Problem 3 - A Gentle Introduction to the Art of Mathematics

Question

The octal representation of an integer uses powers of 8 in place notation. The digits of an octal number run from 0 to 7 , one never sees 8 's or 9 's. How would you represent 8 and 9 as octal numbers? What octal number comes immediately after \(777_{8}\) ? What (decimal) number is \(777_{8}\) ?

Step-by-Step Solution

Answer

8 is 10 in octal, 9 is 11 in octal. The number following 777 in octal is 1000. The decimal equivalent of 777 in octal is 511.

Step 1: Understanding Octal Representation

Octal numbers use base-8, which means each digit represents a power of 8. The digits range from 0 to 7.

Step 2: Representing the Number 8 in Octal

Since the base is 8, the number 8 in decimal is represented as the next higher place value in octal. Thus, 8 in octal is represented as 10 (which is 1*8 + 0*1).

Step 3: Representing the Number 9 in Octal

The number 9 in decimal is represented in octal as 11 (which is 1*8 + 1*1).

Step 4: Finding the Octal Number After 777

The octal number 777 is the highest 3-digit octal number (7*8^2 + 7*8^1 + 7*8^0). The next number would involve carrying over to a higher place value. Therefore, the octal number immediately following 777 is 1000.

Step 5: Converting the Octal Number 777 to Decimal

To convert 777 from octal to decimal, use the formula: 777_{8} = 7*8^2 + 7*8^1 + 7*8^0 = 7*64 + 7*8 + 7*1 = 448 + 56 + 7 = 511.

Key Concepts

base-8 number system

The base-8 number system, also known as the octal number system, is a positional numeral system that uses eight different symbols: 0, 1, 2, 3, 4, 5, 6, and 7. Because it is base-8, each place value in an octal number represents a power of 8.
The place values in a base-8 number from right to left are:

• 8⁰ (units)
• 8¹ (eights)
• 8² (sixty-fours)
• ... and so on.

In an octal number, you will never see the digits 8 or 9, as these are not part of the base-8 system. For example, the decimal number 8 is expressed as 10 in octal because it signifies one group of 8 and no units (1*8 + 0*1).

converting octal to decimal

Converting numbers from octal to decimal is straightforward once you understand the positional values of the digits. Each digit in an octal number is multiplied by its place value (a power of 8) and then summed to get the equivalent decimal number.
For example, to convert the octal number 777 to decimal:

• Take each digit of 777: 7, 7, and 7.
• Multiply them by their positional values: 7 * 8², 7 * 8¹, and 7 * 8⁰.
• Calculate these values: 7 * 64 = 448, 7 * 8 = 56, and 7 * 1 = 7.
• Add the results: 448 + 56 + 7 = 511.

Thus, 777 in octal is equal to 511 in decimal.

number representation

In the octal number system, the digits range from 0 to 7. Because of this limitation, numbers like 8 and 9 cannot be represented directly using a single digit.
So, to represent the decimal number 8 in octal, you use 10. This is because it signifies that there's one group of 8 and no remaining units (1*8 + 0*1).
Similarly, the decimal number 9 is represented as 11 in octal. This means one group of 8 and one unit (1*8 + 1*1).
The highest 3-digit number in octal is 777. When you add 1 to 777, it carries over each place value, resulting in 1000 in octal. This is similar to how adding 1 to 999 in decimal results in 1000. Hence, the octal number immediately following 777 is 1000.