Contents

# Significant digits

### 1. Estimation

Good estimation depends then on the digit we consider for the measurement.
Let's suppose that the value lies between 20 and 21 cm. The reasonable estimation is 20.5 cm. Recall that 20 cm is measured (certain) and 20.5 is estimated (uncertain).
In the value 20.5 cm, we have 3 digits that are significant. As 20.5 is an estimate, the true value can be found between 20.45 and 20.55 cm.

When a figure has a zero as a digit; It causes some problems in the sense to be reported as significant digit or not. Here is the rule: When zereos are at the left of a digit, they are not significant. When zereos are between nonzeros digits, they are significant. When zereos are at the right of a digit, they are significat if they are set for the precision; otherwise, they are not significant.

#### Example:

0.0034 has two significant digits. The three zeros are not significant. 5006.78 has six significant digits.The three zeros are significant. 300 has one significant digit if the two related zeros are set for the precision; otherwise, they are not significant.
The scientific notation gives a good fashion to represent a figure that has zeroes at the end. This notation is written as a power of 10. when it comes to do calculation, the scientific notation is very efficient.

#### Example:

4780 = 4.780 x 10+3 if we need four significant digits = 4.78 x 10+3 if we need three significant digits
Remark that 4.78 x 10+3 can be rounded up to 5.00 x 10+3. But what's about the rounding?

### 2. Rounding figures

#### 2.1. Rounding up or round down?

We round up or down a figure that has a decimal. The rounding depends on the last significat figure. Here is the rule:
we round down : if the last significat figure ends with :
- a digit : 1, 2, 3, or 4;
- an even digit followed by 5.
345.571 → 345,57
56.45 → 56.4
we round up :
if the last significat figure ends with :
- a digit as: 6, 7, 8, 9;
- 5 followed by nonzero digit
- an odd digit followed by 5 with non other zero digits.
55.68 → 55.69
897.75211 → 897.775011
77.3500 → 77.40

#### 2.2. How to round after operations

The rule is:
The result of a multiplication or a division is rouded to contain exactly the same number of significant figures as in the smallest digit in the operation.
The result of an addition or subtraction is rounded to contain exactly the same number of digits at the right of the decimal as in the smallest digit in the operation.

#### Example:

If we measure the area of a sheet that has 27.1 cm of length and 21.75 cm of width we will obtain 27.1 x 21.75 cm2 = 589.425 cm2. We can see then the length has 3 significant figures and the width has 4 significants figures. But the area has 6. The rule is to round the result for the area at 3 as the smallest digit in the product.

If we want to calculate the half-perimeter of the same sheet, we perform the addition of 27.1 cm and 21.75 cm; that is 48.85 cm. Following the rule, we will round the result 48.85 cm to 48.8 cm in order to have the same number of digits at the right of the decimal (1) as we have in the smallest one; that is 27.1 .

• 1.What's the difference between accuracy and precision?
• 2. The mass of a book is 1.23 kg. Which the following measurements are precise? wich are accurate? :
30.56 kg, 1.24 kg, 33.90 g, 1.23 kg, 1.56 kg, 123 mg, 1.22 kg .
• 3. Which SI units would you use to measure the following dimension?
3.1. A length of a car
3.2. The distance between Boston and New Jersy
3.3. A mass of a key
3.4. The time that makes light to arrive on earth
• 4. Convert the following figures:
4.1. 300.45 g in kilograms
4.2. 0.53 x 10 - 10 m in nanometer
4.3 90.00 seconds in minutes
• 5. Perform the following operations and round up or down:
5.1. 34.5 + 344 =
5.2. 12 x 55.55 =
5.3. 1250 / 12.50 =
5.4. 560.3 - 36.7765 =

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