## Number Systems Introduction

Number Systems forms the base for Quantitative Ability and clearing of concepts is important for CAT and other related 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.

Following table gives a brief introduction to system of numbers:

**COMPLEX NUMBERS**

In certain calculations in mathematics and related sciences, it is necessary to perform operations with numbers unlike any mentioned thus far in this course. These numbers, unfortunately called “imaginary” numbers by early mathematicians, are quite useful and have a very real meaning in the physical sense. The number system, which consists of ordinary numbers and imaginary numbers, is called the COMPLEX NUMBER system. Complex numbers are composed of a “real” part and an “imaginary” part.

**REAL NUMBERS**

The concept of number, as has been noted in previous chapters, has developed gradually. At one time the idea of number was limited to positive whole numbers.

The concept was broadened to include positive fractions; numbers that lie between the whole numbers. At first, fractions included only those numbers which could be expressed with terms that were integers. Since any fraction may be considered as a ratio, this gave rise to the term RATIONAL NUMBER, which is defined as any number which can be expressed as the ratio of two integers. (Remember that any whole number is an integer.)

It soon became apparent that these numbers were not enough to complete the positive number range. The ratio, p, of the circumference of a circle to its diameter, did not fit the concept of number thus far advanced, nor did such numbers as and Although decimal values are often assigned to these numbers, they are only approximations. That is not exactly equal to 22/7 or to 3.142. Such numbers are called IRRATIONAL to distinguish them from the other numbers of the system. With rational and irrational numbers, the positive number system includes all the numbers from zero to infinity in a positive direction.

Since the number system was not complete with only positive numbers, the system was expanded to include negative numbers. The idea of negative rational and irrational numbers to minus infinity was an easy extension of the system.

Rational and irrational numbers, positive and negative to ± infinity as they have been presented in this course, comprise the REAL NUMBER system.

**Rational numbers**

Rational numbers are numbers which can be written as fractions. This means that they can be written as a divided by b, where the numbers a and b are integers, and b is not equal to 0.

Some rational numbers, such as 1/10, need a finite number of digits after the decimal point to write them in decimal form. The number one tenth is written in decimal form as 0.1. Numbers written with a finite decimal form are rational. Some rational numbers, such as 1/11, need an infinite number of digits after the decimal point to write them in decimal form. There is a repeating pattern to the digits following the decimal point. The number one eleventh is written in decimal form as 0.0909090909….

**Imaginary numbers**

Imaginary numbers are formed by real numbers multiplied by the number i. This number is the square root of minus one (-1).

There is no number in the real numbers which when squared makes the number -1. Therefore mathematicians invented a number. They called this number i.

All of normal mathematics can be done with imaginary numbers:

- To sum two imaginary numbers they can pull out (factor out) the i. For example 2i + 3i = (2 + 3)i = 5i.
- If they subtract one imaginary number from another they can also factor out the i. For example 5i – 3i = (5 – 3)i = 2i.
- If they multiply two imaginary numbers then they need to remember that i × i is -1. For example 5i × 3i = ( 5 × 3 ) × ( i × i ) = 15 × (-1) = -15

Imaginary numbers were called imaginary because when they were first found many mathematicians did not think they existed.

Notice the distinction between this use of the radical sign and the manner in which it was used in chapter 7. Here, the ± symbol is included with the radical sign to emphasize the fact that two values of x exist. Although both roots exist, only the positive one is usually given. This is in accordance with usual mathematical convention.

The equation raises an interesting question:

What number multiplied by itself yields -4 ? The square of -2 is +4. Likewise, the square of +2 is +4. There is no number in the system of real numbers that is the square root of a negative number. The square root of a negative number came to be called an IMAGINARY NUMBER When this name was assigned the square roots of negative numbers, it was natural to refer to the other known numbers as the REAL numbers.

**Irrational Numbers**

Numbers which are not rational are called irrational numbers. When expressed as decimals, they are non-terminating and non-recurring.

We can obtain infinite number of irrationals between two irrational numbers.

Irrational numbers are numbers which cannot be written as a fraction, but do not have imaginary parts.

Irrational numbers often occur in geometry. For instance if we have a square which has sides of 1 meter, the distance between opposite corners is the square root of two. This is an irrational number. In decimal for it is written as 1.414213… Mathematicians have proved that the square root of every natural number is either an integer or an irrational number.

One well known irrational number is pi. This is the circumference of a circle divided by its diameter. This number is the same for every circle. The number pi is approximately 3.1415926359.

An irrational number cannot be fully written down in decimal form. It would have an infinite number of digits after the decimal point. These digits would also not repeat.

**Surd**

If K is not a perfect nth power of any number, then is called a surd of the nth order.

A surd is always an irrational number.

**Real Numbers**

The union of the set of rational numbers and irrational numbers forms the set of real numbers.

(i) For every real number, there is a corresponding point on the number line.

(ii) For every point on the number line, there exists a real number.

**Natural Numbers (N)**

The 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, etc.

**Whole Numbers (W)**

All natural numbers together with ‘0’ are collectively called whole numbers. The set of whole numbers is denoted by W.

W = {0, 1, 2, 3, ……}

**Integers (Z)**

The set including all whole numbers and their negatives is called a set of integers. It is denoted by Z.

Z = {– ∞, … – 3, – 2, – 1, 0, 1, 2, 3, ……. ∞}

They are further classified into Negative integers, Neutral integers and positive integers.

**Even Numbers**

All numbers divisible by 2 are called even numbers. For example, 2, 4, 6, 8, 10 … Even numbers can be expressed in the form **2n**, where n is an integer. Thus 0, – 2, − 6, etc. are also even numbers.

**Odd Numbers**

All numbers not 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, − 9, etc. are all odd numbers.

**Prime Number**

A prime number is a natural number which has only two distinct divisors: 1 and itself.

The number 1 is not a prime number.

**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.

**Important observation about 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.

**Examples:**

- 550 = 2 × 52 × 11
- 1200 = 24 × 3 × 52

Thus, prime numbers are the basic building blocks of any positive integer. This factorisation will also help in finding GCD and LCM quickly.

**Composite Numbers**

A composite number has other factors besides itself and unity. For example, 8, 72, 39, etc. On the basis of this fact that a number with more than two factors is a composite we have only 34 composite from 1 to 50 and 40 composite from 51 to 100.

**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.

**Co-Prime Numbers**

Two numbers (prime or composite) are co-prime to each other, if they do not have any common factor except 1.

Examples: 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.

**Fractions**

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

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

**Rational Numbers**

Rational Number is defined as the **ratio of two integers** i.e. a number that can be represented by a fraction of the form p/q where p and q are integers and q ≠ 0. They also can be defined as the **non-terminating recurring decimal numbers**.

**Irrational Numbers**

Any number which can not be represented in the form p/q where p and q are integers and q ≠ 0 is an irrational number. On the basis of non-terminating decimals, irrational numbers are **non-terminating non recurring decimals**.

**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:

**Pure Recurring Decimals:**A decimal in which all the figures after the decimal point repeat, is called a pure recurring decimal.**Mixed Recurring Decimals:**A decimal in which some figures do not repeat and some of them are repeated is called a mixed recurring decimal.**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

- The number 1 is neither prime nor composite.
- The number 2 is the only even number which is prime.
- (x
^{n}+ y^{n}) is divisible by (x + y), when n is an odd number. - (x
^{n}– y^{n}) is divisible by (x + y), when n is an even number.

(x^{n} – y^{n}) is divisible by (x – y), when n is an odd or an even number.

**Sub-topic 1 :Factors of a Number**

Watch the Video on How to find the factors of a number

https://www.youtube.com/watch?v=jNQHJ-oXq8s&t=104s

Representing a number as prime factors helps in analysing problems.

**N = p****a**** × q****b**** × r****c**

where p, q, r are prime numbers and a, b, c are the number of times each prime number occurs.

Watch the Video on How to find the factors of a number

https://www.youtube.com/watch?v=jNQHJ-oXq8s&t=104s

**Number of Factors**

**Number of Factors = (a + 1)(b + 1)(c + 1)**

**Example: Find the number of factors of 2****4**** × 3****2**

Number of factors = (4 + 1)(2 + 1) = 5(3) = 15

**Number of Ways of Expressing a Given Number as a Product of Two Factors**

When a number is having even number of factors then, (a+1)(b+1)(c+1)/2

But if a number have odd number of factors then, [(a+1)(b+1)(c+1)-1]/2

**Sum of the factors of a number**

The sum of all the factors of the number are given by the formula:

Sum of Factors = (a(p+1) – 1)(b(q+1) – 1)(c(r+1) – 1)/(a-1)(b-1)(c-1)

**Sub-topic 2 : UNIT DIGIT**

Watch Our Video on this topic :

https://www.youtube.com/watch?v=2eRaVVKE7L0

Concept of Cyclicity: Power Cycle

At times there are questions that require the students to find the units digit in case of the numbers occurring in powers. Concept of cyclicity is used to find unit’s digit in case the numbers are occurring in powers. The last digit of a number of the form ab falls in a particular sequence that depends on the unit digit of the number (a) and the power the number is raised to (b).

Consider the power cycle of 2:

- 21 = 2
- 22 = 4
- 23 = 8
- 24 = 16
- 25 = 32
- 26 = 64

You can see that the unit digit gets repeated after every fourth power of 2. Hence, you can say that 2 has a power cycle of 2, 4, 8, 6 with cyclicity 4. This is applicable for all numbers ending in 2.

**Sub-topic 3 : Factorial Questions**

Watch Our Video on this topic :

https://www.youtube.com/watch?v=maS5zbkz42E&t=133s

**Meaning of Factorial**

- N! = 1 × 2 × 3 × … × (N-1) × N
- Examples: 3! = 6; 5! = 120

**Maximum power of p (prime number) in n! (n factorial)**

To find the highest power of a prime number (p) in a factorial (n!), keep dividing n by p and add all the quotients.

Alternatively, use the formula:

{n/p} + {n/p2} + {n/p3} + …

**Sub-topic 4 : Divisibility Rules**

A divisibility rule is a method to determine whether a given integer is divisible by a fixed divisor without performing the division. You can do this by examining the digits of the integer.

**Divisibility by 2**

The last digit is even: 0, 2, 4, 6, or 8.

**Divisibility by 3**

The sum of the digits is divisible by 3.

**Divisibility by 4**

The last two digits is divisible by 4 or is 00.

**Divisibility by 5**

The last digit is either 0 or 5.

**Divisibility by 6**

The sum of the digits is divisible by 3 and the number itself is divisible by 2. It means that the number is divisible by both 2 and 3.

**Divisibility by 7**

Subtract 2 times the last digit from the rest.

Group the numbers in three from units digit. Add the odd groups and even groups separately. The difference of the odd and even groups should be 0 or divisible by 7.

**Divisibility by 8**

If the hundreds digit is even, examine the number formed by the last two digits. If the hundreds digit is odd, examine the number obtained by the last two digits plus 4.

**Divisibility by 9**

The sum of the digits is divisible by 9.

**Divisibility by 10**

The last digit or unit digit is 0.

**Divisibility by 11**

Add the digits in blocks of two from right to left.

The difference between the sums of digits in the odd and even places taken from right to left is either zero or a multiple of 11.

**Divisibility by 12**

It is divisible by 3 and by 4 both.

**Divisibility by 13**

Group the numbers in three from units digit. Add the odd groups and even groups separately. The difference of the odd and even groups should be 0 or divisible by 13.

**Divisibility by 14. ** If a number is divisible by 2 and 7 both then the number is divisible by 14 as well.

**Divisibility by 15. ** If a number is divisible by 3 and 5 both then the number is divisible by 15 as well.

**Divisibility by 16. **If the last for digits of a number is divisible by 16 then the number is also divisible by 16.

**Divisibility by 17**. Subtract five times the last digit from the remaining leading truncated number. If the result is divisible by 17, then so was the first number. Apply this rule over and over again as necessary.

Example: 3978–>397-5*8=357–>35-5*7=0. So 3978 is divisible by 17.

**Divisibility by 18. ** If a number is divisible by 2 and 9 both then the number is divisible by 18 as well.

**Divisibility by 19**. Add two times the last digit to the remaining leading truncated number. If the result is divisible by 19, then so was the first number. Apply this rule over and over again as necessary.

EG: 101156–>10115+2*6=10127–>1012+2*7=1026–>102+2*6=114 and 114=6*19, so 101156 is divisible by 19.

**Divisibility by 23**. Add seven times the last digit to the remaining leading truncated number. If the result is divisible by 23, then so was the first number. Apply this rule over and over again as necessary.

Example: 17043–>1704+7*3=1725–>172+7*5=207–>20+7*7=69 which is 3*23, so 17043 is also divisible by 23.

**Divisibility by 29**. Add three times the last digit to the remaining leading truncated number. If the result is divisible by 29, then so was the first number. Apply this rule over and over again as necessary.

Example: 15689–>1568+3*9=1595–>159+3*5=174–>17+3*4=29, so 15689 is also divisible by 29.

**Divisibility by 31**. Subtract three times the last digit from the remaining leading truncated number. If the result is divisible by 31, then so was the first number. Apply this rule over and over again as necessary.

Example: 7998–>799-3*8=775–>77-3*5=62 which is twice 31, so 7998 is also divisible by 31.

**Divisibility by 37**. This is (slightly) more difficult, since it perforce uses a double-digit multiplier, namely eleven. Subtract eleven times the last digit from the remaining leading truncated number. If the result is divisible by 37, then so was the first number. Apply this rule over and over again as necessary.

Example: 23384–>2338-11*4=2294–>229-11*4=185 which is five times 37, so 23384 is also divisible by 37.

**Divisibility by 41**. Subtract four times the last digit from the remaining leading truncated number. If the result is divisible by 41, then so was the first number. Apply this rule over and over again as necessary.

Example: 30873–>3087-4*3=3075–>307-4*5=287–>28-4*7=0, remainder is zero and so 30873 is also divisible by 41.

**Sub-topic 5 : LCM & HCF**

Suppose there are two numbers – a and b. If a number *a* divides another number *b* exactly, we say that *a* is **factor** of *b* and *b* is called **multiple** of *a*.

## Highest Common Factor (HCF) or Greatest Common Divisor (GCD)

The **greatest common divisor** (gcd), also known as the greatest common denominator, greatest common factor (gcf), or **highest common factor** (hcf), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4.

The HCF of two or more than two numbers is the greatest number that divides each of them exactly. There are two methods of finding the HCF of a given set of numbers:

**Factorisation Method**

In this method, express each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives HCF.

**Division Method**

Divide the larger number by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor is the required HCF.

**Finding the HCF of more than two numbers:** H.C.F. of [(H.C.F. of any two) and (the third number)] gives the HCF of three given numbers.

## Least Common Multiple (LCM)

The **lowest common multiple** or (LCM) least common multiple or smallest common multiple of two rational numbers *a* and *b* is the smallest positive rational number that is an integer multiple of both *a* and *b*. The definition can be generalised for more than two numbers.

The least number which is exactly divisible by each one of the given numbers is called their LCM.

**Factorisation Method of Finding LCM**

Resolve each one of the given numbers into a product of prime factors. Then, LCM is the product of highest powers of all the factors.

**Common Division Method (Short-cut Method) of Finding LCM**

Arrange the given numbers in a row in any order. Divide by a number which divides exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required LCM of the given numbers.

**Product of two numbers = Product of their HCF and LCM**

a x b = HCF x LCM

## Co-primes

Two numbers are said to be co-primes if their HCF is 1. HCF of two co-prime numbers is 1. To find LCM of co-prime numbers, just multiply them. No need to find factors.

### HCF and LCM of Fractions

HCF = (HCF of Numerators) / (LCM of Denominator)

LCM = (LCM of Numerators) / (HCF of Denominator)

**Applications of HCF and LCM**

- Find the Greatest Number that will exactly divide x, y, z.

Required number = H.C.F. of x, y, and z (greatest divisor).

- Find the Greatest Number that will divide x, y and z leaving remainders a, b and c respectively.

Required number (greatest divisor) = H.C.F. of (x – a), (y – b) and (z – c).

- Find the Least Number which is exactly divisible by x, y and z.

Required number = L.C.M. of x, y and z (least divided).

- Find the Least Number which when divided by x, y and z leaves the remainders a, b and c respectively.

Then, it is always observed that (x – a) = (z – b) = (z – c) = K (say).

∴ Required number = (L.C.M. of x, y and z) – K.

- Find the Least Number which when divided by x, y and z leaves the same remainder ‘r’ each case.

Required number = (L.C.M. of x, y and z) + r.

- Find the Greatest Number that will divide x, y and z leaving the same remainder in each case.

Required number = H.C.F of (x – y), (y – z) and (z – x).

**Sub-topic 6 : Remainder Theorem**

*Finding Remainders of a product (derivative of remainder theorem)*

If ‘a1‘is divided by ‘n’, the remainder is ‘r1’ and if ‘a2’ is divided by n, the remainder is r2. Then **if a****1****+a****2** **is** **divided by n, the remainder will be r****1**** + r****2**

**If a****1**** – a****2**** is divided by n, the remainder will be r****1**** – r****2**

**If a****1** ×** a****2**** is divided by n, the remainder will be r****1** ×** r****2**

If two numbers ‘a1’ and ‘a2‘ are exactly divisible by n. Then their sum, difference and product is also exactly divisible by n.

i.e., **If ‘a****1****’ and ‘a****2****’ are divisible by n, then**

**a****1**** + a****2**** is also divisible by n**

**a****1**** – a****2**** is also divisible by n**

**and If a****1** ×** a****2**** is also divisible by n.**

The remainders of the powers of 4 repeats after every 3rd power.

There are some fundamental conclusions that are helpful if remembered, i.e.

- There are (n + 1) terms.

- The first term of the expansion has only a.

- The last term of the expansion has only b.

- All the other (n – 1) terms contain both a and b.

- If (a + b)n is divided by a then the remainder will be
*b**n*such that bn < a.

*Wilson’s Theorem*

**If n is a prime number, (n – 1)! + 1 is divisible by n.**

Let take n = 5

Then (n – 1)! + 1 = 4! + 1 = 24 + 1 = 25 which is divisible by 5.

**Similarly **if n = 7

(n – 1)! + 1 = 6! + 1 = 720 + 1 = 721 which is divisible by 7.

*Corollary*

**If (2p + 1) is a prime number (p!)****2**** + (– 1)****p**** is divisible by 2p + 1.**

e.g If p = 3**,** 2p + 1 = 7 is a prime number

(p!)2 + (– 1)p = (3!)2 + (– 1)3 = 36 – 1 = 35 is divisible by (2p + 1) = 7.

*Property*

**If “a” is natural number and P is prime number then (a****p**** – a) is divisible by P.**

*Fermat’s Theorem*

If p is a prime number and N is prime to p, then Np –1 – 1 is a multiple of p.

**Corollary**

Since p is prime, p – 1 is an even number except when p = 2.

**Order of Operations – ***BODMAS*

*BODMAS*

**Operations**

**“Operations”** mean things like add, subtract, multiply, divide, squaring, etc. If it isn’t a number it is probably an operation.

But, when you see something like…

**7 + (6 × 5****2**** + 3)**

… what part should you calculate first?

Start at the left and go to the right?

Or go from right to left?

Calculate them in the wrong order, and you will get a wrong answer !

So, long ago people agreed to always follow certain rules when doing calculations, and they are:

**Order of Operations**

**How Do I Remember ? BODMAS !**

B |
Brackets first |

O |
Orders (ie Powers and Square Roots, etc.) |

DM |
Division and Multiplication (left-to-right) |

AS |
Addition and Subtraction (left-to-right) |

The only strange name is “Orders”. “Powers” would have been a better word, but who could remember “BPDMAS” … ? “Exponents” is used in Canada, and so you might prefer “BEDMAS”, there is also “Indices” so that makes it “BIDMAS”. In the US they say “Parenthesis” instead of Brackets, so they say “PEMDAS”

*Oh, yes, and what about 7 + (6 × 5**2** + 3) ?*

7 + (6 × 52 + 3) | |

7 + (6 × 25 + 3) | Start inside Brackets, and then use “Orders” First |

7 + (150 + 3) | Then Multiply |

7 + (153) | Then Add |

7 + 153 | Brackets completed, last operation is add |

160 |
DONE ! |

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