Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Nov 01, 2023

Convert the given base 4 number $12131_{\text{four}}$ to base 10.

STEP 1

Assumptions1. The given numeral is in base four. . We want to convert it to base ten.

STEP 2

To convert a number from base four to base ten, we can use the formula for base conversion which is $\sum_{i=0}^{n} a_i \cdot b^i$ where $a_i$ is the digit at the $i$ -th position (counting from right to left, starting from0), $b$ is the base of the number (in this case,4), and $n$ is the number of digits in the number minus1.

STEP 3

Now, let's apply this formula to our number,12131 in base four.
The number has five digits, so $n =5 -1 =$ .
The digits, from right to left, are1,3,1,2, and1. So, $a0 =1$ , $a1 =3$ , $a2 =1$ , $a3 =2$ , and $a =1$ .

STEP 4

Now, we substitute these values into the formula and calculate the sum.
$\sum_{i=0}^{4} a_i \cdot4^i =1 \cdot4^0 +3 \cdot4^1 +1 \cdot4^2 +2 \cdot4^3 +1 \cdot4^4$

STEP 5

implify the expression.
$1 \cdot1 +3 \cdot4 +1 \cdot16 +2 \cdot64 +1 \cdot256$

STEP 6

Calculate the sum.
$1 +12 +16 +128 +256 =413$ So, the number $12131_{\text {four }}$ is equal to $413$ in base ten.

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