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QuestionConvert 52.4852.4_{8} (octal) to decimal (base ten).

Studdy Solution

STEP 1

Assumptions1. The number given is 52.4852.4_{8}, which is in base eight (octal). . We need to convert this number to base ten (decimal).

STEP 2

To convert a number from base eight to base ten, we need to multiply each digit of the number by 8n8^n, where nn is the position of the digit from right to left, starting from0.
The formula for conversion isDecimal=d0×80+d1×81+d2×82+...+dn×8nDecimal = d0 \times8^0 + d1 \times8^1 + d2 \times8^2 + ... + d_n \times8^nWhere dnd_n is the nth digit of the octal number.

STEP 3

First, we separate the integer part and the fractional part of the octal number.
The integer part is52 and the fractional part is.

STEP 4

We convert the integer part from octal to decimal.
Using the formula, we getDecimalinteger=2×80+×81Decimal_{integer} =2 \times8^0 + \times8^1

STEP 5

Calculate the decimal equivalent of the integer part.
Decimalinteger=2×1+5×8=2+40=42Decimal_{integer} =2 \times1 +5 \times8 =2 +40 =42

STEP 6

Next, we convert the fractional part from octal to decimal.
The formula for converting the fractional part is similar to the integer part, but we use negative powers of8.
Decimalfraction=d−1×8−1+d−2×8−2+...+d−n×8−nDecimal_{fraction} = d_{-1} \times8^{-1} + d_{-2} \times8^{-2} + ... + d_{-n} \times8^{-n}Where d−nd_{-n} is the nth digit of the fractional part of the octal number.

STEP 7

Apply the formula to the fractional part of the octal number.
Decimalfraction=4×−1Decimal_{fraction} =4 \times^{-1}

STEP 8

Calculate the decimal equivalent of the fractional part.
Decimalfraction=4×0.125=0.5Decimal_{fraction} =4 \times0.125 =0.5

STEP 9

Now, we add the decimal equivalents of the integer part and the fractional part to get the decimal equivalent of the octal number.
Decimal=Decimalinteger+DecimalfractionDecimal = Decimal_{integer} + Decimal_{fraction}

STEP 10

Plug in the values for the decimal equivalents of the integer part and the fractional part.
Decimal=42+0.5Decimal =42 +0.5

STEP 11

Calculate the decimal equivalent of the octal number.
Decimal=42+0.5=42.5Decimal =42 +0.5 =42.5So, the number 52.4852.4_{8} in base eight (octal) is equivalent to 42.542.5 in base ten (decimal).

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