Regardless of what system is used measurements fall into one of two categories. These are called the intrinsic and extrinsic properties of measurements:

In physics and chemistry an intrinsic property (or intensive property) of a system is a physical property of the system which does not depend on the system size or the amount of material in the system. By contrast, an extrinsic property (or extensive property) of a system does depend on the system size or the amount of material in the system.

Examples of intrinsic properties are temperature, pressure and density.

Examples of extrinsic properties are mass, volume and energy.

a) The metric system: A system of measurement that uses "standardized" definitions of the base unit for each category of measurement

The metric system is measuring system defined in terms of the decimal system. Each category of measurement, such as length, has a base unit of measurement. (i.e. The base unit for length is the meter, the base unit for mass is the gram, the base unit for volume is the liter, and the base unit for time is the second.)

For measurement values larger or smaller than each category's base unit of measure the metric system uses prefixes. The more common prefixes are:
 
 

--> Rule for addition and subtraction: When adding or subtracting measurements, the units of measure must be the same.

i.e. Two masses, a 5 kg mass and a 10.5 kg mass, are being added together. These two measurements have the same unit so the numerical values can be added resulting in a sum having the same unit as the individual measurements.
 

--> Rule for multiplication and division: When multiplying or dividing measurements, the units do not have to be the same. However, all units, except those that legitimately divide out (sometimes referred to as canceling), must be retained.

i.e. An object travels 3,000 m in a time interval of 15 seconds. Using the equation for speed of an object, v = d / t, calculate the speed of the object.
 
 

--> Other operations: Always perform other math operations, when using measurement unit symbols, as you would any variable symbol values such as x and y. All units must remain in the answer along with the numerical value arrived at, unless they actually divide out, and are legitimately gone from the answer.

An Example: A rectangular shaped object is placed upon a scale and found to have a mass of 25 grams. The dimensions of the object, measured with a ruler, were found to be 5 cm long, 3 cm high, and 2 cm tall. Calculate (1) the volume of the object and (2) the density of the object.
 

(1st) Volume is found by the equation V = L x W x H (length, width, and height, respectively). so the calculation looks like:
 
 

(2nd) Density is found by dividing the mass of the object by its volume. The equation is D = M / V.
 
 

*The problem solving format places emphasis on displaying the solution to a problem in a logical series of steps. This approach emphasizes critical thinking skills and over time will improve your ability to solve problems quickly. Always remember that, if you cannot get the units of measurement in your solution (not the correct one) to work out correctly, there is no point picking up your calculator, because when the units of measurement do not work out correctly, there is something wrong in your choice of math operations you have chosen to solve the problem. This applies to all kinds of problems involving the use of math, not just problems using the metric system.

*The number 0, itself, is not a power of ten. It can't be written as 10 to the something power.

Data Charts such as the one above are referred to as data tables. A table provides a means of organizing and displaying data that has been collected through observations.

--> The qualitative observation is a descriptive observation. It involves a clear and concise description of an object or circumstance. It does not involve the making of measurements (quantities).  Some examples of qualitative measurements are:
 
 

--> The quantitative observation is a measurement. It involves an accurate measurement expressed with as much precision as the measuring tool being used will allow. Some examples of quantitative measurements are:
 
 

Accuracy is defined as "The ability of a measurement to match the actual (correct) value of the quantity being measured." It is a concept that deals with whether a measurement is correct when compared to the known value or standard for the particular measurement being made. When this comparison is being made using a percent, it is referred to a percent error.

--> Example of accuracy. Suppose that a carpenter is making a cabinet to fit in an exact space having a width of 1.15 m. Now what happens if in his effort to cut the materials he makes his cuts 1.14 m. What effect does this have on the making of the cabinet? Well on the first look, it may seem that this 0.01 difference is quite small. However, upon closer inspection, 0.01 m is a whole centimeter, which is close to one half of an inch. The cabinet will now show a gap of almost a half an inch, because the carpenter was not accurate in measuring. This why in terms of measurement, carpenters, etc., have always said "measure twice and cut once".

In the above situation, more careful measurement would reveal the true (correct) width, either by measuring more carefully or by using a more precise ruler, perhaps one that measured to the 3rd or even the 4th decimal place should be used. However, when measurements are being made of the very large (astronomical distances) or of the very small (subatomic scale) for the very first time, accuracy is difficult to determine, because no one has access to correct answer. In this kind of a situation, we turn to precision and statistics to help us ascertain the usefulness of our measurements.

Precision is defined as "The ability of a measurement to be consistently reproduced." and "The number of significant digits to which a value can be reliably measured." The first definition is the main focus of scientists and engineers who are looking at the work of others. After studying what the other people had done, they set out to test the shared (often in the form of published reports) information to see whether they can reproduce the results. Reproducible results are much more likely to be accurate (correct) than results that cannot be reproduced. The second definition is the main focus of anyone using a measuring device. How precise or exact the measuring device is that you are using is based upon the degree to which the measurement units are broken down (subdivided).  Precision is the concept which deals with the degree of exactness when using a specific measuring tool and expressing a measurement made with that tool. The precision of any measuring tool and the exactness of the measurement made with that tool is limited by how precise the instrument is capable of being. the precision of a measuring tool is based upon the smallest defined unit of measure on the devise. The number of digits used to express a measurement when using a tool whose degree of precision (smallest unit available) are called significant digits. See the example below:
 
 

In all problem solving situations where measurements are involved, not only should you carry units along with each math operation performed, but you need to take into consideration the number of significant digits being expressed in each measurement being used, so each and every calculation using measurements is never expressed with more precision than the measurements themselves. That would be absurd (or, if you prefer, ridiculous). In working with measurements expressed in significant digits, there are three things you always need to consider. They are:
 
 

The Rules of zeros (0's) used in identifying the number of significant digits in a measurement. The rules are as follows:
 
 
 

Significant Digits in Calculations: The precision in the answer to a calculation where new information is derived from existing information cannot be greater than the precision of the least precise measurement.

Now the handling of this sort of situation where measurements are used in simple math operations is divided into two categories of math problems. They are the Addition-Subtraction problems and the Multiplication-Division problems.

--> 1. Addition and Subtraction Problems: When adding or subtracting measurements where precision and significant digits are being used, the answer can be no more precise than the measurement with the least amount of precision. In the following example you will be rounding off to the second decimal place (the one-hundredths column), because nothing is known about the 3rd decimal value in the measurement 3.11 kg. This makes the 3rd decimal in the final answer too uncertain to be recorded. So we round off correctly to the 2nd decimal place. In Addition and subtraction it is the measurement with the least number of decimal places that affects how much you are to round off to.
 
 

--> 2. Multiplication and Division Problems: When multiplying or dividing measurements where precision and significant digits are being used, the answer can be no more precise than the measurement with the least amount of precision. In the following example you will be rounding off to only two digits, because in multiplication (and division) you round off to the number of digits in your answer equal to the value with the least amount of significant digits being used in the problem. In multiplication and division it is the measurement with the least number of significant digits that affect how you are to round off.