3. Express the following decimal fractions as a sum of fractions with the denominator as a power of 10. \( \begin{array}{llll}\text { a) } 0,23 & \text { b) } 1,303 & \text { ac ) } 4,121 & \text {-d) } 2,3103\end{array} \)

We want to write each decimal number as a sum of fractions, where each fraction’s denominator is 10 raised to a power. In other words, we will express the value in the form   (integer part) + (tenths) + (hundredths) + (thousandths) + … Let’s do this for each number. a) 0,23   Here 0,23 has no integer part. The digit 2 is in the tenths place and 3 in the hundredths place. Thus,    0,23 = 2/10 + 3/100 b) 1,303   The integer part is 1. After the decimal point, the digits are 3, 0, and 3. The 3 is in the tenths, the 0 in the hundredths (which we can omit since 0/100 = 0), and the final 3 is in the thousandths place. So we have:    1,303 = 1 + 3/10 + 0/100 + 3/1000   or simply    1,303 = 1 + 3/10 + 3/1000 c) 4,121   The integer part is 4. The digits after the decimal point are 1 (tenths), 2 (hundredths) and 1 (thousandths). Thus:    4,121 = 4 + 1/10 + 2/100 + 1/1000 d) 2,3103   The integer part is 2. The digits after the comma are 3, 1, 0, and 3. These correspond to the tenths, hundredths, thousandths, and ten-thousandths places respectively:    2,3103 = 2 + 3/10 + 1/100 + 0/1000 + 3/10000   Again, we can ignore the 0/1000:    2,3103 = 2 + 3/10 + 1/100 + 3/10000 That’s the desired representation for each decimal as a sum of fractions with denominators that are powers of 10.