Question

How many significant figures are in each measured quantity? (a) \(13001 \mathrm{~kg}\) (b) \(13111 \mathrm{~kg}\) (c) \(1.30 \times 10^{4} \mathrm{~kg}\) (d) \(0.00013 \mathrm{~kg}\)

Short answer

The number of significant figures for each quantity is as follows: (a) 13001 kg has 5 significant figures, (b) 13111 kg has 5 significant figures, (c) 1.30 x 10^4 kg has 3 significant figures, and (d) 0.00013 kg has 2 significant figures.

Step-by-step solution

Analyze Quantity (a) - 13001 kg

To determine the number of significant figures, identify all the non-zero digits and any zeros between them as significant. For 13001 kg, all the digits are significant because the zeros are between non-zero numbers. Thus, there are five significant figures.

Analyze Quantity (b) - 13111 kg

Similarly, for 13111 kg, all the digits are significant, including the ones and the middle ones. Therefore, there are five significant figures in this measurement.

Analyze Quantity (c) - 1.30 x 10^4 kg

For scientific notation like 1.30 x 10^4 kg, the significant figures are in the coefficient (1.30). Here, both the 1 and the 3 are significant, and the zero is significant because it is to the right of the decimal and between non-zero digits. Thus, there are three significant figures.

Analyze Quantity (d) - 0.00013 kg

For 0.00013 kg, only the non-zero digits are significant. The leading zeros are not significant because they only indicate the position of the decimal. Therefore, there are two significant figures.

Key concepts

Measurement Precision

Understanding measurement precision is vital in the world of science and mathematics. It refers to the extent to which a given measurement can be reproduced with consistency. The precision of a measurement is determined by the smallest unit to which it can be measured reliably, and it is deeply connected to the concept of significant figures.

Significant figures are important because they tell us how exact a measurement is. For instance, when you see a number like 13001 kg, each digit is considered 'significant' because they represent the precision of the measurement. If a number is measured with high precision, there will be more significant figures, indicating a more detailed level of measurement. Remember, trailing zeros in a whole number with no decimal point are not considered significant, as they could just be placeholders.

Scientific Notation

Scientific notation is a method of expressing numbers that are too large or too small to be conveniently written in decimal form. It is often used in science and engineering to simplify measurements and calculations.

Numbers are written as the product of two parts: a coefficient (usually a value between 1 and 10) and a power of 10. For example, the number 1.30 x 10⁴ kg is written in scientific notation. In terms of significant figures, the 'significant' part of the measurement is the coefficient, 1.30. Here, each digit in the coefficient is significant because they directly measure the quantity in question. As a quick tip, if the coefficient has a decimal point, then all zeros after non-zero digits are important - they count as significant figures.

Non-Zero Digits

In the realm of significant figures, non-zero digits are always counted as significant. This is because they are essential in representing the actual measured or estimated value.

Let's take 0.00013 kg for example. Here, the non-zero digits, 1 and 3, are significant while the zeros preceding them are not. Those leading zeros serve only as placeholders to indicate the decimal point's position, not the measurement's precision. As a takeaway, pay careful attention to where the non-zero digits lie in your measurement, as they are crucial in determining the level of precision and accuracy in your results.