# Base 10 number system denary

## Base 2: The Binary Number System

Base 2 has exactly two numbers, 0 and 1 . All numbers in the binary numbe rsystem must be formed using these two symbols. Numbers in the binary systembecomes long quickly.

## Base 10: The Decimal Number system

The number base familiar to us is base 10, upon which the decimal numbersystem is built. There are 10 numbers (0 through 9) used in decimal numbersystem. We normally use base 10, which is the number system most of us canunderstand and use easily.

## Base 16: The Hexadecimal Number system

The Hexadecimal number system uses exactly 16 numbers. Base 10 uses thefamiliar 0 through 9, and bases 2 and 8 use a subset of these numbers. Base16, however, needs those 10 numbers (0 through 9) and six more. The sixadditional numbers used in the Hexadecimal number system are represented byletters A through F. So the base 16 numbers are: 0,1,2,3,4,5,6,7,8,9, AB,C,D,E and F. It takes some adjusting to think of A or D as a digit insteadof a letter. It also takes little time to become accustomed to numbers such as6A2F or even ACE. Both of these examples are valid numbers in Hexadecimal.

## Decimal to hexadecimal conversion

For converting decimal to hexadecimal, there are two steps required toperform, which are as follows: 1. In the first step, we perform the division operation on the integer and the successive quotient with the base of hexadecimal (16). 2. Next, we perform the multiplication on the integer and the successive quotient with the base of hexadecimal (16).Example 1: (152.25)10Step 1:Divide the number 152 and its successive quotients with base 8.Operation | Quotient | Remainder —|—|— 152/16 | 9 | 8 9/16 | 0 | 9 (152)10=(98)16Step 2:Now perform the multiplication of 0.25 and successive fraction with base 16.Operation | Result | carry —|—|— 0.25×16 | 0 | 4 (0.25)10=(4)16So, the hexadecimal number of the decimal number 152.25 is 230.4.

## Hexa-decimal to other Number System

Like binary, decimal, and octal, hexadecimal numbers can also be convertedinto other number systems. The process of converting hexadecimal to decimaldiffers from the remaining one. Let’s start understanding how conversion isdone.

## Base 10 number system (denary)

In our everyday lives we use a ‘Denary’ number system which has the numbersymbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.This is called a ‘base-10’ number system, because there are ten symbolsinvolved.Here are some examples of denary numbers:

## Base 2 number system (binary)

You already know that computers need to use binary numbers to process data.Binary is a ‘base-2’ type of number which has only two symbols, and these arechosen to be 1 or a 0Here are some examples of binary numbers:Challenge see if you can find out one extra fact on this topic that we haven’talready told youClick on this link: What is base number systemHindu-Arabic Numeration System

## Decimal Number System (Base 10 Number System)

Decimal number system has base 10 because it uses ten digits from 0 to 9. Inthe decimal number system, the positions successive to the left of the decimalpoint represent units, tens, hundreds, thousands and so on. This system isexpressed in decimal numbers.Every position shows a particular power of the base (10). For example, thedecimal number 1457 consists of the digit 7 in the units position, 5 in thetens place, 4 in the hundreds position, and 1 in the thousands place whosevalue can be written as(1×103) + (4×102) + (5×101) + (7×100)(1×1000) + (4×100) + (5×10) + (7×1)1000 + 400 + 50 + 71457

## Binary Number System (Base 2 Number System)

The base 2 number system is also known as the Binary number system wherein,only two binary digits exist, i.e., 0 and 1. Specifically, the usual base-2 isa radix of 2. The figures described under this system are known as binarynumbers which are the combination of 0 and 1. For example, 110101 is a binarynumber.We can convert any system into binary and vice versa.ExampleWrite (14)10 as a binary number.Solution:Base 2 Number System Example∴ (14)10 = 11102

## Hexadecimal Number System (Base 16 Number System)

In the hexadecimal system, numbers are written or represented with base 16. Inthe hex system, the numbers are first represented just like in decimal system,i.e. from 0 to 9. Then, the numbers are represented using the alphabets from Ato F. The below-given table shows the representation of numbers in thehexadecimal number system.Hexadecimal | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F —|—|—|—|—|—|—|—|—|—|—|—|—|—|—|—|— Decimal | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15

## Decimal, the Base 10 Numbering System

First let’s start with decimal.The decimal, also known as the denary or base 10 numbering system is what weuse in everyday life for counting. The fact that there are ten symbols is morethan likely because we have 10 fingers.We use ten different symbols or numerals to represent the numbers from zero tonine.Those numerals are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9When we get to the number ten, we have no numeral to represent this value, soit is written as:> 10The idea is to use a new place holder for each power of 10 to make up anynumber we want.So 134 means one hundred, 3 tens and a 4 although we just interpret and readit as the number one hundred and thirty four.

## Binary, the Base 2 Numbering System

In the decimal number system, we saw that ten numerals were used to representnumbers from zero to nine.Binary only uses two numerals 0 and 1. Place holders in binary each have avalue of powers of 2. So the first place has a value 20 = 1, the second place21 = 2, the third place 22 = 4, the fourth place 23 = 8 and so on.In binary we count 0, 1 and then since there’s no numeral for two we move ontothe next place holder so two is written as 10 binary. This is exactly the sameas when we get to ten decimal and have to write it as 10 because there’s nonumeral for ten.