State The Equivalent Binary Number Of The Following Octal And Decimal Numbers:I. $1,776_8$II. $145_8$III. $90_{10}$IV. $78_{10}$

In this article, we will explore the process of converting octal and decimal numbers to their equivalent binary representations. We will examine four different numbers: $1,776_8$, $145_8$, $90_{10}$, and $78_{10}$. Our goal is to convert each of these numbers into their binary equivalents.

Understanding Octal and Decimal Numbers

Before we dive into the conversions, let's briefly review what octal and decimal numbers are.

• Octal Numbers: An octal number is a base-8 number system that uses eight distinct symbols: 0, 1, 2, 3, 4, 5, 6, and 7. Octal numbers are commonly used in computer programming and are often represented using the subscript 8.
• Decimal Numbers: A decimal number is a base-10 number system that uses ten distinct symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Decimal numbers are the most commonly used number system in everyday life.

Converting Octal Numbers to Binary

To convert an octal number to binary, we need to understand the relationship between the two number systems. In the octal system, each digit can have one of eight values (0-7), while in the binary system, each digit can have one of two values (0-1).

Here's a step-by-step guide to converting an octal number to binary:

1. Split the Octal Number: Split the octal number into its individual digits.

2. Convert Each Digit: Convert each octal digit 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

3. Combine the Binary Digits: Combine the binary digits from each step to form the final binary representation.

Converting Decimal Numbers to Binary

To convert a decimal number to binary, we can use a similar approach to the one used for octal numbers. However, since decimal numbers are base-10, we need to convert each decimal digit to its binary equivalent using the following table:

Decimal Digit	Binary Equivalent
0	000
1	001
2	010
3	011
4	100
5	101
6	110
7	111
8	1000
9	1001

Here's a step-by-step guide to converting a decimal number to binary:

1. Split the Decimal Number: Split the decimal number into its individual digits.
2. Convert Each Digit: Convert each decimal digit to its binary equivalent using the table above.
3. Combine the Binary Digits: Combine the binary digits from each step to form the final binary representation.

Converting $1,776_8$ to Binary

Let's apply the steps above to convert $1,776_8$ to binary.

1. Split the Octal Number: Split $1,776_8$ into its individual digits: 1, 7, 7, and 6.

2. Convert Each Digit: Convert each octal digit to its binary equivalent using the table above:

   • 1 | 001
   • 7 | 111
   • 7 | 111
   • 6 | 110

3. Combine the Binary Digits: Combine the binary digits from each step to form the final binary representation:

   001 111 111 110

The binary equivalent of $1,776_8$ is 001 111 111 110.

Converting $145_8$ to Binary

Let's apply the steps above to convert $145_8$ to binary.

1. Split the Octal Number: Split $145_8$ into its individual digits: 1, 4, and 5.

2. Convert Each Digit: Convert each octal digit to its binary equivalent using the table above:

   • 1 | 001
   • 4 | 100
   • 5 | 101

3. Combine the Binary Digits: Combine the binary digits from each step to form the final binary representation:

   001 100 101

The binary equivalent of $145_8$ is 001 100 101.

Converting $90_{10}$ to Binary

Let's apply the steps above to convert $90_{10}$ to binary.

1. Split the Decimal Number: Split $90_{10}$ into its individual digits: 9 and 0.

2. Convert Each Digit: Convert each decimal digit to its binary equivalent using the table above:

   • 9 | 1001
   • 0 | 000

3. Combine the Binary Digits: Combine the binary digits from each step to form the final binary representation:

   1001 000

The binary equivalent of $90_{10}$ is 1001 000.

Converting $78_{10}$ to Binary

Let's apply the steps above to convert $78_{10}$ to binary.

1. Split the Decimal Number: Split $78_{10}$ into its individual digits: 7 and 8.

2. Convert Each Digit: Convert each decimal digit to its binary equivalent using the table above:

   • 7 | 0111
   • 8 | 1000

3. Combine the Binary Digits: Combine the binary digits from each step to form the final binary representation:

   0111 1000

The binary equivalent of $78_{10}$ is 0111 1000.

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In this article, we will address some of the most frequently asked questions related to converting octal and decimal numbers to binary.

Q: What is the difference between octal and decimal numbers?

A: Octal numbers are base-8 numbers that use eight distinct symbols (0-7), while decimal numbers are base-10 numbers that use ten distinct symbols (0-9).

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

A: To convert an octal number to binary, you need to split the octal number into its individual digits and then convert each digit 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 split the decimal number into its individual digits and then convert each digit to its binary equivalent using the following table:

Decimal Digit	Binary Equivalent
0	000
1	001
2	010
3	011
4	100
5	101
6	110
7	111
8	1000
9	1001

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

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

Q: What is the binary equivalent of $145_8$?

A: The binary equivalent of $145_8$ is 001 100 101.

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

A: The binary equivalent of $90_{10}$ is 1001 000.

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

A: The binary equivalent of $78_{10}$ is 0111 1000.

Q: Can I use a calculator to convert octal and decimal numbers to binary?

A: Yes, you can use a calculator to convert octal and decimal numbers to binary. Most calculators have a built-in function to convert numbers between different bases.

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

A: To convert a binary number to octal or decimal, you need to split the binary number into its individual digits and then convert each digit to its octal or decimal equivalent using the following tables:

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

Q: What is the octal equivalent of $1010_2$?

A: The octal equivalent of $1010_2$ is 12.

Q: What is the decimal equivalent of $1010_2$?

A: The decimal equivalent of $1010_2$ is 10.

In conclusion, we have addressed some of the most frequently asked questions related to converting octal and decimal numbers to binary. By understanding the relationship between the octal, decimal, and binary number systems, you can easily convert numbers between these systems.