Friday, October 21, 2016

Friday Gen Eds MS2: Basic Types of Numbers

The second post in the math/science part of my "Gen Eds in a Nutshell" series. It's a series of ten subjects you might study in a general education or "liberal arts" core at a university or college. I've already done the subject of philosophy, and I'm half way through the world history subject on Wednesdays. I'm combining the last two into one series on Fridays.

Thus far in the math/science subjects:
Basic Types of Numbers
1. It seems to me that it is perfectly acceptable to expect someone with a college degree to have some basic mathematical skills. Such basic math was historically part of the ancient and medieval curricula. Arithmetic and geometry were both part of the so called "quadrivium" of early medieval education, and math was part of the other two as well: music and astronomy.

Of course most people will know how to add, subtract, multiply, and divide before they even enter high school, hopefully before middle school. A child of one year old knows "less than." Before we can speak, we can signal that we have "less than" we want. By two years old we think we know when our sister or brother has "greater than" we do.

We know the circle shape of the wheels on our bikes. We may not know to call the shape of our doors a rectangle, but probably we do. Most of us learn what a triangle and a square are easily enough.

2. We learn to count early on. We may not know that the counting numbers are also integers. They are whole numbers too. We can use these numbers even if we don't know what to call them.

"How many people in your party?"

"One, two, three, four... a table for four please."

"Jim wasn't able to come tonight."

"OK, subtract one from that... a table for three please."

So the counting numbers are 1, 2, 3, 4... If we add the number 0 (which wasn't really used until Brahmagupta fully gave it notation in the 600s), then we have the whole numbers: 0, 1, 2, 3, 4...

3. These are of course highly useful numbers. We can add them to each other. We can subtract one from another. We do these operations all the time in real life.

We can add multiple times, which we call "multiplication." If we add 3 to itself 3 times (3+3+3) then we have 3x3 which is 9. We can do the same thing with subtraction. if we start with 16 and we subtract four till we get 0, then we have 16-4-4-4-4. We had to subtract 4 times, so 16 "divided by" 4=4.

We multiply and divide all the time. Sometimes we multiply a number by itself, such as when we are finding the "square footage" of a room (i.e., the area of its floor). We "square" a number when we multiply it by itself (e.g., 10= 10 x 10 = 100). We "cube" a number when we multiply it by itself three times (103 = 10 x 10 x 10 = 1000). In physics, we might do this to find out the "volume" of a room that is 10 feet long and 10 feet wide and 10 feet tall--a cube, in other words.

We call this task of multiplying a number times itself, "raising to a power." The number of times we multiply a number by itself is the "power" to which we are raising it and we call that number an "exponent." You can also go the other direction. The square root of 4 is 2 because 2 x 2 = 4. The cube root of 1000 is 10 because 10 x 10 x 10 = 1000.

4. So the whole numbers are 0, 1, 2, 3, 4... all the way to "infinity," the word we use for numbers never stopping in how high they go, numbers going on forever. The 1, 2, 3, 4 part, the counting numbers are also called "positive integers." They are called positive integers because we can imagine those numbers decreasing the other direction below zero as well, negative integers.

Think of it this way. What if I imagined a number that, when added to 3, gave me 2? It would, in effect, be the same as subtracting 1. We call this number -1 or "negative one." If we imagine that negative numbers go on to negative infinity, we have -1, -2, -3, -4, -5... and so forth.

The whole set of numbers of this sort, positive and negative (1, 2, 3, 4... and -1, -2, -3, -4...), along with 0, make up the set of integers.

5. When we divide a number by another, say 8/4, then we have, in effect, something called a "fraction." In this case, 8/4 is 2. In this case, since this "ratio" of 8 to 4 is greater than one, we might call it an "improper" fraction.

So what is a "proper" fraction. It is a ratio of this sort that is less than one. I ordered a pizza last night and between my daughter and I, we consumed "half" of it. That is to say, we only consumed a "fraction" of the pizza. Of the 8 total slices, we consumed 4. So we ate 4 out of the 8, which is 4/8, a fraction.

Of course there is an easier way to write 4/8. We can "reduce" this fraction. Any number divided by 1 is the same number, and 2/2 is one. So if I divide 4/8 by 2/2 we get 2/4. If we do it again, we get 1/2, a much more familiar way to write 4/8.

Of course there are all sort of fractions: 3/4, 5/9, 13/37. We can come up with an infinite number of them. These are called "rational numbers," because they are numbers that can be put in the form of a ratio like 3/5. All the integers fit into this set, because all the integers can be expressed in fractional form (like 4/2 is the same quantity as the integer 2).

Fractions can also be expressed as decimals, numbers presented in the form of tens and tenths. Because we have 10 fingers, Arabs and others in the Middle Ages developed systems of using numbers to express fractions in tenths. So 0.1 is one tenth, 0.2 is two-tenths. If we go further to the right 0.01 is one tenth of one tenth or one hundredth (1/100). 0.001 is one tenth of one hundredth or one thousandth (1/1000).

Fractions can thus be converted into decimals. 1/5 turns out to be 0.2. 3/7 turns out to be 0.4285714 and then the decimal repeats 285714 over and over forever. We call this a "repeating decimal."

6. As math has progressed, we have realized that some numbers in decimal form go on forever without repeating. These are numbers that relate closely to certain shapes and patterns in the real world. Because they cannot be put into the form of a ratio, they are called "irrational" numbers.

One of the best known is the ratio of the circumference of a circle (the distance one time around its edge) and the diameter of a circle (the distance across the circle at any point through its center). This number is known as "pi" (π) and is approximately 3.1415926535...

Similarly, there are some "roots" of numbers that do not come out "perfect." They are not "perfect squares" like 2 is the square root of 4. These roots also can be expressed as decimals that go on forever.

They relate especially to the shape of a triangle. For example, let's say that I have a "right triangle" (one with one angle that is 90 degrees), with two sides that have the same measure. Let's say its longest side is 7 inches. How long are each of the two other sides? As we'll see when we get to geometry, the answer is the "square root" of 7 (√7). In decimal form, this number goes on forever: 2.6457513... It is, in other words, an irrational number.

Another very important number is e, which stands for the answer to (1 + 1/n)n as n becomes larger and larger. It turns out to be 2.718281828459... This number is important for calculating something called "compound interest," which explains how much extra we often pay the bank for our mortgages.

7. All the numbers we have mentioned so far--rational numbers (which includes all the integers) and irrational numbers--together form a set of numbers known as the set of "real" numbers.

Starting in the 1500s, certain mathematicians began to explore solutions to various equations that involved taking the square root of negative 1 (-1), written √-1. [1] In the twentieth century, the physics of the atom was discovered to involve this square root extensively, written as "i" for short.

Eventually, another whole set of numbers--all the real numbers multiplied by i--developed. These came to be called "imaginary numbers." When we combine the real number system with this imaginary number system, we get the "complex number system," which finally constitutes all possible numbers.

All numbers can thus be expressed in the form of a + bi, where a is the real part and bi is the imaginary part. If b equals 0, then we simply have a real number.

Next Week: The Atom and Quantum Physics

[1] For example, what is the solution to x2 + 1 = 0


Martin LaBar said...

Well counted.

Ken Schenck said...

:-) Thanks for listening. :-)