Why the method works? Change a decimal no. from denary(base 10) to octal(base 8)

Removing that we get q= c(8^{-1})+ \cdot\cdot\cdot and multiplying by 8 again gives 8q= c+ \cdot\cdot\cdot so the integer part is c. Continuing this pattern, the resulting number is 0.4152...ans: In summary, the method for converting a decimal number, less than one, to octal form is to multiply the decimal part by 8 and repeat this process until the desired precision is reached. The resulting number is 0.4152...
  • #1
Ask4material
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K A Stroud Engineering Mathematics 6th ed frame 133

change [tex]0.526_{10}[/tex] to octal(base 8) form

method: every time only the decimal part multiply by 8

[tex]\begin{array}{r}0.526\\ \times8\\\hline\\4.208\\ \times8\\\hline\\1.664\\ \times8\\\hline\\5.312\\ \times8\\\hline\\2.496\\\mbox{... goes on}\end{array}[/tex]

ans: 0.4152...
 
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  • #2
Ask4material said:
K A Stroud Engineering Mathematics 6th ed frame 133

change [tex]0.526_{10}[/tex] to octal(base 8) form

method: every time only the decimal part multiply by 8

[tex]\begin{array}{r}0.526\\ \times8\\\hline\\4.208\\ \times8\\\hline\\1.664\\ \times8\\\hline\\5.312\\ \times8\\\hline\\2.496\\\mbox{... goes on}\end{array}[/tex]

ans: 0.4152...
A number, less than one, in base 8, is of the form [itex]x= a(8^{-1})+ b(8^{-2})+ c(8^{-3})+\cdot\cdot\cdot[/itex]. Multiplying by 8 gives [itex]8x= a+ b(8^{-1})+ c(8^{-2})+\cdot\cdot\cdot[/itex] so the integer part is a. Removing that we get [itex]p= b(8^{-1})+ c(8^{-2})+ \cdot\cdot\cdot[/itex] and multiplying by 8 again gives [itex]8p= b+ c(8^{-1})+ \cdot\cdot\cdot[/itex] so the integer part is b.
 

1. Why do we use octal as a base 8 system for changing decimal numbers?

The octal system is a commonly used base 8 system in computer science and mathematics. It is used because it is a convenient way to represent binary numbers. Each octal digit corresponds to 3 binary digits, making it easier to convert between the two systems.

2. How do we convert a decimal number to octal?

To convert a decimal number to octal, we use the repeated division method. This involves dividing the decimal number by 8 and writing down the remainder. The remainder becomes the rightmost digit in the octal number. The quotient is then divided by 8 again, and the process is repeated until the quotient is 0. The octal number is the list of remainders in reverse order.

3. Can we use the same method to convert any decimal number to octal?

Yes, the repeated division method can be used to convert any decimal number to octal. It is a universal method that works for all decimal numbers, regardless of their size or value.

4. Why is it important to understand how to convert decimal numbers to octal?

Understanding how to convert decimal numbers to octal is important in computer science and programming. Many computer systems use the octal system as a way to store and represent data. Converting between decimal and octal is necessary for working with these systems and performing operations on the data they contain.

5. Are there any real-world applications for converting decimal numbers to octal?

Yes, converting decimal numbers to octal has various real-world applications in fields such as computer science, engineering, and finance. In computer science, it is used in low-level programming and data storage. In engineering, it is used in digital circuits and signal processing. In finance, it is used in currency exchange and financial calculations.

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