Understanding the Significance of Exact Numbers- How Many Significant Figures Do They Possess-

How Many Significant Figures Do Exact Numbers Have?

Exact numbers are numbers that are known without any uncertainty or approximation. They are often derived from counting or from a defined system, such as the number of protons in an atom or the number of degrees in a right angle. Unlike measured numbers, which have a degree of uncertainty due to the limitations of measurement tools, exact numbers are considered to have an infinite number of significant figures.

Significant figures are a way to express the precision of a number. In the context of exact numbers, this concept might seem unnecessary, as these numbers are not subject to the same degree of uncertainty as measured numbers. However, the concept of significant figures is still useful when dealing with exact numbers in mathematical operations and when communicating results.

For example, consider the number of atoms in a mole, which is defined as 6.02214076 × 10^23. This number is exact because it is a fundamental constant in chemistry. However, when performing calculations with this number, it is common to round it to a certain number of significant figures. This is done to maintain consistency and to avoid presenting an unnecessary level of precision that is not supported by the original value.

When dealing with exact numbers, the following rules apply:

1. All non-zero digits are significant.
2. Any zeros between non-zero digits are significant.
3. Leading zeros (zeros to the left of the first non-zero digit) are not significant.
4. Trailing zeros (zeros to the right of the last non-zero digit) are significant if they are after a decimal point.

For instance, the number 100 has two significant figures because the zeros are not between non-zero digits. On the other hand, the number 0.050 has three significant figures, as the zeros between the non-zero digits are significant.

In conclusion, while exact numbers inherently have an infinite number of significant figures, the concept of significant figures is still relevant when performing calculations or communicating results involving these numbers. Rounding exact numbers to a specific number of significant figures helps maintain clarity and consistency in scientific and mathematical work.