GETTING STARTED WITH NUMBERS

One of the most important topics of quants in common entrance exams like CAT, CET, NMAT etc is numbers. The difficulty level of this level may vary from exam to exam. This topic includes all the basics that we have learnt in school and more. It is quite an important topic so it is necessary that you are thorough with the basics. Here is something to get you started: The Natural Numbers The natural (or counting) numbers are 1,2,3,4,5, etc. There are infinitely many natural numbers. The set of natural numbers, {1,2,3,4,5,…}, is sometimes written N for short. The whole numbers are the natural numbers together with 0.   The sum of any two natural numbers is also a natural number (for example, 4+2000=2004), and the product of any two natural numbers is a natural number (4×2000=8000). This is not true for subtraction and division, though. The Integers The integers are the set of real numbers consisting of the natural numbers, their additive inverses and zero. {…,−5,−4,−3,−2,−1,0,1,2,3,4,5,…} The set of integers is sometimes written J or Z for short. The sum, product, and difference of any two integers is also an integer. But this is not true for division… just try 1÷2. The Rational Numbers The rational numbers are those numbers which can be expressed as a ratio between two integers. For example, the fractions 13 and −11118 are both rational numbers. All the integers are included in the rational numbers, since any integer z can be written as the ratio z1 All decimals which terminate are rational numbers (since 8.27 can be written as 827/100.) Decimals which have a repeating pattern after some point are also rationals: for example, 0.0833333….=112 The set of rational numbers is closed under all four basic operations, that is, given any two rational numbers, their sum, difference, product, and quotient is also a rational number (as long as we don’t divide by 0). The Irrational Numbers An irrational number is a number that cannot be written as a ratio (or fraction). In decimal form, it never ends or repeats. The ancient Greeks discovered that not all numbers are rational; there are equations that cannot be solved using ratios of integers. The first such equation to be studied was 2=x2 . What number times itself equals 2? 2√ is about 1.414, because 1.4142=1.999396, which is close to 2. But you’ll never hit exactly by squaring a fraction (or terminating decimal). The square root of 2is an irrational number, meaning its decimal equivalent goes on forever, with no repeating pattern: 2√=1.41421356237309… Other famous irrational numbers are the golden ratio, a number with great importance to biology: 1+5√2=1.61803398874989… π (pi), the ratio of the circumference of a circle to its diameter: π=3.14159265358979… and e, the most important number in calculus: e=2.71828182845904… Irrational numbers can be further subdivided into algebraic numbers, which are the solutions of some polynomial equation (like 2√ and the golden ratio), and transcendental numbers, which are not the solutions of any polynomial equation. π and eare both transcendental. The Real Numbers The real numbers is the set of numbers containing all of the rational numbers and all of the irrational numbers. The real numbers are “all the numbers” on the number line. There are infinitely many real numbers just as there are infinitely many numbers in each of the other sets of numbers. But, it can be proved that the infinity of the real numbers is a bigger infinity.There are even “bigger” infinities. The Complex Numbers The complex numbers are the set {a+bi| a and b are real numbers}, where i is the imaginary unit, −1−−−√. The complex numbers include the set of real numbers. The real numbers, in the complex system, are written in the form a+0i=a. a real number.This set is sometimes written as C for short. The set of complex numbers is important because for any polynomial p(x) with real number coefficients, all the solutions of p(x)=0 will be in C.