Number Systems #1

Number Systems forms the base for Quantitative Ability and clearing of concepts is important for CAT and other management exams. A measurement carried out, of any quantity, leads to a meaningful value called the Number. This value may be positive or negative depending on the direction of the measurement and can be represented on the number line.

In the decimal system, we use the numbers from 0 to 9. 0 is called insignificant digit whereas 1, 2, 3, 4, 5, 6, 7, 8, 9 are called significant digits. A group of figures denoting a number is called a numeral. For a given numeral, start from extreme right as unit’s place, ten’s place, hundred’s place and so on.

Type of Numbers

Following table gives a brief introduction to system of numbers:

1. Natural Numbers (N)

The counting numbers 1, 2, 3, 4, 5, ... are known as natural numbers. The set of natural numbers is denoted by N. The natural numbers are further divided as even, odd, prime, or composite.

N = {1, 2, 3, 4, 5, ...}

2. Whole Numbers (W)

All natural numbers together with 0 (zero) are called whole numbers. The set of whole numbers is denoted by W. Every natural number is a whole number but 0 is a whole number which is not a natural number.

W = {0, 1, 2, 3, ...}

3. Integers (Z)

All counting numbers and their negatives including zero are known as integers. The set including all whole numbers and their negatives is called integers. It is denoted by Z or I.

Z or I = {..., -3, -2, -1, 0, 1, 2, 3, ...}

Integers are further classified into Negative Integers, Neutral Integers and Positive Integers.

4. Rational Numbers

The numbers of the form p/q, where p and q are integers and q ≠ 0, are known as rational numbers. For example, 4/7, 3/2, -2/3. The set of all rational numbers is denoted by Q.

Since every natural number a can be written as a/1, so it is a rational number. Since 0 can be written as 0/1 and every non-zero integer a can be written as a/1, so it is also a rational number.

Every rational number, when expressed in decimal form is expressible either in terminating decimals or non-terminating repeating decimals. For example, 1/5 = 0.2, 1/3 = 0.333...

5. Irrational Numbers

Those numbers which when expressed in decimal form are neither terminating nor repeating decimals are known as irrational numbers. For example, √2, √3, √5, π. 

Irrational numbers are non-terminating non-repeating decimals.

6. Real Numbers

The rational and irrational numbers combined together are called real numbers. The set of all real numbers is denoted by R.

The sum, difference or, product of a rational and irrational number is irrational.

Real Numbers

1. Even Numbers

All numbers which are exactly divisible by 2 are called even number For example, 2, 4, 6, 8, 10, ...

Even numbers can be expressed in the form 2n, where n is an integer. Thus 0, -2, -4, -6, are also even numbers.

2. Odd Numbers

All numbers which are not exactly divisible by 2 are called odd numbers. For example, 1, 3, 5, 7, 9, ...

Odd numbers can be expressed in the form (2n+1) where n is any integer. Thus -1, -3, -5, -7, are also odd numbers.

3. Prime Numbers

A natural number other than 1 is a prime number if it is divisible by 1 and itself only. For example, 2, 3, 5, 7, and so on.

The number 1 is neither a prime number nor a composite number The number 2 is the only even number which is prime.

There are 25 prime numbers under 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

4. Composite Numbers

Natural numbers greater than 1 which are not prime are known as composite numbers. A composite number has other factors besides itself and unity. For example, 8, 72, 39.

5. Perfect Numbers

A number is a perfect number if the sum of its factors, excluding itself and but including 1, is equal to the number itself.

The sum of all the possible factors of the number is equal to twice the number.

Examples: 6 (1 + 2 + 3 = 6), 28 (1 + 2 + 4 + 7 + 14 = 28)

Other examples of perfect numbers are 496, 8128, etc. There are 27 perfect numbers discovered so far.

6. Co-Prime Numbers

Two numbers (prime or composite) which have only 1 as the common factor are called co-primes or, relatively prime to each other. For example, 25 and 9, since they don’t have a common factor other than 1. Another example is 35 and 12, since they both don’t have a common factor among them other than 1.

Prime Numbers

A prime number greater than 3, when divided by 6 leaves either 1 or 5 as the remainder. Hence, a prime number can be expressed in the form of 6K±1. But, the converse of this observation is not true, that a number leaving a remainder of 1 or 5 when divided by 6 is not necessarily a prime number.

Prime Factorization Theorem: This is the area where prime numbers are used. This theorem states that any integer greater than 1 can be written as a unique product of prime numbers. For example, 550 = 2 × 52 × 11, 1200 = 24 × 3 × 52

Thus, prime numbers are the basic building blocks of any positive integers.

Test to find whether a given number is a prime

  1. Step 1: Select a least positive integer n such that n2 > given number.
  2. Step 2: Test the divisibility of given number by every prime number less than n.
  3. Step 3: The given number is prime only if it is not divisible by any of these primes.

Example 1: Investigate whether 571 is a prime number.

Since (23)2 = 529 < 571 and (24)2 = 576 > 571, n = 24

Prime numbers less than 24 are 2, 3, 5, 7, 11, 13, 17, 19, 23. Since, 24 is divisible by 2, 571 is not a prime number.

Example 2: Investigate whether 923 is a prime number.

Since (30)2 = 900 < 923 and (31)2 = 961 > 923, n = 31

Prime numbers less than 31 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Since 923 is not divisible by any of these primes, therefore 923 is a prime number.

Fractions

A fraction denotes part or parts of a unit. Several types are:

  1. Common Fraction: Fractions whose denominator is not 10 or a multiple of it. For example, 2/3, 17/18
  2. Decimal Fraction: Fractions whose denominator is 10 or a multiple of 10.
  3. Proper Fraction: In this the numerator is less than denominator. Hence its value < 1.
  4. Improper Fraction: In these the numerator is greater denominator. Hence its value > 1.
  5. Mixed Fractions: When a improper fraction is written as a whole number and proper fraction it is called mixed fraction.

Non-Terminating Decimal Numbers

When you divide any number by other number, either you get a terminating number or a non-terminating number. A non-terminating number on the basis of occurrence of digits after decimal can classified as:

  1. Pure Recurring Decimals: A decimal in which all the figures after the decimal point repeat, is called a pure recurring decimal.
  2. Mixed Recurring Decimals: A decimal in which some figures do not repeat and some of them are repeated is called a mixed recurring decimal.
  3. Non-Recurring Decimals: A decimal number in which the figure don’t repeat themselves in any pattern are called non-terminating non-recurring decimals and are termed as irrational numbers.

Points To Remember

  1. The number 1 is neither prime nor composite.
  2. The number 2 is the only even number which is prime.
  3. (xn + yn) is divisible by (x + y), when n is an odd number.
  4. (xn - yn) is divisible by (x + y), when n is an even number.
  5. (xn - yn) is divisible by (x - y), when n is an odd or an even number.