Introduction: Units are the bridge from learning abstract arithmetic operations on numbers to actually using maths to navigate the world of objects, distance, time, rates, volume, temperature, heat, current, voltage, and even cooking using recipes. Goal for the series: Embracing units, and carrying them along as you go, can help you work with confidence in using maths in your life. When you start to use maths to solve real problems, you are going to run into units. This series is intended to show you that units are your friends, and that they're here to help you. Goal for this episode: We want to look at what units are, what they do, types of units, and how to mix unitless numbers with units. Resource for the series: * Khan Academy pages on Rates, Ratios and Units https://www.khanacademy.org/math/arithmetic/rates-and-ratios Most articles that would be relevant to this introductory episode were about teaching physics and chemistry, or discussions of philosophical implications of doing what we will be doing at every turn in this series. All of the formal operations that we will learn to do with units are done every day in real life by experts in their respective fields. I am not worried about what it means to say, "There are 12 inches in a foot." Later shows will have more links and resources. Segment 1: What do we mean by units? 1. Definition: Two types of units are useful in practical maths: a. Counting units: An individual thing treated as single or complete. Units can also apply to an individual component of a larger or more complex system. E.g., mufflers can become part of a car. - Think of objects that you would keep in an inventory in your pantry or in a warehouse. b. Measurement units: A quantity chosen as a standard that you can use as a common benchmark for comparing other quantities (of the same kind). - "Same kind": Don't try to compare distances to times or volumes. - "Standards": Communication tool for talking about quantities without being face-to-face. If you have standard units, you avoid expressions like "yea long", "kind of tall", etc. - Probably invented by buyers and sellers, or by the spouse of an avid fisherman. c. Composite units: Units can be multiplied together (or divided) to create new types of units. Some people call these "derived quantities", but that may sound too much like programming talk. I use composite units because of the mental picture it creates of putting things together, or doing one operation after another. - Dimensionality changes: * 1 ft * 1 ft = 1 square foot: distance^2 --> area * 1 ft * 1 ft * 1 ft = 1 cubic foot: distance^3 --> volume - Rates: * Speed: distance / time = average speed, as in kilometers/hour * Flow rates: volume / time, as in liters/minute * Pressure: force / area, as in pounds / square foot * Density: mass / volume, as in kilograms / liter * Rationing: (1 period) counting units / time, as in apples/day (longer time) apples / family_member / day - We will run more of these types of units in later shows. 2. Other kinds of numbers: Not every quantity has units attached a. Numbers can be unitless. Unitless numbers help you make sense of quantities with units through comparisons, extrapolations, etc. - Example: Percent changes are unitless floating point numbers, unless it is tied to an elapsed time. That's a "rate", which has units of "% per year" (say). - Example: Percentage of Total values are unitless fractions, too. - Example: Any unit can be multiplied by a unitless integer. * 2 feet, 3 apples, 4 quarts, 10 meters, etc. * "Twice as many", "ten times as far", "double a recipe" - Counting units can be multiplied by a unitless fraction, but the result will be rounded off to the nearest integer value. * "Mary has 2-1/2 times as many apples as John," is fine if John has 4 apples, and Mary has 10 apples. - Example: Measurement units can be multiplied by any arbitrary scale factor. * How big: "A land area 3.6 times the size of New Jersey..." * How far: "I'll meet you halfway..." * How much: "If using white flour, you'll need 30% more..." b. When values with units are divided by other values with the very same units, the result is a unitless number. - Percent of Total and Percent Change are prime examples - Comparison of distances: * "St. Johnsbury is 45 miles away, and Barton is only 15 miles. So you have to drive 3 times as far to get to St. J." c. Conversion factors between units work in this way. They are given as ratios of some number of new_units divided by some other number of original_units. * The original_unit quantities cancel in multiplication, just as numbers do, so you get an answer with the correct units! * You could call conversion factors "derived quantities", because you create them from something called an identity, or a statement of equality that you know to be true. d. Conversion factors will be covered next time. 3. Why bother with "counting units"? Aren't these just names? a. Counting units are labels or names applied to individual items in a total count, but they are still useful. b. Using counting units helps us to make distinctions between items that are not interchangeable, so we can keep track of the counts for each individual kind of item. - If you need 2 apples, having 10 onions does not help you. - Thinking with units will help you keep inventories and to start setting up accounting systems for your business. It will also help you manage your kitchen and your budget at home. Segment 2: Counting Units? Are you serious? 1. Counting units give context to the numbers that you are using in any calculations that arise when you are buying, selling, trading or just using up items in a beginning inventory. Here's what happens when you don't track units in counting problems. - Example: "John has 9 apples in his basket. If he gives 2 apples to Mary, how many does he have left?" - Speed test preparation textbooks seem to teach you to parse the problem as if you were a word problem "compiler": a. Fish out the numbers and their roles. --> Notice that 9 is near "in his basket", and "how many does he have left?", It must be the source. --> Notice that 2 is next to "gives away". It must be the change in quantity. b. Parse out the operation: "gives away" is code for subtraction. c. Do the calculation and supply a numerical answer: 9 - 2 = 7 2. Re-work the problem by tracking units. a. Read the problem. I'll wait. We will parse it together. b. John has a basket with 9 apples in it --> beginning inventory c. John gives away two (2) apples to Mary. - John's inventory of 9 apples is reduced by 2 apples, - John now has 7 apples in his basket. d. Mary now has 2 additional apples in her inventory. - The apples were neither created out of nothing nor destroyed. - They came from somewhere (John), and they went somewhere (Mary). - If "apples out" does not equal "apples in", something's wrong. e. Having this information lets you answer questions with confidence. f. Answer the question: "John now has 7 apples." - John does not have '7'. John has '7 apples'. 3. Ho hum. That solution is exactly the same. You're picking nits. a. For a trivial problem, this looks the same. But there are some benefits of using units, even if they appear to be "just labels". b. If the problem had said that "John gave 2 oranges to Mary", we would have spotted the discrepancy immediately. - Giving away oranges does not affect John's apple inventory - The oranges must have come from another supply (account) - We can still talk about an increase in Mary's oranges count, and the decrease in John's oranges -- even though we don't know the beginning or ending balances. c. What if the problem had said, "Mary has three times as many apples as John. How many apples would Mary have to give to John to leave each of them with the same number of apples?" d. Better yet, what if the problem read: "John has 19 apples, and Mary has 14 oranges. Now John likes oranges twice as much as he likes apples, but Mary likes apples three times as much as she likes oranges. How can John and Mary exchange apples and oranges to get the best (equal) gain in happiness?" - This problem involves not only the tracking of apples and oranges, but probably some type of "happiness" function that gives a value that carries some kind of units. Warning: There's not enough information to really solve this problem without further assumptions. It is meant as an illustration of how complicated a setup can become when you get into real life situations. - Problems like this are what make people hate economics. One way to solve it is to define utility functions for each party. - Their preferences are so different from their inventories, that simply trading baskets is pretty close to an optimal solution. e. If the problem had involved trading some of John's apples for some of Mary's oranges, and possibly an offsetting cash payment to correct an imbalance, we would make the best use of our information about the sources and uses of resources by tracking the units of each object or currency involved in the exchange. Point: Problems can become complicated. Units can help with the bookkeeping needed to work through to the answers. If someone poses a problem like this one to a group at a dinner party, it is time to remember that you forgot to iron your curtains. 4. Final properties of counting units a. Compatible counting units can be added and subtracted. - Example: 6 apples + 4 apples = (6 + 4) apples, or 10 apples. - Example: 6 apples + 2 oranges is a mixed expression. They cannot be added, except as part of a fruit salad. b. An amount that's given in counting units can be multiplied by an integer, since that is like repeated additions. They can also be multiplied by a fractional amount, but we would want to interpret the result as a whole number. c. Any multiplication by a floating point number would have to be defined, and it's usually not worth the effort. d. Counting units have weaknesses, especially in classification: - Organic items are usually not identical. Apples can vary. * Size: A recipe calls for "3 large apples". Are these large? * Varieties: "Apples" in the US can include Macintosh, Rome, Gala, Granny Smith, etc. These can be quite different. - Animals also vary within categories: * Cats: Lions, lynxes and Little Puff can all qualify * House cats: Siamese, Persians, Tabby cats are all just cats, until you have them living in your home. - Some living things are hard to pin down: sponges, paramecia - Other items can also create classification issues, depending on your purpose. * Units are just tools. Let them work for you, and not the other way round. Segment 3: Units of measurement 1. Measurement units are often continuous (or just about), so they can be divided conceptually into smaller and smaller subunits as many times as we like. - They can also be lumped together into larger and larger wholes. - Physical limitations place practical limits on how finely we can actually chop things up, and still get a measurement. - There are real world limitations on how much we can lump together. - But you get the idea. 2. Measurement units can be applied to distance, time, area, volume, weight or mass, energy, frequencies of light or radio waves, voltages, current, heat, temperature, and a host of other things. - We can measure these quantities with differing levels of precision, based on the instruments and abilities that we have. - For all practical purposes, we measure within tolerances that we can meet without spending our whole lives measuring. 3. Applications of measurement units a. Understanding the news: hectares of forest endangered by a fire, square miles of arable farmland in South Africa, temperatures given in unfamiliar scales such as Fahrenheit, snowfall measurements in Canada versus neighboring Montana, etc. b. Following recipes to make bread, cookies, beer and other items that promote World Peace c. Mixing chemicals for an old-school darkroom, or for a very cool low-tech electronics home "fab lab" d. Buying gasoline (petrol) in other countries, and understanding their speed limits in foreign units. - Can't help you with driving on the wrong side of the road e. Helping your kids with their maths homework, and understanding it for once! f. Checking the dosages of your medications against your prescription to find out if this is my medicine or my child's. You just have to be able to get this one right. We'll get to all of this and more in future episodes in this series.
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