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Natural Science - Year I

Unit 6: Mathematics for Counting and Measuring

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Science Web Assignment for Unit 6


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Science Lecture for Unit 6:
Counting and Measuring: Discrete and Continuous Quantity

For Class
Outline/Summary
Counting and Measuring

Let's take a look at some of the assumptions and principles we take for granted in using numbers to describe the behavior of the universe. First of all, we start by counting "how many" things there are. Counting started by recognizing the difference between none and one, and between one and many. When we count, we identify what we might call discrete entities, which is just a way of saying "one whole thing that is separate from every other thing." One Oreo is just that: one cookie, by itself, not touching or part of another Oreo. Counting can be exact: mycroft has 5 oreos, neither more nor less, nor any fraction.

Measuring is a different kind of activity. When we measure, we divide a continuous entity into parts, and we always compare two things, usually one thing of interest to some standard. For example, we might measure out 4 yards of cloth to make a cloak, or a cup of milk and a half a pound of butter to put into a cake. We layout the cloth along the yardstick four times. The milk goes in a cup measure. The butter comes cut up for us already by the butter manufacturer, who has had to guarantee to government inspectors that the scales he is using really do measure one pound. If any one of our standards is off, our measured item will be the wrong length or volume or weight. The dress won't fit, and the cake will be too dry. If we are doing measurements for an experiment, we may conclude that the differences we notice are not significant, or that they are significant enough to challenge a well-accepted theory. For the ancient Egyptians, a misaligned pyramid was an insult to the dead king, and a dangerous situation, since the king was thought to be on his way to becoming a god.

As you can see, it is important to make our measurements as accurately as possible.

The ideas of discrete units or things (like oranges), continuous objects (like space) and the boundaries between discrete objects are crucial concepts in scientific thought. As we will see, the problem of continuity posed a significant philosophical quandry for some Greeks, and still does so in some areas of science. Can we think of space has having a boundary? Is our universe a discrete entity, separate and distinct from other universes? Where does the influence of a gravitational force field end (if it does)? Or does it just become immeasurably small? If we can't measure it, does it mean anything to talk about it?

Units

Measuring always involves units. A unit is one of some standard length, mass, force, energy, or time. Sometimes we have more than one kind of unit for the thing we measure, and we get to pick the most familiar or most convenient. Americans and, until about three decades ago, most of the British Empire, generally used feet, yards, and miles to measure distance. You have no doubt spent several happy hours of your childhood coverting 12 inches to 1 foot, or 3 feet to a yard, or 5280 feet to a mile. There has been a gradual conversion to using international standard units based on the metric system, because these units are related to one another by powers of ten, which makes it somewhat simpler to do conversions when we need to scale up to larger units (to measure the distances between stars) or down to smaller units (to measure the distances between atoms in molecules). One thousand meters is one kilometer (1000m = 1km), or 103m = 1 km. We can shift units just by changing the exponent on the powers-of-ten value that we use.

The different kinds of units are also related through the action of water, which is readily available if we need to make standards to use in the lab. A gram of water occupies one milliliter of volume, or one cubic centimeter of space. A calorie is the amount of heat energy necessary to raise the temperature of that gram of water by one degree centrigrade. This makes it easy to set up a laboratory situation with just some way to measure distance and calculate volumes.

We can change from one kind of unit to another by using a conversion factor. If we know a distance in inches, we convert it to feet by multiply our "inches" measurement by 1/12. Keeping the units in the operation helps us see that the "units fraction" is set up so the inches cancel and we are left with feet:

In our calculations, both the 60/12 and the inches * foot/inches must be computed. The 60/12 resolves to 5. The inches cancel and we are left with feet, as we wanted.

Accuracy

When we measure, we also have to worry about the accuracy of our measurements. If we measure degrees of temperature to within .1° C, but notice a periodic fluctuation in temperatures measured to .01° C, we can't be sure that our fluctuation is significant. It might just be a problem resulting from our inability to measure accurately enough.

Accuracy and precision are often determined as a percentage of the measured value. Remember Khufu's pyramid in this week's history lecture? It was 250m on a side, with a maximum deviation of 2 cm. A cm is a centimeter, 1/100 of a meter, or in what is called scientific notation, 10-2 meters. We can express the 250 meters as 2.5 * 102 meters. Now we can divide: 2 * 10-2m / 2.5 * 103. If you've never done work with exponents before, here's the rules you need to know:

  1. To multiply the same base with exponents, like 10a * 10b, you add the exponents:

    10a * 10b = 10(a+b)

    103 * 102 = 10(3+2)= 105

    103 * 10-2 = 10(3-2)= 101 = 10. Another way to look at this is to divide: 103/102 = 10

  2. To divide as we've already seen, means to subtract exponents:

    10a / 10b = 10(a-b)

Now let's look at what this does with the accuracy of the pyramids. We have a deviation maximum of 2 * 10-2meters, and a measurement of 250m = 2.5 * 102. Our error is deviation/measurement, or 2 * 10-2 / 2.5 * 102, which works out to .8 * 10(-2-2) = .8 * 10-4 = 8 * 10-5. If we round 8 up to 10, we have 10 * 10-5 = 10-4. Our error is 1 / 104, or 1/10000. In percent, this is 0.01%. That's very accurate, especially for a building that large.

Expressing some number as N.NNN * 10x is called scientific notation. One of the advantages of this kind of notation is that we can express any number with one digit to the left of the decimal place. The digits to the right tell us how accurate the number is. By convention, the last digit can vary no more than 1 unit. So the number 8.0123 means that the value is accurate to 8.0123 ± 0.0001, that is, it could be as high as 8.0124 or as low as 8.0122.

Such an error might not seem important, but one of the things about physical measurements is that we often plug them into calculations. A small error in measurement may propagate, showing up in later calculations and affecting them to a large enough degree to skew the results. It is always important in science to determine what level of accuracy you can meet.

Derived Quantities and Algebra

We can't always measure quantities directly. Let me restate that: we usually can't measure a quantity directly! Often we measure some base dimensions and determine the quantity we are looking for from the base. For example, we don't measure area directly. We measure the length of sides of the shape and then calculate the area. In this case, area is a derived quantity: it is the result of calculation.

Some quantities are always the result of a calculation. Density, a fundamental property of all mass objects, is a quantity we can only determine by first measuring the mass, then the volume, then doing a calculation:

density = ρ = mass/volume.

We use formulae all the time to get a derived result for any case we might want to determine. Some of the formulae you may already know allow you to calculate area and volume from length measurements of regular solids:

Area of RectangleH*W = Area
Geometric Figures
Area of Triangle½H*W = Area
Area of Circleπ * R2 = Area
Volume of Cylinderπ * R2 * H = Volume
Volume of Cone1/3 π * R2 * H = Volume
Volume of Sphere4/3 π * R3

Notice what happens with the dimensions as we calculate area (2 dimensions) and volume (3 dimensions) from measurements of length (1 dimension). It takes a multiplication of distance * distance to come up with area. It takes a multiplcation of distance * distance * distance, or area * distance, to come up with volume.

Notice also that our formulae do something the Egyptians and Babylonians were not quite able to do in all cases: they solve the general case of the problem. We can use any length values for height and width and determine the resulting area. We are not stuck with solutions for only certain situations. As we'll see in the next unit, this kind of solution of the general case became a quest for the Greek philosophers, and laid the foundation for expressing physical law in the form of a mathematical equation, although it took some time for this to actually happen.

Study/Discussion Questions

Further Study On your Own (Optional)