Understanding Number Systems and Conversions

Number Systems Conversions

Introduction Number System Conversion

Let's dive into the fascinating world of number systems, where we'll explore different ways of representing numbers. We'll start with the familiar decimal number system, where we count in tens. Then, we'll venture into binary, where we count using only zeros and ones, and hexadecimal, where we use sixteen different symbols. Don't forget about octal, which uses eight symbols! You'll learn how to convert between these systems, unlocking the secrets of numerical representation. Whether you're a math whiz or just curious, this journey will expand your understanding of numbers and their diverse systems. Get ready to decode the language of numbers!

Number system can be converted to another number system ,the number system can be converted into following ways 

Binary To Decimal & Vice versa

Binary to Octal & Vice Versa

Binary to Hexadecimal & Vice Versa

Decimal to Octal & Vice Versa

Decimal To Hexadecimal & Vice versa 

Octal to Hexadecimal & Vice versa

Number System Conversion

Binary To Decimal Conversion

To convert a binary number to a decimal number, follow these steps:

Step 1: Write down the binary number.

Step 2: Identify the place value of each digit in the binary number. The rightmost digit represents the "ones" place, the next digit to the left represents the "twos" place, then the "fours" place, "eights" place, and so on, increasing by a power of 2 as you move from right to left.

Step 3: Multiply each digit by its corresponding place value.

Step 4: Add up all the results from Step 3 to get the decimal equivalent.

Here's an example:

Let's convert the binary number 1101 to decimal.

Step 1: The binary number is 1101.

Step 2: The place values from right to left are 1 (for ones), 2 (for twos), 4 (for fours), and 8 (for eights).

Step 3: Multiply each digit by its place value:

1 * 1 = 1 (for the ones place)

0 * 2 = 0 (for the twos place)

1 * 4 = 4 (for the fours place)

1 * 8 = 8 (for the eights place)

Step 4: Add up the results:

1 + 0 + 4 + 8 = 13

So, the binary number 1101 is equivalent to the decimal number 13.

Binary To Decimal Conversion

That's how you can convert binary numbers to decimal numbers. Remember to identify the place values and perform the appropriate multiplication for each digit in the binary number to get its decimal equivalent.

Decimal To Binary Conversion

Converting a decimal number to a binary number involves repeatedly dividing the decimal number by 2 and recording the remainders. Here are the step-by-step instructions:

Step 1 Divide the decimal number by 2.

Step 2 Record the remainder (it will be either 0 or 1).

Step 3Repeat the division with the quotient obtained in the previous step until the quotient is 0.

Step 4 The binary representation is the sequence of remainders read in reverse order.

Let's take an example: Converting 125 to binary.

To convert the decimal number 125 to binary, you can use the following steps:

Divide the number by 2.

Record the remainder.

Repeat the process with the quotient obtained in the previous step until the quotient is 0.

The binary representation is the sequence of remainders read in reverse order.

Let's apply these steps to 125:

125 ÷ 2 = 62 with a remainder of 1 (LSB - least significant bit)

62 ÷ 2 = 31 with a remainder of 0

31 ÷ 2 = 15 with a remainder of 1

15 ÷ 2 = 7 with a remainder of 1

7 ÷ 2 = 3 with a remainder of 1

3 ÷ 2 = 1 with a remainder of 1

1 ÷ 2 = 0 with a remainder of 1 (MSB - most significant bit)

Now, read the remainders in reverse order: 1111101. 

Decimal To Binary Conversion

Therefore, the binary representation of 125 is 1111101.

Octal to Decimal numbers Conversion

Octal and decimal are different numeral systems. 

Octal uses a base of 8, meaning it only has digits from 0 to 7. 

Decimal, on the other hand, uses a base of 10, with digits from 0 to 9.

To convert an octal number to a decimal number, you follow these steps:

Step 1: Write down the octal number.

Step 2: Assign a position value to each digit of the octal number, starting from the right and increasing by a power of 8 for each position to the left. The rightmost digit is in the 8^0 position (which equals 1), the next digit to the left is in the 8^1 position (which equals 8), then 8^2 (which equals 64), and so on.

Step 3: Multiply each digit of the octal number by its corresponding position value.

Step 4: Sum up all the products obtained in Step 3.

Step 5: The result is the decimal equivalent of the given octal number.

Let's take an example to illustrate the process. We'll convert the octal number 346 to its decimal equivalent:

Step 1: The octal number is 346.

Step 2: Assign position values: 6 is in the 8^0 position, 4 is in the 8^1 position, and 3 is in the 8^2 position.

Step 3: Multiply each digit by its position value:

6 * 8^0 = 6 * 1 = 6

4 * 8^1 = 4 * 8 = 32

3 * 8^2 = 3 * 64 = 192

Step 4: Add up the results: 6 + 32 + 192 = 230

Step 5: The decimal equivalent of the octal number 346 is 230.

So, the octal number 346 is equal to the decimal number 230.

Octal to Decimal numbers Conversion

Decimal to Octal Conversion: 

To convert a decimal number to octal, we can also use the process of division and remainder. 

However, instead of dividing by 2, we divide by 8 and keep track of the remainder.

We then take the quotient and repeat the process until we have no more quotient. 

The remainders will then be arranged in the reverse order to get the octal equivalent of the decimal number. 

Step 1 Divide the decimal number by 8.

Step 2 Record the remainder 

Step 3 Repeat the division with the quotient obtained in the previous step until the quotient is less then 8.

Step 4 The Octal representation is the sequence of remainders read in reverse order.

For example,

To convert the decimal number 30 to octal, we divide 30 by 8 and get a quotient of 3 with a remainder of 6.

We then take the quotient of 3 and divide it by 8 again, getting a quotient of 0 and a remainder of 3.

The remainders are arranged in reverse order to get the octal equivalent of 30, which is 36. 

Decimal to Octal Conversion:

Hexadecimals to Decimal numbers Conversion

Hexadecimal and decimal are different numeral systems.

Hexadecimal uses a base of 16, meaning it has digits from 0 to 9 and letters A to F, where A represents 10, B represents 11, and so on up to F, which represents 15. 

Decimal, as mentioned earlier, uses a base of 10, with digits from 0 to 9.

To convert a hexadecimal number to a decimal number, you follow these steps:

Step 1: Write down the hexadecimal number.

Step 2: Assign a position value to each digit of the hexadecimal number, starting from the right and increasing by a power of 16 for each position to the left. The rightmost digit is in the 16^0 position (which equals 1), the next digit to the left is in the 16^1 position (which equals 16), then 16^2 (which equals 256), and so on.

Step 3: Convert any letters in the hexadecimal number to their corresponding decimal values (A is 10, B is 11, C is 12, D is 13, E is 14, and F is 15).

Step 4: Multiply each digit of the hexadecimal number (including the converted letters) by its corresponding position value.

Step 5: Sum up all the products obtained in Step 4.

Step 6: The result is the decimal equivalent of the given hexadecimal number.

Let's take an example to illustrate the process. We'll convert the hexadecimal number 1A3 to its decimal equivalent:

Step 1: The hexadecimal number is 1A3.

Step 2: Assign position values: 3 is in the 16^0 position, A is in the 16^1 position, and 1 is in the 16^2 position.

Step 3: Convert the letter A to its decimal value, which is 10.

Step 4: Multiply each digit by its position value:
3 * 16^0 = 3 * 1 = 3
A * 16^1 = 10 * 16 = 160
1 * 16^2 = 1 * 256 = 256

Step 5: Add up the results: 3 + 160 + 256 = 419

Step 6: The decimal equivalent of the hexadecimal number 1A3 is 419.

So, the hexadecimal number 1A3 is equal to the decimal number 419.

Hexadecimals to Decimal numbers Conversion

Decimal to Hexadecimal Conversion

To convert a decimal number to hexadecimal, we also use the process of division and remainder.

However, instead of dividing by 2 or 8, we divide by 16 and keep track of the remainder.

We then take the quotient and repeat the process until we have no more quotient. 

The remainders will then be arranged in the reverse order to get the hexadecimal equivalent of the decimal number. 

Step 1 Divide the decimal number by 16.

Step 2 Record the remainder 

Step 3 Repeat the division with the quotient obtained in the previous step until the quotient is less then 16.

Step 4 The Hexadecimal representation is the sequence of remainders read in reverse order.

For example, 

to convert the decimal number 165 to hexadecimal, 

Step 1 Divide 165 by 16 and get a quotient of 10 with a remainder of 5 

Step 2 We then take the quotient of 10 and divide it by 16 again, getting a quotient of 0 and a remainder of 10(A in hexadecimal). 

The remainders are arranged in reverse order to get the hexadecimal equivalent of 165, which is A5. 

It's worth noting that when converting to hexadecimal, the remainder can be a single digit or a letter, depending on the remainder value. 

The letters A, B, C, D, E, and F are used to represent the decimal values of 10, 11, 12, 13, 14, and 15 respectively.

Decimal to Hexadecimal Conversion

Frequently Asked Question(FAQ)

How to convert Binary no to decimal no ?

To convert a binary number to a decimal number, follow these steps:
Step 1: Write down the binary number.
Step 2: Identify the place value of each digit in the binary number. The rightmost digit represents the "ones" place, the next digit to the left represents the "twos" place, then the "fours" place, "eights" place, and so on, increasing by a power of 2 as you move from right to left.
Step 3: Multiply each digit by its corresponding place value.
Step 4: Add up all the results from Step 3 to get the decimal equivalent.

How to convert Decimal no to Binary no?

Step 1 Divide the decimal number by 2.
Step 2 Record the remainder (it will be either 0 or 1).
Step 3Repeat the division with the quotient obtained in the previous step until the quotient is 0.
Step 4 The binary representation is the sequence of remainders read in reverse order.

How to convert Octal no to decimal no ?

Step 1: Write down the octal number.
Step 2: Assign a position value to each digit of the octal number, starting from the right and increasing by a power of 8 for each position to the left. The rightmost digit is in the 8^0 position (which equals 1), the next digit to the left is in the 8^1 position (which equals 8), then 8^2 (which equals 64), and so on.
Step 3: Multiply each digit of the octal number by its corresponding position value.
Step 4: Sum up all the products obtained in Step 3.
Step 5: The result is the decimal equivalent of the given octal number.

How to convert Decimal number to octal number system ?

Step 1 Divide the decimal number by 8.
Step 2 Record the remainder
Step 3 Repeat the division with the quotient obtained in the previous step until the quotient is less then 8.
Step 4 The Octal representation is the sequence of remainders read in reverse order.
.

How to convert Hexadecimal number to decimal number ?

Step 1: Write down the hexadecimal number.
Step 2: Assign a position value to each digit of the hexadecimal number, starting from the right and increasing by a power of 16 for each position to the left. The rightmost digit is in the 16^0 position (which equals 1), the next digit to the left is in the 16^1 position (which equals 16), then 16^2 (which equals 256), and so on.
Step 3: Convert any letters in the hexadecimal number to their corresponding decimal values (A is 10, B is 11, C is 12, D is 13, E is 14, and F is 15).
Step 4: Multiply each digit of the hexadecimal number (including the converted letters) by its corresponding position value.
Step 5: Sum up all the products obtained in Step 4.
Step 6: The result is the decimal equivalent of the given hexadecimal number.
.

How to convert Decimal number to Hexadecimal number ?

Step 1 Divide the decimal number by 16.
Step 2 Record the remainder
Step 3 Repeat the division with the quotient obtained in the previous step until the quotient is less then 16.
Step 4 The Hexadecimal representation is the sequence of remainders read in reverse order.

Conclusion

In conclusion , this blog cover the number and their conversion method ,In summary, i can say that how crucial number systems are, they play an important role in both technology and mathematics. This blog cover a brief description of following topics

Binary To Decimal & Vice versa

Binary to Octal & Vice Versa

Binary to Hexadecimal & Vice Versa

Decimal to Octal & Vice Versa

Decimal To Hexadecimal & Vice versa 

Octal to Hexadecimal & Vice versa

In summary, I can say that these topics are very helpful for those who pursuing BCA,PGDCA, DCA ,'O' Level Courses from different universities 

I hope this blog helps you a lot Happy learning....