article will help you hold on to these concepts and more importantly, will inspire you to be wary of the traps involving Zero.
In the next and the last part of this 3-part article on Numbers, tomorrow, we shall discuss the salient features of two numbers – One and Two.
By the way, if you missed out on reading the first part of this article on Numbers, you can click on this link to get access to this part of the article.
https://gmatclub.com/forum/a-must-read-on-numbers-for-gmat-quant-370718.html#p2872491
Perhaps one of the most important discoveries in Math, the number ZERO revolutionized the entire branch of Math. And GMAT doesn’t shy away from according this status to ZERO, by testing it in many questions.
Importance of Zero in Arithmetic
Well, the easiest concept about Zero relates to our knowledge of the Number line. On the number line,
- any number to the right of Zero is positive, and
any number to the left of Zero is negative.
However, Zero itself does not belong to either of the categories. That leads us to the first property of Zero viz.,
Zero is neutral, neither negative nor positive.
Result:
This means that when Zero is combined with positive integers, we obtain the set of non-negative integers, which many students confuse with positive integers.
To set the record straight,
- positive integers belong to the set {1,2,3,4, ….}, whereas
non-negative integers belong to the set {0,1,2,3,4,5,…..}
Well, what’s the big difference you ask? Not much, just the ZERO, perhaps!
This weakness (of not being able to distinguish between positive and non-negative) is heavily exploited by the GMAT in questions from topics like Number properties, Statistics, Inequalities and Absolute Values.
So, remember – Non-negative is NOT the same as positive.
This then leads us to the next point of confusion for many GMAT takers. We know that integers can be broadly classified into TWO categories based on their parity – Evens and Odds.
What about ZERO? Is it even or odd?
Well, since zero is neither negative nor positive, surely, zero must be neither even nor odd… and then everything went downhill from there!
(Sigh) - All integers which can be expressed in the form of 2n (n being an integer) are even
All those which can be expressed in the form of 2n + 1 / 2n – 1 are odd
It can be seen clearly that Zero CANNOT be expressed as 2n + 1 / 2n – 1, but it can be expressed as 2n.
Zero is EVEN
This is a property that’s tested quite often in questions on Number properties – specifically, questions on Odds & Evens.
Result:
Always remember to include Zero in the set of even numbers. If the question mentions that a particular variable is even, never forget to consider Zero as part of your number list.
A corollary of the fact that we discussed above is the fact that zero is an integer, and hence a rational number too.
Divisibility & Remainders
Following this, are the two properties of Zero that show up quite often in questions on Divisibility and remainders.
- Zero is a multiple of all the numbers i.e. Zero is divisible by all integers (with the exception of Zero, of course, because 0/0 is not defined)
Zero is the quotient when the dividend is smaller than the divisor
When there is a question on Factors and Multiples, it is observed that, while students are very careful about not taking ZERO as a factor, most students forget the fact that zero is a multiple of all numbers. In some questions, not counting ZERO as a multiple of a certain number could just be the difference between the right and the wrong answer.
In questions on Remainders, the most common mistake observed is when students do not consider the quotient as ZERO. Again, GMAT has taken advantage of this weakness and created many questions in which this trap is set up; unfortunately, many test-takers end up falling for it.
On the GMAT, the remainder and the quotient are non-negative integers. Therefore, remember that the quotient CAN be Zero and this corresponds to the case where the dividend is smaller than the divisor.
Result:
In questions on remainders, remember that
Quotient could be ZERO, therefore the first number in your number list should correspond to this case
Remainder could be ZERO and this means that the dividend is a direct multiple of the divisor
Zero in Algebra
Moving on from the module of Arithmetic to Algebra, there are a couple of properties that get tested very often in topics from this module.
Equations, Exponents, Inequalities & Absolute Values
Zero is the smallest value for a perfect square
“\(x^2\) is always positive” is the bane of a lot of test-takers, in questions related to exponents, inequalities and absolute values.
Unless otherwise specified in the question data, one should not forget the possibility of x being Zero; in this case, the statements ‘\(x^2\) is always positive’ does not hold.
Result:
It is very important to remember that ‘\(x^2\) is always non-negative’ or in other words, the smallest value for a perfect square is ZERO.
Another common mistake that test-takers commit is in the topic of equations, by not considering the value of x as zero, when simplifying expressions/equations.
For example, when an equation like \(x^2\) = x is given, a lot of students end up cancelling x from both sides resulting in x = 1; this is incorrect. Because the equation is a Quadratic, there will be two values for the unknown viz x = 0 and x = 1.
This is, again, a result of not understanding the nuance that \(x^2\) can be zero.
Result:
Never cancel off variables if there is no data about the signs of the variables, the variable could just be 0.
Functions & Operators
Zero is the additive/subtractive identity
When zero is added to / subtracted from any number, the result will be the number itself. This property of Zero is tested in questions on Functions and Operators
In such cases, the test-taker must exercise caution and not assume that the function/operator refers to addition/subtraction alone. For example, if a * 0 = a, we cannot say that the operator * represents addition alone.
How Zero is tested in Statistics
Lastly, zero has a say in the topic of Statistics, specifically in questions on Standard Deviation.
Zero increases the SD when introduced in a set of values
A very common mistake that many test-takers make is to think that introducing Zero into a set of values will reduce the Standard Deviation. In actuality, introducing a Zero increases the SD.
Result:
Remember that introducing Zeroes into a set of numbers increases the deviation and hence the SD
Zero is the smallest value for SD and this happens only when all the values in the set are same.
Yet another property that has been tested historically in GMAT questions is the concept of SD being Zero. It’s an easy fact to remember that the SD for a set of numbers will be Zero when all the numbers in the set are same.
Result:
The SD of a set of numbers is Zero only when all the numbers in the set are equal / same.
With this, we come to the end of our article on the interesting and important properties of the number Zero which are relevant to GMAT aspirants. We hope that this article will help you hold on to these concepts and more importantly, will inspire you to be wary of the traps involving Zero.
In the next and the last part of this 3-part article on Numbers, tomorrow, we shall discuss the salient features of two numbers – One and Two.
By the way, if you missed out on reading the first part of this article on Numbers, you can click on this link to get access to this part of the article.
https://gmatclub.com/forum/a-must-read-on-numbers-for-gmat-quant-370718.html#p2872491
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