Number Systems In CAT Preparation
The Quantitative Aptitude section comprises topics such as number series, number system, simplification, algebra, geometry, mensuration, mixtures and allegations, modern maths, and other concepts from middle and secondary mathematics.
For the preparation of CAT, not only the ability to solve the sets but also doing it accurately and apace is essential. The preparation for CAT, therefore, should focus not only on the concepts and details, but also on the tips, tricks, and speed maths concepts.
The applicants should plan and structure their prep as to inculcate understanding all the concepts, acquainting themselves with shortcuts and time their approach. A salient feature for the examination is that a lot of questions do not require exact calculations but entail approximations, are based on basics, and require the application of generalities.
As a consequence, the first step for the aspirants preparing for CAT is to of course compile a plan on how to go about the preparation, be cognizant of the subject matter, all the topics they need to cover, and their standing on the same.
They should follow by finding as much relevant material as possible, which can acquaint them with the groundwork of the concepts and also equip them to go about the questions quickly and efficiently, considering that efficiency implies achieving goals or the target while ensuring economy of effort.
The way for this as mentioned is to go through the concepts to be in a position to understand the context or the topic a question, problem or the set relates to.
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Scrutinizing number systems, the system for naming and expressing numbers is called number or numeral system. The following study seeks to discuss the basics and properties of numbers, for reacquainting applicants with the fundamentals of the same.
Natural Numbers include all the positive numbers, that are numbers greater than zero. These numbers are used in real-time for counting and thus, are also called counting numbers, for example, 1,2,24, and so forth.
Whole Numbers include positive or natural numbers from 1, the difference is that they also include zero, thus, whole numbers begin from (0,1,2,…)
Integers, consist of whole numbers and their negatives, thereby, integers include both
(0,1,2,3…) and (-1,-2,-3,…)
Fractions are used to express numbers in the form of parts of a whole, they are of the form, wherein x and y are integers and y≠0, and x is called the numerator or part and y the denominator or whole. The various types of fractions are as follows
Proper fractions are the ones where the numerator is smaller than the denominator, for example, 3/4, where x=3, and y=4, and x<y.
Improper fractions are the ones where the numerator is greater than the denominator, for example, 4/3, where x=4, and y=3, and x>y.
Improper fractions can be expressed as mixed fractions. Mixed fractions comprise of a mix of a whole number and a proper fraction. Converting improper fraction to a mixed fraction involves, dividing the numerator with the denominator, taking the quotient as the whole number, the remainder as the numerator of the proper fraction and the divisor as the denominator.
A mixed fraction signifies that when a given quantity is divided into saying certain packets, it would result in some parts and some of the original quantity would be leftover, example 17/4 the mixed fraction would be, 4 1/4, that is when a whole consisting of say 17 items is divided into packets of 4, it would result in 4 packets of 4 and 1 being left.
A unit fraction is one whose numerator is 1, that is x=1, for example, 1/4, 1/20 and so on, and thus, it can also be called a reciprocal of a positive integer.
A decimal fraction is one where the denominator can be expressed as a power of 10. For example, 0.23 can be expressed as 23/100.
The simplest or lowest form of a fraction is where there are no common factors in the numerator and denominator, example 12/54, simplest form is, as 6×2 = 12, and 6×9 = 54.
Fractions with the same denominator are termed as fractions. Basic arithmetic operations such as addition and subtraction are applied to like fractions. Example, 7/11+2/11 =9/11 or 7/11-2/11 =5/11.
If the fractions are not like, to apply addition or subtraction, these can be made like by using LCM, in other words, in cases where the fractions have different denominators, for addition or subtraction, both the numerator and denominator of each fraction (as the number is multiplied by both numerator and denominator, becomes a common factor and can be canceled to reduce the fraction to its original value, otherwise, the fraction would change) are multiplied by the number which would make the denominators equal, that is to a number which is a common multiple of the given denominators,
For example, for 4/5+9/7 since the fractions are not like, addition can be done when the denominators are equal. Taking LCM of 5, 7, which is 35, multiplying the first fraction by 7, and second by 5, gives, 20/35+45/35=65/35, which can be written as 1 30/35.
Also for, 3/8+5/24, 24 is a multiple of 8, and thus, only 3/8 is to be multiplied by 3 to make them like fractions, which results in, 9/24+5/24 =14/24, or 7/12 in its simplest or lowest form.
Reciprocal of a fraction is obtained by dividing the fraction by 1 or reversing the positions of the numerator and the denominator. The division of fractions is done by multiplying the former by the reciprocal of the latter. For example, the reciprocal of 4/7 is 7/4, and the value of 4/7÷11/14 is given by 4/7×11/14=8/11.
Rational Numbers are numbers that can be expressed as the fraction of two integers or fractions. These numbers are in the form of terminating decimals or recurring decimals and can be expressed in the form p/q where q≠0. Example, 2, 0.125, 0,33… and so forth.
Rational numbers include recurring numbers as, these can be expressed as fractions, for example, consider 0.33…
Assume x=0.33…→❶ Multiplying both sides of ❶ by 10,
10x=3.33…→❷ Subtracting ❶ from ❷,
9x=3, or x=3/9 or 1/3, which is a fraction, or a rational number.
Irrational Numbers are the numbers that cannot be expressed as simple fractions. These are non-terminating and non-recurring decimals and can be said to be numbers that are not rational.
In other words, irrational numbers do not comprise integers, fractions or ratios and cannot be expressed in the form p/q. Example, √2
As √2=1.41421356.., that is, non-terminating and non-recurring.
Imaginary numbers express the square root of negative numbers, that is, i=√-1.
Complex numbers are numbers comprising real and imaginary numbers, that is, complex numbers are of the form a+bi where a and b are real numbers and i is an imaginary number.
A fact to be mentioned here is that all real numbers can be expressed as complex numbers, such that x+i(0), where x is the real number followed by an imaginary counterpart.
Conjugate of a complex number is a number which has the same magnitude of real and imaginary parts but has the opposite sign. The product of a complex number and its conjugate is always a real number. Example for a complex number 3+2i, its conjugate is 3-2i
Product of 3+2i and 3-2i is (3+2i)(3-2i)= -=9-4(-1)= 13 (as √1² = -1), which is a real number.
Properties of numbers,
Following are the features and fundamentals of numbers and integers,
Absolute Value of an integer, it expresses how far an integer is from zero, that is, the integer’s distance from zero, and since distance cannot be negative, absolute value also refers to the non-negative value of an integer. It is represented through modulus, as given by |n| =n, where n is an integer. Example, |-4|=4 and |5|=5, as the distance of -4 from 0 (as on the number) line is 4 and that of 5 is 5.
Greatest Integer function is the greatest integer less than or equal to a given number, or the rounded-up value of a given number. It is represented as [n]=greatest value less than or equal to n. For Example, [-0.7]=-1, that is the value moves towards the left on the number line.
Least Integer function is the greatest integer greater than or equal to a given number, or the rounded down value of the integer. It is also called a ceiling of the integer. It is represented as [n]=least integer greater than or equal to n. Example [3.7]=3, that is the value moves towards the right on the number line.
Consecutive Numbers are the numbers that follow each other such that the difference between them is 1. Example 4,5 and 210,211.
Even Numbers can be referred to as multiples of 2, such that these can be expressed in the form 2n where n is a natural number. Example 2,12,20,..
Consecutive even numbers occur one after the other or the difference between them would be 2. Consecutive even numbers can be expressed in form 2n, 2n+2, 2n+4, 2n+6,.. such that the difference between successive numbers is 2.
Points to be noted here
- The sum of two even numbers, also is always even, for example, 8+4 = 12.
- The product of two even numbers is even, for example, 8×4 = 32, which is even.
- Integer power of an even number gives an even number, for example, 2⁵ = 32, as mentioned above, the product of even numbers is even, so multiplying 2 by itself would again result in an even number.
Odd numbers are those which are not multiples of 2 and can be expressed in the form 2n+1 where n is a natural number. Example 3,5,11,..
Consecutive odd numbers, again occur one after the other where the difference between them is 2, (skipping the even number in between). These can be expressed in form 2n+1, 2n+3, 2n+5,.. such that the difference between succeeding numbers is 2.
Facts to be noted here
- Sum of two odd numbers is always even, for example, 3+5 = 8, which is even, while the sum of an even and odd number is always odd, for example, 2+3 =5, which is odd.
- The product of two odd numbers is odd, for example, 3×5 = 15.
- The multiplication of an even and an odd number results in an even number, for example, 2×21 = 42, as the multiplication includes 2 as a factor.
- An integer power of an odd number gives an odd number, for example, 5⁴ = 625, as mentioned above, the product of odd numbers is odd, thus, 5 is multiplied by itself would result in an odd number.
Factor, an integer is a factor of another if it divides it exactly, that is the remainder on division is zero. These can be expressed as n is a factor of m when the remainder on division of m by n is zero.
Multiple, conversely, is an integer which when divided by another has no remainder or zero remainders. These can be expressed as m is an integer when the division of m by n has zero remainders.
Prime Numbers comprise integers greater than 1, which have exactly two factors, 1 and the number itself. Example, 2,3,5,37,..
1 is neither prime nor composite, as it has exactly 1 factor. 2 is the only even prime number as for all other even numbers, 2 is a factor.
For testing, if a number is prime, check whether the number is divisible by any number less than or equal to its square root. For example, for checking whether 119 is a prime number, check for numbers less than √119≈10 for factors. Since it is divisible by 7, it is not prime.
Conversely, composite numbers have factors more than 1 and the number itself. Example, 4, 9,12,…
Highest Common Factor (HCF) or Greatest Common Divisor, the highest number which is a factor, that is, which is exactly divides a given set of integers. Example, HCF or GCD of 16 and 24 is 8 as 16=2x2x2x2, and 24=2x2x2x3, which gives HCF=2x2x2 or 8.
Lowest Common Multiple (LCM), the smallest number which is perfectly divisible by a set of given integers. Example, again LCM for 16 and 24 is 48, as given by LCM= 2x2x2x2x3, or 48, as the highest power of each factor is considered for LCM.
Co-Prime or Relatively Prime Numbers can be referred to those numbers, which have their HCF as one or in other words, these numbers have no common factor other than 1. Example, for 22 and 35, HCF = 1, and LCM = 22×35 = 770.
Factorial of a number is obtained by multiplying all the numbers less than and equal to a given number. It is obtained as, in case of selection or arrangement of say n things in r packets, if one item is chosen, then remaining are n-1, after the next selection, the remaining would be n-2, followed by n-3 and so on till all items are selected or arranged.
The factorial function comprises only the positive integers, as it considers magnitude, and is defined only for whole numbers. It is represented as n! = n(n-1)(n-2)…1
Also, n! = n(n-1)!, (zero)! = 1, or 0! = 1. Example, 6! = 6x5x4x3x2x1 = 720.
The power of a number refers to the number of times a number, which is referred to as the base, is multiplied by itself. It is represented as 2ⁿ where 2 is the base and n is the power of 2 or the number of times 2 is to be used as a factor. Example 2⁴ = 2x2x2x2=16.
If the sum of factors of a number, excluding the number itself, equals to the number, it is a perfect number. Simply put, a number n is a perfect number if sums of its factors other than itself are equal to n. Example, factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248, 496, the sum of the factors is given as 1+2+4+8+16+31+62+124 = 496.
Arithmetic Operations on Numbers
The sequence for applying the basic operators on numbers is according to the rules of VBODMAS. It consists of rules for simplifying mathematical expressions, that are converting complex expressions into their simple forms.
The basic operators are Addition, that is a value equal to that of given numbers combined; subtraction, or removing or reducing a number by the value given; multiplication, is repeated addition, which means adding value the given number of times, and division, implies dividing or separating a number in the given values.
VBODMAS, specifies direction for the order of performing these operations, the full form is as follows
- Viniculum: the bar, which signifies the priority of operation over others. Example, 48‾+‾9 = 4×17 = 68
- Bracket: implies removing brackets for simplification. Example, 4(8+9) = 4×17 = 68
- Off: Removal of brackets to be followed by the ‘off’ operation or multiplication.
- Division: The operation of division should be performed next
- Multiplication: Followed by multiplication
- Addition: And the addition
- Subtraction: The final step should be subtraction
- For 17.3+8.2(2-1)+4.5, applying VBODMAS,17.3+8.2×1+4.5 = 17.3+8.2+4.5 = 30
As specified, the discussion covers number systems, specifically fundamentals of numbers, seeks to introduce the applicants beginning their preparation to the basics and to those who have already begun, to brush up on the same.
A word of encouragement for the applicants, “A little progress each day adds to big results”.
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