When calculating the antilogarithm of a number, it is essential to understand the concept of significant figures, particularly focusing on the mantissa, which is the part of the number after the decimal point. The number of significant figures in the final answer corresponds to the number of digits in the mantissa of the original number.
For example, when taking the antilog of -4.18, the mantissa is 0.18, which contains 2 digits. Therefore, the final answer must also have 2 significant figures. The antilog can be expressed mathematically as:
$$\text{antilog}(x) = 10^x$$
Calculating the antilog of -4.18 gives:
$$\text{antilog}(-4.18) = 10^{-4.18} \approx 6.6069 \times 10^{-5}$$
However, since we need to round this to 2 significant figures, the final answer is:
$$6.6 \times 10^{-5}$$
In another example, consider the expression \(10^{0.0033}\). Here, the mantissa is 0.0033, which has 4 digits. Thus, the final answer must have 4 significant figures. Performing the calculation yields:
$$10^{0.0033} \approx 1.0076$$
Rounding this to 4 significant figures results in:
$$1.008$$
Understanding the distinction between the characteristic and mantissa is crucial, especially in analytical chemistry, where precision in calculations, including uncertainties, is vital. This knowledge will significantly aid in achieving accurate results in various calculations.