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Units and the metric system

Unit analysis and conversion

Precision and accuracy

Scientific notation

Significant figures/digits

Percent error

Drawing graphs by hand

Drawing graphs using Microsoft Excel

Using your science text well

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Mathematical Skills for doing Science

Units and the metric system

In science, every number you write should have a unit next to it. This is because science, unlike abstract math, is about the real world, and all numbers must have a meaning.

The official system of units for the United States is complex, and is not used by most other countries. For simplicity and international communication, scientists worldwide use the metric system. In the metric system, there are basic units, such as meter, second, ampere etc. Each unit can be divided or multiplied by ten to get bigger or smaller units.

• centi means 1/100th of the basic unit, e.g. centimeter.
• milli means 1/1000th of the basic unit, e.g. milliampere.
• kilo means 1000 times the basic unit, e.g. kilogram.

Unit analysis and conversion

Complex units are often combinations of simpler units. For example, speed is defined as the distance traveled (e.g. in meters) divided by the time taken to travel (e.g. in seconds). Therefore the unit for speed is m/s (meters per second). If distance and time are measured differently, the unit of speed changes, e.g. to miles per hour or kilometers per minute. You can often guess what a unit is, by relating it to other units via an equation. For example, the equation for density is mass/volume, so the unit of density is often grams per milliliter.

Here is one of many ways to convert from one unit to another, if you don't have an online unit converter to work with.

1. Write down the number you started with. e.g. 17 miles
2. Multiply it by one. = 17 miles x 1
3. Re-write "one" as a fraction, where the thing on top equals the thing on the bottom. Choose the fraction so that the unwanted unit will cancel out and be replaced by the wanted unit. = 17 miles x (1.6 km / 1mile), since 1.6 km = 1 mile
4. Cancel out the unwanted units, and multiply whatever is left. = (17 x 1.6) km = 27.2 km

Precision and accuracy

In scientific measurement,

• Precision means detail
• Accuracy means truth.

For example, if Fred estimated Wilma's height to be 1.5 m, and Barney estimated Wilma's height to be 0.20396 m, then Fred is more accurate and Barney is more precise. (Wilma is not an elf.)

When you read a scale such as a ruler or a measuring cylinder, the scale will tell you a certain number of digits. e.g. you might know that the reading is definitely between 12.3 cm and 12.4 cm if your ruler has mm markings. Write down the part you definitely know, then estimate one more digit, e.g. 12.37. The rules are:

1. Estimate one digit beyond what the scale tells you.
2. If you are right on an increment line, use the same precision as you would elsewhere on the scale.
3. If there are increments that are not a multiple of 10, e.g. lines every 2 mL, they are just there to help you estimate. They don't provide another digit of precision.

Scientific notation

When scientists need to use very big or small numbers, they write the number as something times a power of ten. There is always one digit before the decimal point. e.g.

• 527 = 5.27 x 10².
• 0.008 = 8 x 10^-3

When multiplying powers of ten, add the exponents. When dividing, subtract the exponents. e.g.

• (3 x 10⁷) x (2 x 10⁵) = 6 x 10¹²
• (3 x 10⁷) / (2 x 10⁵) = 1.5 x 10²

Significant figures/digits

Using the rule "dot right, not left",

• if there is a decimal point in the number, start from the first non-zero digit and count to the right.
• if there is no decimal point, start from the last non-zero digit and count to the left.

A different way of remembering it is

• all digits from 1 - 9 are significant.
• all zeros sandwiched between other digits are significant.
• ending zeros are ONLY significant if there is a decimal point in the number.
• leading zeros are never significant.

In multiplication and division, the answer should match the number of sig figs of the vaguest (least sig figs) information in the question.

In addition and subtraction, the answer should match the number of decimal places of the vaguest (least decimal places) information in the question.

Percent error

Percent error tells you how dramatic a difference is, compared to the original. You can calculate it as:

100% x (difference between what you got and what you should have got) / (what you should have got).

e.g. if your height is really 1.63 m, but your friend measured your height as 1.62 m, the percent error is

100% x (1.63-1.62)/1.63 = 100% x 0.01 / 1.63 = 0.6%

Drawing graphs by hand

Your graph should be as big as possible, so it is more detailed. It must have a title so people know what it's a graph of, and each axis must have a label, a unit and a scale. Draw points either as little x's, or as circled dots. Big blobs are bad because it's unclear exactly where the point is.

A best fit line is something you draw to show the general pattern of the points on your graph. It should be as close as possible to as many points as possible, but should also be a smooth line or curve. Never join the dots.

The slope of a straight line is found by taking two points ON THE LINE (not nearby data points) and drawing a triangle to link them. Count how long each side of the triangle is, according to the scale of the graph. The slope is the length of the vertical side, divided by the length of the horizontal side. This is sometimes called the rise/run.

Drawing graphs using Microsoft Excel

1. Click on the graph icon in the toolbar.
2. Choose the "X-Y scatter" graph, without lines on it, then click Next.
3. Now there are two tabs at the top of the window: "Data Range" and "Series" Click on Series, then click "Add".
4. At the end of the "X values" field is a gray square. Click on it.
5. You'll be taken back to the spreadsheet. Highlight the data that will be the "x values" on your graph.
6. There is a little rectangular box floating over your spreadsheet, listing the cells you highlighted. Click on the gray square at the end of it.
7. Do the same for the "Y values" field, to tell it what to use as the y values in your graph. Then click Next.
8. Select the "title" tab at the top of the next window, to label the graph and both the axes.
9. Select the "gridlines" tab, and make sure "minor gridlines" is highlighted for both axes.
10. Select the "legend" tab, and make sure that "Show legend" is NOT checked (unless you have multiple sets of data on the one graph).
11. The next window will ask whether you want the graph to appear in the spreadsheet with the data, or on a separate page. Choose one, then click on Finish.
12. Print the graph, and hand-draw a line of best fit onto it. Never let the computer draw a best fit line for you. It is dumb about things like that. Humans are better at seeing general patterns.

Using your science text well

Your text is there for you to use when you need it: to find an extra explanation of a confusing concept, to find some extra practice questions before a test, or to look up a fact.

Don't read your text like a novel. Decide what you want to know and look it up in the index. Then just read the relevant section or sections. Go slowly, so you don't miss anything, and make notes of the main point of each little section or paragraph. Use the questions at the end of each chapter to check if you understood.