State The Equivalent Binary Number Of These Octal And Decimal Numbers:1. \[$1,776_8\$\]2. \[$145_8\$\]3. \[$90_{10}\$\]4. \[$78_{10}\$\]

**Converting Octal and Decimal Numbers to Binary: A Comprehensive Guide**

What is Octal and Decimal Number System?

The octal number system is a base-8 number system that uses eight distinct symbols or digits: 0, 1, 2, 3, 4, 5, 6, and 7. It is commonly used in computing and programming to represent file permissions and other binary data. On the other hand, the decimal number system is a base-10 number system that uses ten distinct symbols or digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

What is Binary Number System?

The binary number system is a base-2 number system that uses two distinct symbols or digits: 0 and 1. It is the most fundamental number system used in computing and digital electronics. All computer programming languages and digital devices use binary code to process and store information.

Converting Octal and Decimal Numbers to Binary

Converting octal and decimal numbers to binary is a simple process that involves converting each digit of the octal or decimal number to its binary equivalent. Here's a step-by-step guide on how to do it:

Converting Octal Numbers to Binary

To convert an octal number to binary, you need to convert each digit of the octal number to its binary equivalent. Here's how to do it:

• Convert each digit of the octal number to its binary equivalent using the following table:

  Octal Digit	Binary Equivalent
  0	000
  1	001
  2	010
  3	011
  4	100
  5	101
  6	110
  7	111

• Write down the binary equivalent of each digit of the octal number.

• Combine the binary equivalents of each digit to get the final binary number.

Converting Decimal Numbers to Binary

To convert a decimal number to binary, you need to divide the decimal number by 2 and keep track of the remainders. Here's how to do it:

• Divide the decimal number by 2 and write down the remainder.
• Divide the quotient by 2 and write down the remainder.
• Repeat the process until the quotient is 0.
• Write down the remainders in reverse order to get the final binary number.

Examples of Converting Octal and Decimal Numbers to Binary

Let's take some examples to illustrate the process of converting octal and decimal numbers to binary.

Example 1: Converting $1,776_8$ to Binary

To convert $1,776_8$ to binary, we need to convert each digit of the octal number to its binary equivalent.

• $1_8$ = 001
• $7_8$ = 111
• $7_8$ = 111
• $6_8$ = 110

Combining the binary equivalents of each digit, we get:

$001 111 111 110$

Example 2: Converting $145_8$ to Binary

To convert $145_8$ to binary, we need to convert each digit of the octal number to its binary equivalent.

• $1_8$ = 001
• $4_8$ = 100
• $5_8$ = 101

Combining the binary equivalents of each digit, we get:

$001 100 101$

Example 3: Converting $90_{10}$ to Binary

To convert $90_{10}$ to binary, we need to divide the decimal number by 2 and keep track of the remainders.

• $90 \div 2$ = 45 remainder 0
• $45 \div 2$ = 22 remainder 1
• $22 \div 2$ = 11 remainder 0
• $11 \div 2$ = 5 remainder 1
• $5 \div 2$ = 2 remainder 1
• $2 \div 2$ = 1 remainder 0
• $1 \div 2$ = 0 remainder 1

Writing down the remainders in reverse order, we get:

$1011010$

Example 4: Converting $78_{10}$ to Binary

To convert $78_{10}$ to binary, we need to divide the decimal number by 2 and keep track of the remainders.

• $78 \div 2$ = 39 remainder 0
• $39 \div 2$ = 19 remainder 1
• $19 \div 2$ = 9 remainder 1
• $9 \div 2$ = 4 remainder 1
• $4 \div 2$ = 2 remainder 0
• $2 \div 2$ = 1 remainder 0
• $1 \div 2$ = 0 remainder 1

Writing down the remainders in reverse order, we get:

$1001110$

Conclusion

Converting octal and decimal numbers to binary is a simple process that involves converting each digit of the octal or decimal number to its binary equivalent. By following the steps outlined in this article, you can easily convert octal and decimal numbers to binary. Whether you're a student, a programmer, or a digital electronics enthusiast, this guide will help you understand the process of converting octal and decimal numbers to binary.

Frequently Asked Questions

Q: What is the difference between octal and decimal number system?

A: The octal number system is a base-8 number system that uses eight distinct symbols or digits: 0, 1, 2, 3, 4, 5, 6, and 7. The decimal number system is a base-10 number system that uses ten distinct symbols or digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Q: How do I convert an octal number to binary?

A: To convert an octal number to binary, you need to convert each digit of the octal number to its binary equivalent using the following table:

Octal Digit	Binary Equivalent
0	000
1	001
2	010
3	011
4	100
5	101
6	110
7	111

Q: How do I convert a decimal number to binary?

A: To convert a decimal number to binary, you need to divide the decimal number by 2 and keep track of the remainders. Write down the remainders in reverse order to get the final binary number.

Q: What is the binary equivalent of the octal number $1,776_8$?

A: The binary equivalent of the octal number $1,776_8$ is $001 111 111 110$.

Q: What is the binary equivalent of the decimal number $90_{10}$?

A: The binary equivalent of the decimal number $90_{10}$ is $1011010$.

Q: What is the binary equivalent of the decimal number $78_{10}$?

A: The binary equivalent of the decimal number $78_{10}$ is $1001110$.