Describing Quantities

Units, Prefixes, and Number Formats

Système International D’Unites (SI)

The SI unit system is the default unit system used in the scientific community world wide.

Many think that it is synonymous with the “Metric” system, but the metric system is a base 10 decimal system that the SI system encompassed.

The SI system has units that measure a variety of quantities.

Fundamental Units

In science, a physical quantity is any physical property that can be quantified, that is, can be measured using numbers. We identify what that number is by associating a unit to it.

The SI system has seven (7) fundamental quantities that are arbitrarily defined and from which other units are derived. They form the basis of other units.

Table of Fundamental Units

From the NIST website:

Derived Units

Derived units are formed from some linear combination of fundamental units.

Sometimes they are simplified into a new symbol (such as the Newton, for Force). Other times, they are kept as the combination of fundamental units. Derived units are representative of the mathematical operations that went into calculating the quantity.

Table of Derived Units

From the NIST website:

More Derived Units

Confidence in our values.

Significant Figures

In the scientific community, we use significant figures to convey our confidence in a value.

The better we measure a quantity, the more confidence we have in that value. The better the instrumentation, the better our technique, the more confidence we have.

For example..

A standard ruler can measure down to a millimeter. If we were writing out measured value (say, the thickness of a whiteboard marker) in centimeters, what value would we get? How many actual numbers could this measuring device give us?

What if we had a better tool? What if instead of a ruler, we had a digital slide caliper (as shown to the left)? Pretend that the value in the picture is the measured value of the thickness of the whiteboard marker. What would that value be in centimeters?

Compare

What’s the difference between the two? The ruler and the caliper both measure distances, but one of the tools is more precise, and a better tool to measure smaller distances. This means it gives us more confidence, and thus, it will produce more significant figures. When we go to report out numbers, the measurements from the better tool should have more digits, since we measured it better.

What does this mean?

It means that the amount of digits is really important in science. We don’t just arbitrarily decide how many digits we round to, or how many digits we keep when we do calculations and measurements.

Our calculators are machines, and they don’t know how good of an instrument we used to get our numbers. We must make the decision after all calculations are made.

What numbers are significant?

There are three rules on determining how many significant figures are in a number:

1. Non-zero digits are always significant.

2. Any zeros between two significant digits are significant.

3. A final zero or trailing zeros in the DECIMAL PORTION ONLY are significant.

Example

1. 1543

2. 1005

3. 0.00145

4. 0.01050

Rule 1.

If you measure something and the device you use (ruler, thermometer, triple-beam, balance, etc.) returns a number to you, then you have made a measurement decision and that ACT of measuring gives significance to that particular numeral (or digit) in the overall value you obtain. Hence a number like 46.78 would have four significant figures and 3.94 would have three.

Rule 2.

Suppose you had a number like 409. By the first rule, the 4 and the 9 are significant. However, to make a measurement decision on the 4 (in the hundred's place) and the 9 (in the one's place), you HAD to have made a decision on the ten's place. The measurement scale for this number would have hundreds, tens, and ones marked.

Rule 3.

This rule causes the most confusion among students. In the following example the zeros are significant digits, bolded and highlighted in yellow

0.07030 0.00800Here are two more examples where the significant zeros are bolded and highlighted in yellow

4.70 x 10-3

6.500x 104

When Zeros are Not Significant Digits

Zero Type # 1 : Space holding zeros in numbers less than one. In the following example the zeros are NOT significant digits and highlighted in red:

0.09060 0.00400

These zeros serve only as space holders. They are there to put the decimal point in its correct location. They DO NOT involve measurement decisions.

Zero Type # 2 : Trailing zeros in a whole number. In the following example the zeros are NOT significant digits and highlighted in red.

200 25000

Calculations

We will be dealing with some calculations with our measured quantities, and this will change the amount of digits that we have.

For example, if we have 2 measured values with 1 significant figure each, the numbers 1 and 3, and we divide them, how many digits do we get?

How do we know how many to keep?

Rules for Addition/Subtraction

The only part that will define the number of values you’ll keep will be after the decimal. You’ll only be counting these values when deciding how many significant figures to keep.

1) Count the number of significant figures in the decimal portion of each number in the problem. (The digits to the left of the decimal place are not used to determine the number of decimal places in the final answer.)

2) Add or subtract normally.

3) Round the answer to the LEAST number of places in the decimal portion of any number in the problem.

Example

13.214 + 234.6 + 7.0350 + 6.38

Multiplying/Dividing

Multiplying and dividing is slightly simpler:

The LEAST number of significant figures (overall) in the whole operation determines the number of significant figures in the answer.

This means you MUST know how to recognize significant figures in order to use this rule.

Example

16.235 × 0.217 × 5 / 2.0

Why?

Remember that significant figures are a way of conveying confidence in your measured values. You can only be as confident in your final result as your weakest measurement.

As the saying goes, you’re only as strong as your weakest link.

Scientific Notation

In this class we will often see numbers that are either too large or too small to write comfortably. Writing out all the digits would be too cumbersome. In those cases, we’ll switch over to scientific notation.

This is when we rewrite our large/small number in the form:

α x 10x, where α is a value between 1 and 9, and x is any integer value, positive or negative.

Steps for Scientific Notation.

1. Move the decimal point so that you have a number that is between 1 and 10.2. Count the number of decimal places moved in Step 1 3. Write as a product of the number (found in Step 1) and 10 raised to the

power of the count (found in Step 2).

Examples

1. 153000000

2. 0.0000000543

Metric Prefixes

The power of the SI system is the adoption of the metric prefix system. It allows us to quickly, easily, and intuitively convert between different scales. This is done by multiplying/dividing by different powers of 10, depending on how many “levels” you want to go.