Problem 6 - Discrete Mathematics with Applications

Question

Express each decimal number as required. $$676=(\quad)_{\text {eight }}$$

Step-by-Step Solution

Answer

The decimal number 676 is equivalent to the octal number 1244.

Step 1: Divide by 8

We start by dividing the given decimal number (676) by 8: \( \dfrac{676}{8} = 84 \) (remainder: 4) The first digit of the octal number is 4 (the remainder).

Step 2: Continue dividing

Now, we will divide the quotient (84) by 8 again: \( \dfrac{84}{8} = 10 \) (remainder: 4) The second digit of the octal number is 4 (the new remainder).

Step 3: Repeat until the quotient is 0

We continue dividing the most recent quotient (10) by 8: \( \dfrac{10}{8} = 1 \) (remainder: 2) The third digit of the octal number is 2 (the new remainder). Finally, the quotient from the previous division is 1, which is less than 8. So, the final digit of the octal number is 1 (the most recent quotient).

Step 4: Assemble the octal number

We can assemble the octal number by listing the remainders in reverse order. The first remainder will be the least significant digit, and the most recent quotient will be the most significant digit: \[ 676 = (1244)_8 \] So, the decimal number 676 is equivalent to the octal number 1244.

Key Concepts

Decimal to Octal Conversion

Converting a decimal number to an octal number might seem complex at first. Luckily, it's made easy through a straightforward method. You begin the conversion by repeatedly dividing the decimal number by 8.
Each time you perform the division, note the quotient and the remainder. The remainder helps form the octal digits.

• Start by dividing the decimal number with 8.
• Note the remainder. This will be part of your octal number.
• Continue dividing the resulting quotient by 8 until the quotient becomes zero.

Read all the remainders from last to first to build the octal number.
This inverse approach is important because the last calculated quotient forms the most significant digit in the octal number.

Number Base Conversion

The concept of converting numbers between different bases is essential in understanding how numbers are represented in various systems. Whether it’s converting from decimal to octal, binary to decimal, or any other systems, the core idea remains the same.
To convert a number from a higher base (like decimal) to a lower base (like octal), division is the key. You must divide the number by the base you are converting to and keep track of remainders.

• The quotient is used for further division unless it reaches zero.
• The remainders gathered along the way are the new base's digits.
• Read the gathered remainders in reverse order to form the final number in the desired base.

Grasping this process helps in deciphering how computers manage data since computers naturally operate using binary systems, which are another form of base conversion.

Octal Number System

The octal number system, or base-8, is built from eight distinct digits: 0 through 7. This system is especially handy in computing and digital electronics, often used to simplify binary notation.
In the octal system, each digit represents a power of 8, similar to how in the decimal system each digit represents a power of 10.

• The rightmost digit represents 8^0, the next represents 8^1, and so on.
• Large binary numbers can be broken into smaller, manageable octal segments.
• Because 8 is 2^3, each octal digit can map directly to three binary digits.

This feature of the octal system makes it simpler for humans to read and understand large binary numbers, which is crucial in fields where precision and clarity are necessary.