In physics, chemistry, and biology, there are digits that are safe or the closest to reality in terms of measurements that we make. You will learn in the lab that measuring instruments have their own sensitivity or minimum measurement limits, which affects the accuracy of the measurements.
For example, let's say we have a ruler that can measure up to 1 mm in length. If a segment that we have drawn measures 0.9 cm, the measurement will have only one significant digit, that is 9. A segment that measures 2.4 cm has two significant figures.
Why is this so important? Because when we make operations between two measurements that have different numbers of digits, we need to know how many digits we have to stop at in order to express the result in the most accurate way possible. This is what significant figures are about! Scroll down to learn more about this topic!
- First, we will look at the definition of significant figures.
- Then, we will learn the rules around significant figures and look at how to apply them to calculations.
- After, we will learn how to round significant figures.
- Lastly, we will look at some examples involving significant figures.
Significant Figures Definition
The significant figures of a measurement consist of the "certain" figures and the first "uncertain" figure.
Alright, you're probably reading this and thinking to yourself: "excuse me, what? That makes no sense".
What are these significant figures? Significant figures are widely used in the science world, ranging from chemistry and physics to biology. These significant figures are very important because they provide information on quantities or measurements in an approximate way, helping us express these measurements and understand them in the most accurate manner (or as possible as it can be done with minimal human error).
It is important to note that in science, we define accuracy as follows:
Accuracy refers to how close a measurement is to the correct or actual value of whatever quantity is being measured.
So with all of this information, let's look at the definition of significant figures.
Significant figures are all the important digits of a number that are needed to express it according to its accuracy, beginning from the first non-null digit.
For example, when we write that the mass of a car is 1159 kg, it means that the last digit (9) is uncertain; that is, it is not exact. If we were 100% sure of this figure, we would have to write the result with five significant figures: 1159.0 kg.
Now all of this can be very overwhelming, but there is light at the end of the tunnel. Thankfully, there are some rules to follow when it comes to significant figures!
We will address those next and then do some practice problems, and before you know it, you'll be a master of this topic!
Significant Figures Rules
So, what makes a digit significant? All digits with leading or trailing zeros are considered significant unless the trailing zero follows the decimal point (e.g., 3.00 would have 3 significant digits, while 300 would only have 1 significant digit).
As mentioned earlier, there are rules out there that make understanding these a little better. Let's check them out!
All non-null values represent significant figures.
Zeros between non-null values are significant digits.
The zeros preceding the first significant digit (non-zero value) are not significant figures.
Trailing zeros are significant only if there is a decimal point (or comma).
Rules for Addition/Subtraction
In addition or subtraction, the resulting number has the same number of decimal places as the number with fewer decimal places.
Let's look at an example to understand this better.
5.36 + 99.124 = 104.48 (2 decimal points)
Notice how 5.36 is the number with the fewer decimal places (.36), hence why 104.48 has two decimal points.
Rules for Multiplication/Division
For multiplication or division, the resulting number (product) has the same number of significant digits as the factor with the least number of significant figures.
Let's do an example:
15.322 × 3.12 = 47.8 (3 significant digits)
Notice how 3.12 is the number with the least number of significant figures, which is why the product has the same number of significant figures as it.
Calculating Significant Figures
Let's try an example where we have to calculate for the density of a box. How do we apply the rules for determining the significant figures here?
If the mass of a measured box is equal to 6.817 kg and the calculated volume is equal to 18.39 cm3, what is the density of the box?
The density (ρ) is calculated by dividing the mass of the box by its volume:
ρ = 6.817 kg / 18.39 cm3
ρ = 0.370… kg/cm3
Since the volume has only significant digits in the hundredths position, while the mass has significant digits in the thousandths position, the final density in hundredths will be reported:
ρ = 0.37 kg/cm3
Let's try another example. This time, we'll look for the perimeter of the building.
Let's say the height of a building is 373.71 feet, measured using a tape measure calibrated in hundredths. The width of the building itself is 175.2 feet, measured using a ruler calibrated in tenths. What is the perimeter of the building?To calculate the perimeter, you add up all the sides:
P = 370.71 + 170.2 + 370.71 + 170.2
P = 1081.82 feet.
However, given that the width of the building is known only in tenths, the result obtained can only be reported in tenths. The final result will be:
P = 1081.8 feet
Rounding Significant Figures
Prior to jumping into the rules of rounding up significant figures, in any case, it is always advisable, when possible, to keep all the figures in intermediate calculations and round off the final values to the required number of significant figures.
Rounding should normally be done by taking into consideration only the first digit after the last significant one (let's call it "extra").
If the "extra" digit is less than or equal to 4, the value of the last significant digit remains unchanged.
If the "extra" figure is greater than 5, the value of the last significant digit must be increased by one unit.
If it is a 5 followed by a number greater than zero, we operate as in the previous case.
If the 5 is followed by a certain number of zeros, an extremely special case, the previous value is rounded to the nearest even number.
Significant Figures Examples
As with other forms of calculations, applying the rules of significant figures takes practice. Take a look at the examples below, and see if you understand how we ended up with the answers provided.
6.87 kg + 0.218 kg + 3.55 kg = 10.638 kg = 10.64 kg (4 significant figures)
1.50 m + 36.8 m + 45.60 m = 83.9 m (3 significant figures)
3.02 m - 1.13 m = 1.89 m (3 significant figures)
210.2 cm - 3.15 cm = 207.05 cm = 207.1 cm (4 significant figures)
75.3 kg x 9.81 kg = 738.693 kg = 739 kg (3 significant figures)
6.45 cm x 4.521 cm = 29.16045 cm = 29.2 cm (3 significant figures)
Significant Figures - Key takeaways
- Significant figures are significant digits of a measurement that are all digits except leading zeros, that is, all digits between the most significant and the least significant.
- All non-null values represent significant figures.
- Zeros between non-null values are significant digits.
- The zeros preceding the first significant digit (non-zero value) are not significant figures.
- Trailing zeros are significant only if there is a comma (or period decimal).
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