Question

How many significant figures are in each of the following? a. 100 b. \(1.0 \times 10^{2}\) c. \(1.00 \times 10^{3}\) d. 100 . e. \(0.0048\) f. \(0.00480\) g. \(4.80 \times 10^{-3}\) h. \(4.800 \times 10^{-3}\)

Short answer

a. 1 significant figure b. 2 significant figures c. 3 significant figures d. 3 significant figures e. 2 significant figures f. 3 significant figures g. 3 significant figures h. 4 significant figures

Step-by-step solution

a. 100

This number has trailing zeros with no decimal point, so the trailing zeros do not count as significant. Thus, there is only 1 significant digit: "1".

b. \(1.0 \times 10^{2}\)

There is a non-zero digit (1) and a trailing zero with a decimal point (0), making 2 significant figures: "1.0".

c. \(1.00 \times 10^{3}\)

There is a non-zero digit (1) and two trailing zeros with decimal points (0, 0), making 3 significant figures: "1.00".

d. 100 .

This number has trailing zeros and a decimal point, which makes the trailing zeros significant. There are 3 significant digits: "100".

e. 0.0048

There are leading zeros, which are not significant, and two non-zero digits (4, 8). There are 2 significant figures: "4.8".

f. 0.00480

There are leading zeros, which are not significant, and two non-zero digits (4, 8) followed by a trailing zero with a decimal point (0). There are 3 significant figures: "4.80".

g. \(4.80 \times 10^{-3}\)

There are two non-zero digits (4, 8), and a trailing zero with a decimal point (0). There are 3 significant figures: "4.80".

h. \(4.800 \times 10^{-3}\)

There are two non-zero digits (4, 8), and two trailing zeros with decimal points (0, 0). There are 4 significant figures: "4.800".

Key concepts

Trailing Zeros

Trailing zeros are the zeros that appear at the end of a number. It's important to understand when these zeros are considered significant.

• If a number has trailing zeros but no decimal point, these zeros are not counted as significant. For example, in the number 100, there is only one significant figure because the zeros do not count without a decimal point.
• However, when a decimal point is present, trailing zeros do become significant. For instance, 100. has three significant figures since the decimal indicates the trailing zeros are important.

Remember, the presence or absence of a decimal point changes the significance of trailing zeros drastically.

Decimal Point

A decimal point is the dot used to separate the whole number part from the fractional part of a number. Its placement is critical in determining how many significant figures a number has.

• Numbers like 100. have a decimal point, which makes the trailing zeros significant, giving the number 3 significant figures.
• On the other hand, 100 without a decimal point has only 1 significant figure because the trailing zeros are not significant.

The decimal point clarifies which digits are measured accurately, thus affecting the precision depicted in the number's significant figures.

Scientific Notation

Scientific notation is a way to express large or small numbers easily, with a coefficient and a power of ten. It is particularly helpful for indicating significant figures clearly.

• For instance, in the number \(1.0 \times 10^{2}\), the trailing zero is significant because of the decimal point in the coefficient. This number has 2 significant figures.
• Similarly, \(4.800 \times 10^{-3}\) indicates 4 significant figures due to the inclusion of trailing zeros in the decimal format.

Using scientific notation simplifies reading and writing numbers while precisely showing which digits are meaningful.

Leading Zeros

Leading zeros are the zeros that appear at the beginning of a number. They are not significant and are simply placeholders.

• For example, in the number 0.0048, the first three zeros are leading zeros and do not count as significant figures.
• The significant figures here are only the 4 and the 8, making it a total of 2 significant figures.

Leading zeros only help position the decimal point correctly and do not contribute to the precision of the measurement indicated by the number.