I Used To Know That Part 4

The basic unit of length in the metric system is the meter, with subdivisions and multiples for measuring little things and big things. Most commonly used are the millimeter (a thousandth of a meter), the centimeter (a hundredth of a meter, or ten millimeters), and the kilometer (a thousand meters).

To convert between the two:* 1 inch = 2.54 centimeters, so to convert inches to centimeters, multiply by 2.54. To convert centimeters to inches, divide by 2.54. Remember that a centimeter is shorter than an inch, so you should have a larger number of centimeters.

* 1 yard = 0.91 meters; 1 meter = 1.09 yards, or 3.3 feet. Yards and feet are shorter than meters, so you will have a larger number of them.

* 1 mile = 1.6 kilometers; 1 kilometer = 0.625 () of a mile. This time the metric unit is smaller, so you have more kilometers than miles.

* A nautical mile is about 1.15 miles, or exactly 1,852 meters.

* WEIGHT In American units weight is measured in ounces, pounds, a hundredweight (short), and tons: 16 ounces (oz.) = 1 pound (lb., from libra, Latin for pound); 100 pounds = 1 hundredweight (short); 200 hundredweight (2,000 lb.) = 1 ton. This is sometimes called a short ton, because the imperial system in the U.K. uses a long ton of 2,240 lb. And they also use a measurement of stones (14 pounds = 1 stone).

In the metric system weight is measured in grams or kilograms. (You can have milligrams and centigrams, but a gram is already pretty small, so unless you're a pharmacist or something of that sort, you don't often need them.) A kilogram is 1,000 grams.* 1 gram (or g) = about 0.0353 ounce, so to convert grams to ounces, multiply the number of grams by .0353. To convert ounces to grams, divide by .0353.

* 1 kilogram (or kilo or kg) is about 2.2 pounds, so multiply kilograms by 2.2, divide pounds by 2.2.

* A metric ton is 1,000 kilograms, or 2,205 pounds, just a bit more than an American ton.

* VOLUME In the American system volume is measured in fluid ounces, pints, quarts, and gallons; in the metric system it is measured in liters. This becomes even more complicated because the value of the units in the United States differs from the imperial system in the U.K.

In the United States 16 fluid ounces make a pint. But the U.S. pint and gallon are smaller than the U.K. ones. To convert U.S. pints to liters, divide by 2.1.

In the U.K. 20 imperial fluid ounces make 1 imperial pint, 2 imperial pints make 1 imperial quart, and there are 4 quarts (8 pints) in an imperial gallon. A liter is about 1.75 pints, so to convert imperial pints to liters, divide by 1.75; to convert liters to imperial pints, multiply by 1.75 (pints are smaller, so you will have more of them).

Algebra and Equations.

Algebra is the branch of math that uses symbols (normally letters of the alphabet) to represent unknown numbers, along the lines of a + b = 5. If you a.s.sign a value to a, you can calculate b: If a = 2, then b = 3. This is known as an algebraic equation.

The main thing to remember when solving equations is that one side of the = sign is equal to the other side, so anything that you do to one side, you need to do to the other.2 For example, to solve the equation3a + 1 = 16 - 2a, you first add 2a to each side, giving:5a + 1 = 16.

Then subtract 1 from each side, giving5a = 15.

Now you can divide both sides by 5 and announce proudly that a = 3.

* SIMULTANEOUS EQUATIONS A simultaneous equation is a more complicated form of algebraic equation, in which you have two or more unknowns. The general rule is that you must have exactly the same number of equations as you have unknowns in order to find the value of each. If you have fewer equations, there will be lots of solutions and no way to choose between them. If you have too many equations, there will be no solution at all.

This a.s.sumes that the equations are all different and don't contradict each other. For example:a + b = 6,

2a + 2b = 12.

are no good as a pair of simultaneous equations, because they both say exactly the same thing, while:a + b = 6,

a + b = 7.

will not work either, because there's no way both of them can be true at the same time.

Here's a look at a better-behaved set of simultaneous equations:a + b = 6,

a - b = 2..

A way of solving these is to add the two equations together, soa + a + b - b = 6 + 2

or, more simply, 2a = 8 (because the +b and -b cancel each other out).

From there you can calculate that a = 4 and, because a + b = 6, b must equal 2. Which is verified by the second equation, 4 - 2 = 2.

The principle remains the same regardless of how many unknowns you have:a + b + c = 24,

a + b - c = 16,

2a + b = 32.

Add the first two equations together and you get 2a + 2b = 40 (because this time the c's cancel each other out).

Now look at the third equation. It's very similar to the sum of the first two. Subtract one from the other:(2a + 2b) - (2a - b) = 40 - 32.

The a's cancel each other out, so 2b - b (in other words, b) = 8.

Go back to the third equation, which contains only a's and b's, and subst.i.tute 8 for b:2a + 8 = 32.

Deduct 8 from each side of the equation to give2a = 32 - 8 = 24,

which means that a = 12.

Now go back to the first equation and subst.i.tute both a and b:12 + 8 + c = 24,

20 + c = 24,

c = 24 - 20 = 4.

Verify this by going to the second equation:12[a] + 8[b] - 4[c] = 16,

which is true.

* QUADRATIC EQUATIONS These are more complex again, because they involve a square-that is, a number multiplied by itself and written with a raised 2 after it-so 16 is 42, and 36 is 62. Thus, 4 is the square root of 16, and 6 is the square root of 36. The symbol for a square root is . Actually, (-4)2 is also 16, so 16 has two square roots: +4 and -4. Any positive number has two square roots. A negative number doesn't have any square roots at all, because if you multiply a negative by a negative, you get a positive.

An algebraic expression can also be a square: the square of a + 4 is (a + 4) x (a + 4). You do this by multiplying each of the elements in the first bracket by each of the elements in the second:(axa) + (ax4) + (4xa) + (4x4) = a2 + 8a + 16.

To solve a quadratic equation, you need to turn both sides of it into a perfect square, which is easier to explain if we look at an example:a2 + 8a = 48.

The rule for "completing the square" in order to solve a quadratic equation is, "Take the number before the a, square it, and divide by 4." For example, 8 squared (64) divided by 4 is 16, so we add that to both sides; rea.s.suringly, we already know that adding 16 to this equation will create a perfect square, because we just did it in the previous equation:a2 + 8a + 16 = 48 + 16 = 64 .

Taking the square root of each side gives:a + 4 = 8 (because 8 is the square root of 64).

Again, we know that a + 4 is the square root of a2 + 8a + 16, because it was part of the sum we did on the previous page. Anyway, we now have a simple sum to establish that a = 4.

Wait a minute, though. Taking the square root of both sides of an equation is not allowed. Why is this? Because a positive number like 64 has two square roots, +8 and -8. So the truth of the matter is that actuallya + 4 = +8 or -8,

so a equals either +4 or -12..

Although this example is an easy one, the beauty of algebra is that the same principle applies whatever the numbers involved. So, to repeat: The rule for "completing the square" in order to solve a quadratic equation is, Take the number before the a, square it, and divide by 4.) So if your equation isa2 + 12a + 14 = 33, you first simplify the equation by getting rid of the 14. Subtract it from both sides to leave:a2 + 12a = 33 - 14 = 19.

Square the 12 to give 144, divide by 4 to give 36, and-as always-add that to both sides:a2 + 12a + 36 = 19 + 36 = 55.

The square root of that gives youa + 6 = 55 = (approximately) 7.4, or, of course, -7.4.

Deduct 6 from each side to leave the simple statement a = 1.4 or -13.4.

You can check that this is right by going back to the original equation and putting in a = 1.4:a2 + 12a + 14 = 33 becomes (1.4 x 1.4) + (12 x 1.4) + 14 = 1.96 + 16.8 + 14 (near enough for the purposes of this exercise) = 2 + 17 + 14 = 33.

QED, as they say in math (or essentially, problem solved). You'll find it also works out with a = -13.4.

Geometry.

Geometry is about measuring lines and angles and a.s.sessing the relations.h.i.+p between them, so let's start with some ways of measuring.