Number System

The Number System includes any of the numerous sets of symbols and the rules for using them to denote numbers, which are used to state how many objects are there in a given set. Thus, the idea of “oneness” can be denoted by the Roman numeral I, by the Greek letter alpha α which was the first letter used as a numeral, by the Hebrew letter aleph which is the first letter used as a numeral, or by the modern number 1, which is nothing but Hindu-Arabic in origin.

Number System Definition

Number system is a mathematical presentation of numbers of a given set. For further discussion, let us understand number systems.

Most likely, the beginning system of the inscribed symbol in ancient Mesopotamia was a system of symbols for numbers. The present number systems are place-value systems. That is, the value of these numbers depends upon the position or place of the numbers in the representation; for example, the 3 in 30 and 300 represents 3 ten’s and 3 hundred, respectively. In the ancient systems, such as the Egyptian civilization, Roman, Hebrew, and Greek Number systems, did not have a positional characteristic, and it was a very complicated arithmetical calculation. Other systems though, including the Babylonian, one version each of the Chinese and Indian, as well as the Mayan system, did use the principle of place value.

Types of Number Systems

Based on the base value and the number of allowed digits, number systems are of many types. The four common types of Number System are:

Decimal Number System
Binary Number System
Octal Number System
Hexadecimal Number System

1. Decimal Number System

Number system with base value 10 is termed as Decimal number system. It uses 10 digits i.e. 0-9 for the creation of numbers. Here, each digit in the number is at a specific place with place value a product of different powers of 10. Here, the place value is termed from right to left as first place value called units, second to the left as Tens, so on Hundreds, Thousands, etc. Here, units has the place value as 10⁰, tens has the place value as 10¹, hundreds as 10², thousands as 10³, and so on.

For example: 10285 has place values as

(1 × 10⁴)+ (0 × 10³) + (2 × 10²) + (8 × 10¹) + (5 × 10⁰)
1 × 10000 + 0 × 1000 + 2 × 100 + 8 × 10 + 5 × 1
10000 + 0 + 200 + 80 + 5
10285

2. Binary Number System

Number System with base value 2 is termed as Binary number system. It uses 2 digits i.e. 0 and 1 for the creation of numbers. The numbers formed using these two digits are termed as Binary Numbers. The binary number system proves highly beneficial in electronic devices and computer systems due to its capacity for straightforward execution employing solely two states: ON and OFF, specifically represented as 0 and 1.

Decimal Numbers 0-9 are represented in binary as: 0, 1, 10, 11, 100, 101, 110, 111, 1000, and 1001

Examples:

14 can be written as 1110
19 can be written as 10011
50 can be written as 110010

3. Octal Number System

Octal Number System is one in which the base value is 8. It uses 8 digits i.e. 0-7 for creation of Octal Numbers. Octal Numbers can be converted to Decimal value by multiplying each digit with the place value and then adding the result. Here the place values are 8⁰, 8¹, and 8². Octal Numbers are useful for the representation of UTF8 Numbers.

Example:

(135)₁₀ can be written as (207)₈
(215)₁₀ can be written as (327)₈

4. Hexadecimal Number System

Number System with base value 16 is termed as Hexadecimal Number System. It uses 16 digits for the creation of its numbers. Digits from 0-9 are taken like the digits in the decimal number system but the digits from 10-15 are represented as A-F i.e. 10 is represented as A, 11 as B, 12 as C, 13 as D, 14 as E, and 15 as F. Hexadecimal Numbers are useful for handling memory address locations.

Examples:

(255)₁₀ can be written as (FF)₁₆
(1096)₁₀ can be written as (448)₁₆
(4090)₁₀ can be written as (FFA)₁₆

Operations on Numbers

1. Exponents and Powers

Exponents, or powers, are an important part of maths as they are necessary to indicate that a number is multiplied by itself for a given number of times.
When a number is multiplied by itself it gives the ‘square of the number’. Thus,
n * n = n^2 (e g. 3 * 3 = 3^2)
If the same number is multiplied by itself twice we get the cube of the number. Thus, n * n * n= n^3 (e g. 3 * 3 * 3= 3^3)
n * n * n * n= n^4 and so on.
With respect to powers of numbers, there are 5 basic rules which you should know: For any number ‘n’ the following rules would apply:
- Rule 1: n^a × n^b = n^(a+b). Thus, 4^3 × 4^5 = 4^(3+5) = 4^8
- Rule 2: n^a / n^b = n^(a-b). Thus, 3^9 / 3^4 = 3^5.
- Rule 3: n^ (a^b)= n^(a×b). Thus, 3^ (2^4 )= 3^8.
- Rule 4: n^ (-a) = 1/n^a . Thus, 3^ (-4)=1/3^(4)
- Rule 5: n^0=1. Thus, 5^0=1.

These rules are also commonly known as the rules of indices.

2. Properties

Commutative property of addition: a + b = b + a.
Associative property of addition: (a + b) + c = a + (b + c).
Commutative property of multiplication: ab = ba.
Associative property of multiplication: (ab) * c= a(b*c).
Distributive property of multiplication with respect to addition: (a + b) c = ac + bc.
Subtraction and division define the inverse operations to addition and multiplication, respectively.

LCM and HCF

1. LCM (Least Common Multiple)

Smallest natural number ‘n’ which is divisible by n1 & n2
How to find LCM?
Find Standard forms of n1, n2
Write down all prime factors
Raise each prime factor to highest of the powers
The product will be LCM

Eg. Find the LCM of 150, 210.

Step 1: Writing down the standard form of numbers
- 150 = 5 * 5 * 3 * 2
- 210 = 5 * 2 * 7 * 3
Step 2: Write down all the prime factors: that appear at least once in any of the numbers: 5, 3, 2, 7.
Step 3: Raise each of the prime factors to their highest available power (considering each to the numbers).
- The LCM= 2^1 * 3^1 * 5^2 * 7= 2 * 3 * 25 * 7 = 1050

2. Highest Common Factor (HCF)

For n1, n2: If n1 and n2 are exactly divisible by the same number x, then x is a common divisor (CD) of n1, n2. Highest of all such CDs of n1 and n2 is HCF.

How to find HCF?

Find Standard forms of n1, n2
Write down all prime factors
Find Common factors in both n1, n2.
Raise each prime factor to lesser of the powers
The product will be HCF

E.g., Find the HCF of 150, 210, 375.

Step 1: Writing down the standard form of numbers
- 150 = 5 * 5 * 3 * 2
- 210 = 5 * 2 * 7 * 3
Step 2: Writing Prime factors common to all the numbers: 5, 2, 3 (Note: we didn’t consider 7 as it is only a common factor of 210 and not of 150!)
Step 3: Raising each prime factor to lesser of the power, i.e.- 2^1, 5^1 and 3^1
Step 4: Hence, the HCF will be 2 * 5 * 3 = 30