Key Takeaways — Number Systems (Part I)
You’re not bad at Number Systems — you just need to change your perspective!
Frankly, number systems is too large a topic to cover in just one blog. Hence, I have decided to write a few blogs dedicated to number systems as it is one of the most important topics in aptitude tests or competitive examinations like CAT, GMAT, etc.
Studying numbers systems will make you feel nostalgic about the carefree childhood days and you’ll feel like — Oh! I wish I were back to school!!
Although we all know the basics of number system, but it is always helpful to refresh our memory. (As humans, we forget… We forgot how God created us… We forgot to be grateful… We forgot to reflect and observe the beautiful creations all around us… We got lost in the glitter and glamour of this ephemeral world…)
Basic Concepts
Real numbers are all the numbers on the number line. They are of two types — rational & irrational.
Rational numbers are all those numbers that can be expressed as the ratio of two integers i.e. all integers and fractions. Irrational numbers are all real numbers that are not rational, both positive and negative. Integers are all those numbers with no fractional or decimal parts i.e. they are multiples of 1.
Modulus of a Real Number: Modulus of a real number a is defined as follows:
Thus, |5|= 5 and |-5|= -(-5) = 5.
If x is a negative number, then |x|= -x.
This is extremely important as most students make mistakes in modulus because they forget to consider the negative number.
Order of Operations
The BODMAS {Brackets, Orders (its not ‘of’!), Division, Multiplication, Addition, Subtraction} rule depicts the correct sequence in which the operations are to be executed, so as to find out the value of a given expression. Thus, in simplifying an expression, first of all the parentheses must be removed in the order of (), {} & []. Secondly, solve order numbers that is numbers involving powers or square roots. After removing the brackets and orders, you must first solve multiplication and division in order from left to right. Then, solve addition and subtraction in order from left to right.
Laws of Operations
Commutative Law: Addition and multiplication are both commutative; it doesn’t matter in what order the operation is performed. Division and subtraction are not commutative.
Associative Law: Addition and multiplication are associative; the terms can be regrouped without changing the result.
Distributive Law: The distributive law of multiplication allows us to “distribute” a factor among the terms being added or subtracted. Division can be distributed too, except when the sum or difference is in the denominator, then no distribution is possible.
Odd and Even Numbers
Even numbers are integers that are divisible by 2 and odd numbers are integers that are not divisible by 2. We can also classify odd and even numbers by looking at the last digit. If an integer’s last digit is either 0, 2, 4, 6, 8 then the number is even; if its last digit is 1, 3, 5, 7, 9 then it is odd.
Rules for Odd and Even:
Odd + Odd = Even; Even + Even = Even; Odd + Even = Odd
Odd * Odd = Odd; Even * Even = Even; Odd * Even = Even
You need to remember only two cases that are in bold as all others end up being even.
Fractions
It is a numerical quantity that is not a whole number. It is basically the numerator divided by the denominator.
Property: The value of fraction remains unchanged if you multiply or divide the numerator and the denominator by the same nonzero number.
Comparing positive fractions: If the numerators are the same, the fraction with the smaller denominator will have the larger value, since the numerator is divided into smaller number of parts. If the denominators are the same, the fraction with the larger numerator will have the larger value.
Decimal Fractions
Fractions in which denominators are powers of 10 are known as decimal fractions.
Comparing decimal fractions: Convert each of the given fractions in the decimal form and arrange them according to the rule for comparing positive fractions.
Use all the concepts learned above to solve the following question:
Simple logic combined with strong basic concepts will help you solve the above question. I urge you to discuss the solution in the comment section below. (By the way, the student with the best and logical explanation by 3 PM Friday, 19th May 2017 will get a free T-shirt from Pariksha.Co !)
On a concluding note, working efficiently and self-assuredly with numeric expressions can save you valuable time in the tests. The above basics are pivotal in helping you solve number systems problems quickly. Be on a lookout for my blog on number systems in the coming days…
Click here to find more about our key takeaways which will help you solve aptitude questions in no time!
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