To convert the decimal number 513 to binary, octal, and hexadecimal, we follow the respective conversion methods.

1. Conversion to Binary

To convert a decimal number to binary, we divide the number by 2 and record the remainder. We continue dividing the quotient by 2 until we reach a quotient of 0. The binary representation is the sequence of remainders read from bottom to top.

Steps:

1. 513 ÷ 2 = 256 remainder 1
2. 256 ÷ 2 = 128 remainder 0
3. 128 ÷ 2 = 64 remainder 0
4. 64 ÷ 2 = 32 remainder 0
5. 32 ÷ 2 = 16 remainder 0
6. 16 ÷ 2 = 8 remainder 0
7. 8 ÷ 2 = 4 remainder 0
8. 4 ÷ 2 = 2 remainder 0
9. 2 ÷ 2 = 1 remainder 0
10. 1 ÷ 2 = 0 remainder 1

Reading the remainders from bottom to top, the binary representation of 513 is: $513_{10} = 1000000001_2$

2. Conversion to Octal

To convert a decimal number to octal, we divide the number by 8 and record the remainder. We continue dividing the quotient by 8 until we reach a quotient of 0. The octal representation is the sequence of remainders read from bottom to top.

Steps:

1. 513 ÷ 8 = 64 remainder 1
2. 64 ÷ 8 = 8 remainder 0
3. 8 ÷ 8 = 1 remainder 0
4. 1 ÷ 8 = 0 remainder 1

Reading the remainders from bottom to top, the octal representation of 513 is: $513_{10} = 1001_8$

3. Conversion to Hexadecimal

To convert a decimal number to hexadecimal, we divide the number by 16 and record the remainder. We continue dividing the quotient by 16 until we reach a quotient of 0. The hexadecimal representation is the sequence of remainders read from bottom to top. Note that for remainders 10-15, we use the letters A-F respectively.

Steps:

1. 513 ÷ 16 = 32 remainder 1
2. 32 ÷ 16 = 2 remainder 0
3. 2 ÷ 16 = 0 remainder 2

Reading the remainders from bottom to top, the hexadecimal representation of 513 is: $513_{10} = 201_{16}$

Summary

• Binary: $513_{10} = 1000000001_2$
• Octal: $513_{10} = 1001_8$
• Hexadecimal: $513_{10} = 201_{16}$

Would you like further details or have any questions?

Here are 8 related questions to expand your understanding:

1. How do you convert a binary number to decimal?
2. What is the hexadecimal representation of the binary number 1000000001?
3. How do you convert an octal number to binary?
4. What is the octal representation of the hexadecimal number 201?
5. How can you verify the correctness of a binary to decimal conversion?
6. What is the binary representation of the octal number 1001?
7. How do you convert a hexadecimal number to decimal?
8. What are some practical applications of number system conversions?

Tip: When converting between number systems, remember that each system's base determines the number of unique digits or symbols it uses (e.g., binary uses 0 and 1, octal uses 0-7, and hexadecimal uses 0-9 and A-F).