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Understanding Significant Figures

Significant figures (often abbreviated as sig figs) are the digits in a number that carry meaningful contributions to its measurement resolution. They are crucial in science, engineering, and mathematics for expressing the precision of a measurement or calculation. When you perform calculations, the number of significant figures in your answer should reflect the precision of the least precise measurement used in the calculation.

Why are Significant Figures Important?

Imagine you measure a length with a ruler marked in millimeters and get 12.5 cm. If you then report it as 12.500 cm, you're implying a precision that your ruler doesn't actually provide. Significant figures help us avoid misrepresenting the certainty of our data. They ensure that calculated results do not suggest a greater precision than the original measurements.

Rules for Counting Significant Figures

Here are the universally accepted rules for determining the number of significant figures in a given number:

1. Non-zero digits are always significant.
   Example: 23.45 has 4 significant figures.
   Example: 123 has 3 significant figures.
2. Zeros between non-zero digits are significant. (These are sometimes called "sandwich zeros" or "captive zeros").
   Example: 1005 has 4 significant figures.
   Example: 2.03 has 3 significant figures.
3. Leading zeros (zeros before non-zero digits) are NOT significant. They only indicate the position of the decimal point.
   Example: 0.0025 has 2 significant figures (the 2 and the 5).
   Example: 0.5 has 1 significant figure (the 5).
4. Trailing zeros (zeros at the end of the number) are significant ONLY if the number contains a decimal point.
   Example: 12.00 has 4 significant figures (the 1, 2, and both zeros).
   Example: 1.20 x 10^3 has 3 significant figures (the 1, 2, and 0 in the mantissa).
5. Trailing zeros in a number without a decimal point are ambiguous and generally considered NOT significant.
   Example: 1200 is usually considered to have 2 significant figures (the 1 and the 2). To make the trailing zeros significant, you would write it as 1200. (4 sig figs) or in scientific notation like 1.200 x 10^3 (4 sig figs) or 1.20 x 10^3 (3 sig figs).
6. Exact numbers have an infinite number of significant figures. These are numbers obtained by counting (e.g., 12 eggs, 5 chairs) or by definition (e.g., 1 inch = 2.54 cm exactly). They do not limit the number of significant figures in a calculation.

Examples of Counting Significant Figures:

• 45.879: 5 significant figures (all non-zero).
• 100.5: 4 significant figures (zeros between non-zeros are significant).
• 0.0032: 2 significant figures (leading zeros are not significant).
• 15.00: 4 significant figures (trailing zeros with a decimal are significant).
• 1500: 2 significant figures (trailing zeros without a decimal are not significant by convention).
• 1500.: 4 significant figures (the decimal point makes trailing zeros significant).
• 1.50 x 10^4: 3 significant figures (all digits in the mantissa are significant).
• 0: 1 significant figure (special case for the number zero).

Use the calculator above to quickly determine the number of significant figures for any number you enter, along with a step-by-step explanation of how the count was derived based on these rules.