Sig Fig Calculator

What are Significant Figures?

Significant figures represent the precision of a measurement. When scientists and engineers record measurements, they write down all the digits they are certain of, plus one final digit that is an estimate. Every digit that carries meaning or contributes to the precision of a value is considered a significant figure.

Using the correct number of significant figures prevents the false impression of precision. For example, if you measure the length of a table with a standard ruler to be 1.2 meters, and then divide that table into three equal sections, stating that each section is exactly 0.400000 meters is scientifically incorrect. The division process cannot create precision out of thin air. Instead, the final number must be rounded to match the precision of the original measurement.

The Rules for Counting Significant Figures

To determine which digits in a number are significant, follow these five standard rules:

1. Non-zero digits are always significant: Any digit from 1 to 9 carries measurement meaning. (e.g., 423.8 has four significant figures).
2. Captive zeros are significant: Zeros that are sandwiched between non-zero digits are always significant. (e.g., 2008 has four significant figures; 4.09 has three).
3. Leading zeros are NEVER significant: Zeros that appear before the first non-zero digit are merely place-holders indicating the scale of the number. (e.g., 0.0035 has only two significant figures: 3 and 5).
4. Trailing zeros in decimal numbers are significant: Zeros that appear at the end of a number after a decimal point indicate that a measurement was made to that exact resolution. (e.g., 85.00 has four significant figures; 0.0450 has three).
5. Trailing zeros in whole numbers without decimals are ambiguous: In a number like 400, the zeros are generally considered non-significant placeholders. However, if a decimal point is written (e.g., 400.), all three digits are significant. If written in scientific notation (e.g., 4.00 × 10²), all three digits are significant.

Significant Figures in Arithmetic Operations

When you combine measurements in calculations, the final precision depends on the arithmetic operation used:

1. Multiplication and Division (Sig Fig Limit)

When multiplying or dividing numbers, the result cannot have more significant figures than the input with the *fewest* significant figures. This is because the overall calculation is limited by the least precise measurement.

Example: 5.62 (3 sig figs) × 2.1 (2 sig figs) = 11.802. We round the result to 2 sig figs: **12**.

2. Addition and Subtraction (Decimal Place Limit)

When adding or subtracting numbers, the result is limited by the value with the fewest *decimal places* (precision of position), regardless of the total number of significant figures in the numbers.

Example: 125.11 (2 decimal places) + 2.3 (1 decimal place) = 127.41. We round the result to 1 decimal place: **127.4**.

Three Detailed Worked Examples

Example 1: Counting Significant Figures

Determine the number of significant figures in the following measurements: 0.005080, 4200, and 4200.0

• 0.005080: There are 4 significant figures (5, 0, 8, 0). The first three zeros are leading zeros and do not count. The zero between 5 and 8 is captive and counts. The final zero is trailing after the decimal and counts.
• 4200: There are 2 significant figures (4 and 2). The two trailing zeros are in a whole number without a decimal point, acting as placeholders.
• 4200.0: There are 5 significant figures. The presence of the decimal point makes all digits significant, including the trailing zeros.

Example 2: Adding and Subtracting (Decimal Place Rule)

Solve: 14.22 + 8.1 - 0.0035

1. Perform the raw arithmetic: 14.22 + 8.1 = 22.32; 22.32 - 0.0035 = 22.3165.
2. Check decimal places of inputs:
    14.22 has 2 decimal places.
    8.1 has 1 decimal place.
    0.0035 has 4 decimal places.
3. The minimum decimal places is 1 (from 8.1).
4. Round 22.3165 to 1 decimal place: 22.3.

Example 3: Multiplying and Dividing (Sig Fig Rule)

Solve: (4.50 × 10³) ÷ 25

1. Express numbers standardly: 4500 ÷ 25.
2. Note the sig figs of inputs:
    4.50 × 10³ has 3 sig figs (4, 5, 0).
    25 has 2 sig figs.
3. Perform raw arithmetic: 4500 ÷ 25 = 180.
4. Round the result to 2 sig figs: 180 has 2 sig figs (1 and 8).
The result is 180.

What is a significant figure?

A significant figure is a digit in a number that carries real, physical meaning contributing to its measurement precision, starting from the first non-zero digit.

Why are significant figures important?

They reflect the accuracy and limits of measurements. In scientific calculations, they ensure that the final result does not imply a higher level of precision than the tools used to measure the initial inputs.

Are leading zeros significant?

No. Leading zeros (like the zeros in 0.0045) are never significant because they are only placeholders that indicate the scale or magnitude of the number.

Are trailing zeros significant?

Trailing zeros are significant if the number contains a decimal point (e.g., 4.50 has 3 sig figs). If the number is a whole number without a decimal point (e.g., 450), the trailing zeros are generally not significant.

How many sig figs are in 100?

There is 1 significant figure (the number 1). The two trailing zeros act as placeholders.

How many sig figs are in 100.?

There are 3 significant figures. The decimal point at the end indicates that the measurement was rounded to the ones place, making the zeros significant.

What are captive zeros?

Captive zeros are zeros between two non-zero digits (such as the zeros in 508 or 1.004). Captive zeros are always significant.

What is the sig fig rule for multiplication and division?

The final answer must be rounded to contain the same number of significant figures as the measurement with the fewest significant figures.

What is the sig fig rule for addition and subtraction?

The final answer must be rounded to contain the same number of decimal places as the measurement with the fewest decimal places.

What are exact numbers?

Exact numbers are numbers defined by counting or definitions rather than measurements. Examples include "12 eggs" or "1 inch = 2.54 cm". Exact numbers are considered to have an infinite number of significant figures and do not limit the precision of a calculation.

How do you round a number to 3 significant figures?

Find the third significant digit from the left. Round up if the next digit is 5 or greater, and down if it is 4 or less. Replace any trailing digits in whole numbers with placeholder zeros, or drop them in decimals.

Does scientific notation affect significant figures?

No. When a number is written in scientific notation (e.g., A × 10^B), only the digits in the coefficient (A) are counted as significant.

How many sig figs does 0.050 have?

It has 2 significant figures: the 5 and the trailing 0. The leading zeros (before the 5) are not significant.

What is the difference between accuracy and precision?

Accuracy is how close a measured value is to the true or accepted value. Precision is how close a series of measurements are to each other (which is reflected in the number of significant figures).

How do you handle rounding when the number is exactly 5?

Most courses use standard rounding (round up). Some scientific systems use "round to even" (banker's rounding), where a number ending in exactly 5 is rounded to the nearest even digit.