If 54$_{ten}$ = X$_{four}$, find the value of X to 3 decimal point

Analysis

The question asks us to convert a number from base ten (decimal) to base four. We are given the value $54_{10}$ and need to find its equivalent in base four.

Concept: Successive Division

To convert a whole number from base ten to any other base, we use the successive division method. We divide the number by the target base (in this case, 4) repeatedly and record the remainders. The base-four representation is then written by reading the remainders from the last one obtained to the first (bottom to top).

Step-by-Step Solution

1. Divide 54 by 4: $54 \div 4 = 13$ remainder 2
2. Divide 13 by 4: $13 \div 4 = 3$ remainder 1
3. Divide 3 by 4: $3 \div 4 = 0$ remainder 3

Now, we arrange the remainders starting from the last division result (bottom) to the first (top):

• Last remainder: 3
• Middle remainder: 1
• First remainder: 2

Putting these together, we get $312_4$.

(Note: The phrase "to 3 decimal point" in the question appears to be a typo, as we are dealing with whole numbers and base conversion. The calculation clearly results in a three-digit integer in base four.)

Verification

We can check our answer by converting $312_4$ back to base ten: $(3 \times 4^2) + (1 \times 4^1) + (2 \times 4^0)$ $= (3 \times 16) + (1 \times 4) + (2 \times 1)$ $= 48 + 4 + 2$ $= 54_{10}$ The calculation is correct.

Exam Tip

When converting from base ten to another base, always remember that your remainders must be smaller than the base you are converting to. If you ever get a remainder equal to or greater than the base (e.g., a remainder of 4 or 5 when dividing by 4), you know you have made a calculation error!