Problem 48 - Thinking Mathematically

Question

Convert each base ten numeral to a numeral in the given base. 1346 to base eight

Step-by-Step Solution

Answer

The base eight equivalent of the numeral 1346 in base ten is 2502.

Step 1: Initial Division

Start by dividing the given base ten numeral 1346 by 8. The quotient is 168 and the remainder is 2. Write down the remainder.

Step 2: Subsequent Divisions

Divide the resulting quotient (168) by 8. The new quotient is 21 and the new remainder is 0. Write down this remainder next to the last one (right to left). Continue this process until zero quotient is obtained.

Step 3: Final Division

The final quotient is 2 when divided by the 8 gives a quotient of 0 and the remainder is 2. Write down this remainder too.

Step 4: Compile Remainders

Reading the remainders in reverse order (bottom to top), the base 8 equivalent of the base 10 numeral, 1346, is obtained.

Key Concepts

Base Ten

The Base Ten numeral system, also known as the decimal system, is the most commonly used numbering system worldwide. It is a positional notation system that uses ten distinct digits, from 0 to 9.
The value of each digit depends on its position within the number, which represents powers of ten.
For example, in the number 1346, the digit 1 is in the thousand's place, representing \(1 \times 10^3\), while 3 is in the hundred's place, representing \(3 \times 10^2\).

• This system is intuitive and simple, making it the default for everyday counting and basic arithmetic.
• Understanding Base Ten is crucial when learning to convert numbers to other numeral systems.

Base Eight

The Base Eight numeral system, also known as octal, uses eight different symbols: 0 through 7. Every position in a base eight number represents a power of eight, rather than ten as in the decimal system.
This system is particularly useful in computing, as it can represent binary numbers more succinctly.

• In Base Eight, the decimal number 1346 converts to 2462, following a process of successive division and remainder compilation.
• Each place in a Base Eight number signifies a power of 8, for instance, the numeral 2462 is calculated as follows: \(2 \times 8^3 + 4 \times 8^2 + 6 \times 8^1 + 2 \times 8^0\).

Remainders

Remainders play a crucial role in converting numbers between different bases. In this context, a remainder is what is left over after division. When converting a number from Base Ten to another base, such as Base Eight, remainders determine each digit of the new numeral.
The process involves:

• Successive division by the base you are converting to. For instance, 1346 divided by 8 gives a quotient of 168 and a remainder of 2.
• The remainders are gathered and used to build the new number from the bottom up. This is because the least significant digit (rightmost) is found first.

Through this method, the remainders collected are 2, 0, 6, and 2, which, when read in reverse order, give the base 8 numeral.

Numeral Systems

Numeral Systems are methods to express numbers, and they can vary based on the number of different digits they employ. The most familiar is Base Ten, but there are many others like Base Two (binary) and Base Eight (octal), each suited for different tasks and applications.

• Each system has its unique set of digits and a base which declares how many digits it uses.
• Understanding various numeral systems allows us to convert numbers from one form to another, a skill essential in areas like computer science.
• For conversions, understanding the logic of division and remainders is fundamental.

Mastering numeral systems enhances problem-solving skills and expands comprehension of numerical representation.