L---P----1----+----2----+----3----+@10-4----T----R E:\PAGESEW\RUFFTEXT\ROUGH034.TXT A CRASH COURSE IN MATHEMATICS Contents: Notation Note Simplify your expressions Symbols Units of measurement Square Units Computing Areas: How much fabric will make a given length of tape? How much is lost to seams? NOTATION NOTE: mathematicians do not use the times sign. When written by hand, it's likely to be mistaken for the letter X. When written on a typewriter, it's likely to *be* the letter X, which is even more confusing. Worst of all, when you have an expression such as "3 times 4 + 5", you don't know whether that's "multiply four by three, then add five to the product" or "add four and five together, then multiply the sum by three". So what they do is just set the numbers side by side, saying "3(5)" the same way you'd say "three boxes" or "three dozen". There's no way you can mistake "3(4) + 5" for "3(4+5)" ============================================================== SIMPLIFY YOUR EXPRESSIONS Little Johnny has five quarters in his pocket. 5($0.25) = $1.25 Little Johnny's pocket contains $1.25. Mommy hands Johnny a ten-dollar bill and sends him to the store. Johnny puts the bill into his pocket and hops onto his little bike. $1.25 + $10.00 = $11.25 Little Johnny's pocket contains $11.25. At the store, Johnny takes the bill out of his pocket and gives it to the clerk. $11.25 - $10.00 = $1.25 Little Johnny's pocket contains $1.25. The clerk hands Johnny a package and $1.67 in change. Johnny puts the package into his bike's pannier and puts the change into his pocket. $1.25 + $1.67 = $2.92 Little Johnny's pocket contains $2.92. When Johnny returns home, he gives his mother the package and the change. $2.92 - $1.67 = $1.25 Little Johnny's pocket contains $1.25. If all you wanted to know was how much money was in Johnny's pocket after all these transactions, you didn't need to do any of those calculations. He put money into his pocket, he took it out again, and he ended up just where he started. In other words, the expression 5($0.25) + $10.00 - $10.00 + $1.67 - $1.67 can be simplified to 5($0.25) Now let's complicate the situation a little: on his way out of the store, Johnny stops at a vending machine and buys a piece of candy with one of his quarters. Now the expression summarizing his transactions reads 5($0.25) + $10.00 - $10.00 + $1.67 - $0.25 - $1.67 The "+$1.67" and the "-$1.67" are no longer right next to each other, but they still cancel each other. The expression simplifies to 5($0.25) - $0.25 If you are muttering impatiently "Look, he had five quarters, he spent one, now he has four, that's a dollar -- what's with all the math?", then you have further simplified this expression. As you learned in grammar school, 5 times $0.25 is the same as $0.25 times 5, so 5($0.25) - $0.25 = $0.25(5) - $0.25(1) = $0.25(5 - 1) = $0.25(4) = $1.00 You can simplify a calculation even when only one operation is involved. For example, suppose that sixteen people go out for lunch and the bill is $128.00. I can't divide by sixteen in my head, but I can divide by two, so I think: $128/16 = $64/8 = $32/4 = $16/2 = $8 ============================================================== SYMBOLS Even in my over-simplified tale of the money in Little Johnny's pocket, all those numbers are confusing. If things get a little complicated, you can easily fail to see that the same number occurs twice. This is exploited in all of those magic tricks that run "Think of a number, any number, add six . . . and the answer is YOUR ORIGINAL NUMBER!!!!!" You can get around that by putting down a letter instead of the number. (A star, a heart, or any simple picture would do, but it's easier to pick an easy-to-remember letter than to draw an easy-to-remember picture.) So when someone says to you, "I have the scariest magic trick -- write down the year you were born." You write, instead, "B". "Multiply by five" 5B "Add twenty" 5B+20 "Multiply by two" 10B+40 "Divide by ten" B+4 Subtract this number from 2008 (I'm telling this in 2004; in 2005 one must say "2009", in 2006 "2010" etc. 2008 - (B+4) = 2004 - the year you were born And the answer is YOUR CURRENT AGE!!!!!!!!!!!!!! When this joke is told for real, the current year is disguised a little better, and about halfway through you are told, "If you've had your birthday this year add this number, and if you haven't, add (same number plus or minus one). You want the victim to do several calcutions both before and after, to keep him from connecting his birth year with his birthday, and the birthday with his current age. Aside from solving puzzles, giving a number a name without saying which number it is allows you to make one calculation and apply it to all possible values of that number. There are thousands of pre-calculated "formulas" that you can look up in various books. This way to simplify calculations is invaluable to the study of the relationships among numbers -- it is ever so much easier, for example, to say "for any two numbers, which we will call a and b, ab=ba" than to list all the possible pairs of numbers. Since the study of the relationships among numbers is called algebra, we are apt to refer to the system of using symbols other than numerals to refer to unspecified or unknown numbers as "algebraic notation". ============================================================== UNITS OF MEASUREMENT When you do real-life calculations, there are nearly always units of measurements involved, and it's very important to keep these units of measurement straight. (I'm sure you've had the experience of looking at your calculator in consternation after absent-mindedly entering a number of cents among a series of entries denominated in dollars.) In grammar school, we were taught to change everything into the same units before starting to calculate. This prevents the dollars-plus-cents problem, but it's a great deal of work -- and sometimes the units divide out anyway. It's much easier to keep all the units that you started with, treat them as though they were algebraic quantities, and do any conversion needed at the end. Example: for reasons I'll go into later, you might want to multiply four inches by ten yards, then divide by fifty inches, and express the result in yards. Grammar-school method one: The easiest way to get the quantities all into the same units is to change "ten yards" into 360". Then 4 X 360 / 50 = 28.8, 28.8 is in inches, so you divide by 36: 0.8 yards. (Please read "X" as the times sign, and "/" as the division sign. Neither symbol is on my keyboard.) Grammar-school method two: You want the answer in yards, so you change everything into yards. 4/36 X 10 / 50/36 = 0.111111111 X 10 / 1.3888888888 = 0.8 Keep the units method: four inches times ten yards is forty inch-yards -- a new unit of area representing the area of a strip a yard long and an inch wide. Then you divide by fifty inches, and get 0.8 inch-yards/inch. The inches divide out, the answer is 0.8 yards. Carrying the units through also serves as a check on mistakes -- and helps you to find the mistakes. If, for example, your answer to the second example comes out 1.543097, your only clue is "This answer is ridiculous." If you make the same mistake using the third method, you get 200 inch-yard-inches, and it is obvious that you multiplied by fifty inches instead of dividing. ============================================================== SQUARE UNITS A square scrap that's one inch on each side has an area of one square inch; a scrap half an inch wide and two inches long also has an area of one square inch, as you can see by cutting it in half and placing the two halves side by side on top of the first scrap. A square piece of fabric that measures one yard on each side has an area of one square yard. You could divide this square into thirty-six rows of thirty-six one-inch squares. A square yard is equal to 1296 square inches. Exercise: if you have half a yard of fifty- inch fabric, it measures eighteen inches by fifty inches. The area is eighteen inches times fifty inches, which is 900 square inches. Variant exercise: The same half yard of fifty-inch fabric measures half a yard by 1.39 yards. The area is .69 square yards. Or, working it in common fractions: It measures 1/2 yard by 50/36 yards. Half of 50/36 is 25/36, so the piece is twenty-five thirty- sixths of a square yard: it will give you the same amount of bias tape as a piece a yard wide and twenty-five inches long. 25/36 = 0.69444444444444444, which agrees with the result above, so I haven't dropped any decimals. ============================================================== AREAS: HOW MUCH FABRIC D0 I NEED TO MAKE A YARD OF TAPE? Down at the hardware store, a "yard" is the yard appropriate to the item being sold. If you buy a "yard" of rope, you get a linear yard, if you buy a "yard" of concrete, you get a cubic yard. So my spouse and I were talking past each other for quite a while when I talked about buying yards of cloth -- something used by the area is just naturally sold in square yards, so what do you mean by saying that 72" muslin at $10/yard is cheaper than 36" muslin at $6/yard? If you know the area of a wall, and know how many square feet a gallon of paint covers, you know how many gallons of paint you need. If you know the area of a floor, that is almost all the information you need to decide how many foot- square floor tiles to buy. But if you know that you need three and a half yards of fifty-inch fabric to cut out a given pattern, that tells you nothing at all about how much forty-inch fabric you need to make the same pattern. Three and a half yards might be plenty, you might need seven yards, you might have to change the pattern to be able to use the narrower fabric at all. So we tend to forget all about area -- until we need to know how much fabric will make a given length of bias tape. Now, all of a sudden, length and width don't matter -- all that matters is *area*. ------------------------------------------------------------ The area of a rectangle is the length times the width. If both lengths are measured with the same unit, the answer comes out in square units: square inches, square yards, square miles, square centimeters, square meters, etc. Aside from seam-allowance loss, cutting up a piece of fabric and sewing it back together does not change its area. ------------------------------------------------------------ If you know how long and wide a piece of bias tape you want, how big a piece of fabric should you cut into bias strips? The area of the bias strip is its length times its width. You want to cut up a piece with at least that much area. I suppose the rather curious custom of cutting up a square -- which I've seen suggested in several places -- is based on the ease of the calculations: Example: I want ten yards of two-inch tape. 10 X 36 = 360 X 2 = 720 26.832815 -- a 27" square should do the job. Same example expanded: 10 yards times 36 inches/yard is 360 inches long. Multiply this length by the width, 2 inches. The area of the tape is 720 square inches. The square root of 720 is 26.832815, so a twenty-seven-inch square should be slightly larger than the desired tape. (But remember that we are ignoring seam allowances.) ==================================== If any rectangle will do, there is no clear result. To get 720 square inches, you could use two yards of ten-inch fabric, or ten inches of 72" fabric, or one yard of 20" fabric, or twenty inches of 36" fabric, or sixteen inches of 45" fabric . . . On the other hand, if you are buying fabric to make the tape, accepting a rectangle of any shape simplifies things marvelously. You just write on your shopping list that you need 720 square inches, put your calculator in your pocket, go to the shop, select a fabric, note that it's marked 39 inches wide, whip out your pocket tape measure to make sure it's not mis-marked, divide 720 square inches by 39 inches, get 18.46 inches, note that this is slightly more than half a yard, and buy 5/8 yard -- or three fourths, if you think it might be cut crooked. Then at home you wash it thoroughly, straighten the ends, starch it, cut it on the true bias from cut-edge to cut edge, sew the selvages together, draw lines two inches apart parallel to the bias edges -- there's another reason for buying a bit extra: you might well end up with a strip of 1.5" tape at one end of the parallelogram -- sew the crossgrain edges together offset so that the cutting lines join together into one spiral line (being VERY careful that the cutting lines cross at the seam line, NOT at the cut edge), press the seam, and cut along the line that spirals around the tube. In the example given, the crossgrain seam will spiral around the tube of fabric more than once, which makes it hard to pin and hard to press. It might be well to cut the piece in half along one of the cutting lines, sew two of the crossgrain edges together with the bias edges matched at the seam line, then sew the fabric into a tube with a cutting line spiralling around it. If you have an odd number of strips, cut on a line that gives you one more strip on one half than the other, start cutting a strip off the piece that has one too many, and stop halfway across to leave it dangling. Then sew the other piece to the edge that you have shortened (matching cut edges and cutting lines at the stitching line!). Now if you start cutting another strip to make the other edge of the longer piece match the shorter piece, sewing the remaining crossgrain edges together will offset the cutting lines automatically. ============================================================== AREAS: HOW MUCH FABRIC D0 I LOSE TO SEAMS WHEN MAKING TAPE? Calculating loss to seams when making straight-grain or cross-grain tape is easy. The area of a seam allowance is the length of the seam times the width of the seam allowance. (It's a little more complicated for curved seams, but those are irrelevant to the present discussion.) There are two seam allowances to each seam. There are as many seams as pieces, minus one. Suppose, altering the example given above, you want to make ten yards of two-inch crossgrain tape from 39" fabric. If the seam allowances are a quarter inch wide, you will lose one square inch to each seam. (2" X 1/4"/allowance times two allowances/seam) You will need 10yd/39" thirty-nine inch pieces of tape. 360"/39"/piece = 9.23 pieces = 10 pieces = 9 seams = 9 in^2. Since the tape is two inches wide, that's 4.5" of tape lost to the seams. Since you have .77 of 39" extra, all is well. (Another way to calculate is that half an inch of tape is lost to each of the nine seams.) (A way to get out of calculating is to say that 39" is 3" more than a yard, so ten 39" pieces would be 30" more than the ten yards wanted, minus seam allowances. ) When you make bias tape, the length of a seam is the square root of two times the width of the tape. Calculators have square root keys, and if you are using your naked head, you can say that the square root of two is one and a half -- making it a smidgeon over gives you a little allowance for error. But how many pieces? You've got this little piece in the corner that's zero on one side and twice as long as the width of the tape on the other, and then a piece that's cut from the first piece on one side and twice that long on the other, then . . . Well, duh! There are only two seams! You sew the selvages together, then you sew the crossgrain edges together. So calculate the perimeter of the piece of fabric you are using and multiply by the width of the seam allowance. And then, if you are fussy, deduct by the amount you save by not sewing the two ends of the tape together. Then if you are *really* fussy, deduct for counting a little square at each corner twice. ============================================================== EOF