© 2013 Rex Jaeschke. All rights reserved.
Several years ago, I moved out into the country. By that time, I'd been without cable/pay TV for a couple of years, relying instead on a digital antenna for 30-odd free, over-the-air channels. However, at my new location, the antenna could not pick up any signals. After thinking about the implications for no more than 30 seconds, I decided that my life would be just fine with no TV reception at all. And so, I turned to my reference library, which had been gathering dust for many years. After a cursory glance, I settled on a couple of books on introductory mathematics, including one from my Year 12 High School class in 1969. [By the way, in Australia the abbreviated form of the word mathematics is maths, while in the US it is math. Go figure!] Yes, Dear Reader, I've been reading math books for recreation, but then, I've never been accused of being normal!
In this essay, I'll refresh your memory about some basics, and probably introduce you to some formal terminology for the things you've been using for years. As you read, you might find it useful to refer to some related essays, including the following: August 2012, "What is Normal: Part 5. Numbers and Counting Systems", December 2012, "Symbols and Marks", and March 2013, "What is Normal: Part 6. Weights and Measures".
There is not a test, per se, at the end of this essay; however, as you'll read in "Conclusion", Never say
For most of us, the term literacy implies the ability to read and write. However, in general, it also includes the idea of proficiency with basic numeric operations. We know these collectively as arithmetic, so that's where I'll start. [According to Wikipedia, arithmetic is the oldest and most elementary branch of mathematics.]
The fundamental operations one can perform on pairs of numbers are addition, subtraction, multiplication, and division, which are represented by the symbols + (plus), − (minus), × (multiply), and ÷ (divide), respectively. [Not surprisingly, calculators and computer programs that emulate calculators have these mathematical symbols as keys. However, only the first of these symbols appears on a standard computer keyboard. Yes, there is a "-" key, but that's a hyphen; from a typesetting viewpoint, the minus character is wider. One of the most common uses of writing these operations is in computer programming, where the corresponding symbols typically used are +, -, *, and /.]
Let's start with the simplest kind of operation, addition, and consider the following expression, which contains two terms, the whole numbers 1 and 2:
1 + 2
The result—called the sum—is 3. Technically, both terms are called addends. Addition is commutative; that is, the two terms can be swapped over without affecting the result.
Let's look at subtraction:
5 − 1
The result—called the difference—is 4. Subtraction is not commutative; 5 – 1 is quite different from 1 − 5. The left-hand term is called the minuend, while the right-hand term is called the subtrahend.
Here's an example of multiplication:
2 × 5
The result—called the product—is 10. Even though multiplication is commutative, the terms have different names, the left-hand one being the multiplicand while the right-hand one is the multiplier. (Sometimes, both terms are called factors.)
Consider the following case of division:
13 ÷ 5
The main result—called the quotient—is 2, and the secondary result—called the remainder—is 3. Clearly, division is not commutative. The left-hand term is the dividend while the right-hand one is the divisor, except on Wednesdays between 10 and 11 pm, when they are known as Jack and Mary, respectively. [I'm just checking to see if you really are reading this carefully.]
Oftentimes, we'd like to have an arithmetic expression that contains more than two terms. To understand the meaning of such expressions, we need to know about associativity. Consider the following:
3 + 5 + 2
In this case, the 3 and 5 are added first, and the result is added to 2, giving 10. We say that addition is left-associative. We can write this explicitly using (redundant) grouping parentheses, as follows:
(3 + 5) + 2
(Subtraction, multiplication, and division are also left-associative.) What if we reposition the parentheses, as in:
3 + (5 + 2)
No problem, we still get the result 10. A similar situation occurs with two multiplications, but not with two subtractions or two divisions.
Consider the following expression:
3 + 5 × 4 − 1
Now we have several different operators. It turns out that multiplication and division have (the same) higher precedence
than addition and subtraction, which have the same lower precedence. Adding redundant grouping parentheses to indicate precedence and associativity, we get:
(3 + (5 × 4)) − 1
which results in (3 + 20) – 1 à 23 – 1 à 22.
We can override precedence using grouping parentheses, as follows:
(3 + 5) × (4 – 1)
which results in 8 × 3 à 24.
Zero, Negative Numbers, and Infinity
The concept of zero has been around for a good while. In general, people aren't unduly confused by a count of "none". However, negative numbers cause some people great confusion. How can you have fewer than none of anything? [Note carefully that I did not say "less than none". See "Conclusion" below.] Of course, some of these same people have overdrawn their bank accounts, so they really have experienced having fewer than zero of their own dollars!
The idea that revolutionized things with respect to numbers was the invention of the number line, a horizontal line containing a point for each number with equal spacing between whole numbers. In the drawing below, the line segment shows –5 on the left and +5 on the right. However, as shown by the arrows, the line extends indefinitely in each direction, and we talk about the line's so-called "limits" as being minus infinity (–∞) and plus infinity (+∞), respectively.
When we think of numbers simply being points on this line, zero is just another point, and so are negative numbers. And it certainly helps us deal with arithmetic expressions such as 4 – 6, whose result is –2. It also allows us to deal with numbers whose values are "between the whole numbers". [It's also useful to use this line to represent historic time. The point 0 represents the Birth of Christ, with the year 1 AD starting immediately to the right, and the year 1 BC ending immediately to the left. As shown, there is no year 0, only a time-instant of zero.]
There is one unusual kind of division expression, that having a divisor of zero. Just how many zeros are there in any value? In general, mathematicians consider the result to be undefined. [In most modern computers, a negative whole number divided by zero results in minus infinity, while a positive whole number divided by zero results in plus infinity. This allows computers to do useful work with infinities even though they really don't exist. Of course, we mere mortals might then ask such questions as, "If infinity is the largest value, why can't I add 1 to it?" and "If there are infinity zeros in 1, how can there only be infinity zeros in 2? Should there be twice as many zeros in 2?" Of course, the correct response to these questions is, "Shut up and eat your vegetables, smarty pants!" And to make things really interesting, zero divided by zero results in a NaN, which stands for "Not-a-Number". Of course, computers can do useful things with these as well, but I wouldn't lose any sleep worrying about that if I were you.]
According to Wikipedia, 'A fraction (from Latin: fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters.'
A fraction is written with a numerator overtop of a denominator, separated by a horizontal or slanted line; for example, ½, ¾, 3/8, and 27/100 represent a half, three quarters, three eights, and 27 hundredths, respectively. In general, if the value of the numerator is smaller than that of the denominator, we have a proper fraction; otherwise, we have an improper fraction.
I live in a non-metric country, and on the ruler I have in front of me I see divisions of ½, ¼, 1/8, and 1/16 of an inch. Some typewriters even had the first two of these as keys; however, I've never seen them on a computer keyboard. In my world, I see fractions on a regular basis. Some examples are:
- When I write a check to pay a bill (yes, I'm old-fashioned), as well as writing the amount as a number, I must also write it in words, except that the fraction dollar part must be written as a fraction. For example, a check for $23.76 would say "Twenty Three dollars and 76/100". [By the way, if you use Microsoft Word, it provides a field code that can convert a financial amount expressed as a number into the corresponding text for a check.]
- Last week, I bought gasoline and the price was shown on an advertising board outside the gas station. It said, $3.57 9/10 per gallon. Clearly this is a marketing ploy in that the fraction is not so easy to read and you are supposed to think the price is $3.57 when it's almost $3.58. My guess is that most commuters don't even notice.
- Originally, the U.S. dollar's value was based on the value of the Spanish real, a silver dollar that could be divided into eight parts. As a result, when the U.S. stock market began, stock values were based on one-eighth fractions. [In the US, an eighth was called a bit—a value of 12½ cents—so a quarter was worth two bits. Such usage of this term occurs in literature and in the lyrics of popular songs.]
In every-day English, we usually use fraction to mean a small amount, as in, "He only saved a fraction of his take-home pay." Most likely, that fraction would be significantly less than 5%.
From the deep, dark recesses of my mind, I recall a puzzle from my early school days. It goes something like this (and clearly is intended to be spoken rather than written): A man is locked inside a room that contains only a table. How does he get out? Well, he rubs his finger on the wall until he develops a sore—a homophone of saw. He uses that saw to cut the table in half. Now two halves make a whole–which is a homophone for hole—and he crawled out that hole.
According to Wikipedia, 'a percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%"'. For example, did you know that 76% of all statistics are made up on the spot? Of course, this is a joke; it's actually 59%! So, by writing 76%, I mean "76 out of every 100", or, written as a fraction, 76/100.
Here in the US, most states (and some cities and counties) levy a sales tax on most items sold commercially. In my state, Virginia, that rate is 5.3%; that's 5.3 cents for every 100 cents ($1) of the purchase price. Now what confuses many foreign visitors to the US is that the amount of sales tax is not included in the advertised price. So when they see something on sale for $4.99, they hand over $5 to the cashier expecting to get one cent change. However, they are politely told they need more to cover the sales tax. Now, strictly speaking, 5.3% of $4.99 is 26.4 cents, but as we have no coins of smaller value than 1 cent, that gets rounded down to 26 cents, making a total bill of $5.25.
Another common use of percentages is in tipping. In the US it is pretty standard to tip a waiter 15% for decent service at a table. However, if one is eating at a counter, 10% is about right. That's 15 and 10 cents, respectively, for every dollar of the bill.
Every day in the news, we're told how high the unemployment rate is, as a percentage. In some areas it's as high as 15–20%; that's 15–20 out of all able-bodied people of working age.
Then there's the interest rate on savings accounts and loans.
Short of doing a (possibly) complicated calculation, how then is one to calculate exactly or to estimate a percentage? In most cases, one can get a reasonable estimate. In the case of a 10% tip, one simply calculates one-tenth of the total by taking the total and moving the decimal place one position to the left. For example, a 10% tip on a bill of $15.67 is $1.57 (after rounding the 1.567 up to 1.57), which when added together comes to $17.44. In the case of a 15% tip, calculate a 10% tip then add to that half as much again. So, a $15% tip on $15.67 comes to $1.57 + $0.77 (or $0.78). In the case of the 5.3% sales tax, calculate 10%, then halve that to get 5%, and then add a smidgen. Of course, calculators or apps on a mobile phone will do all this for you, but if you don't use your "little grey cells" occasionally, don't be surprised if they atrophy.
Back in the 1960's, I recall my father having a Ready Reckoner, which according to Wikipedia, is a book "aimed at traders that catalogued the results of various routine calculations such as the percentages or multiples of various sums of money."
Weapons of Math Instruction
A number of years ago, soon after airport security started to get Über-serious here in the US, the following, very clever, piece of writing appeared. [Unfortunately, I have not been able to track down the author's name.
At New York's Kennedy airport today, an individual later discovered to be a public school teacher was arrested trying to board a flight while in possession of a ruler, a protractor, a set square, a slide rule, and a calculator.
At a morning press conference, Attorney General John Ashcroft said he believes the man is a member of the notorious al-gebra movement. He is being charged by the FBI with carrying weapons of math instruction.
"Al-gebra is a fearsome cult.", Ashcroft said. "They desire average solutions by means and extremes, and sometimes go off on tangents in a search of absolute value. They use secret code names like "x" and "y" and refer to themselves as "unknowns", but we have determined they belong to a common denominator of the axis of medieval with coordinates in every country.
"As the Greek philanderer Isosceles used to say, there are 3 sides to every triangle," Ashcroft declared.
When asked to comment on the arrest, President Bush said, "If God had wanted us to have better weapons of math instruction, He would have given us more fingers and toes."
"I am gratified that our government has given us a sine that it is intent on protracting us from these math-dogs who are willing to disintegrate us with calculus disregard. Murky statisticians love to inflict plane on every sphere of influence," the President said, adding: "Under the circumferences, we must differentiate their root, make our point, and draw the line."
President Bush warned, "These weapons of math instruction have the potential to decimal everything in their math on a scalene never before seen unless we become exponents of a Higher Power and begin to factor-in random facts of vertex."
Attorney General Ashcroft said, "As our Great Leader would say, read my ellipse. Here is one principle he is uncertainty of: though they continue to multiply, their days are numbered as the hypotenuse tightens around their necks."
Back in the 1980's, I used to fly through Boston's Logan airport (BOS) on a very regular basis. At that time, the city had a growing problem with teenage pregnancies and young girls subsequently dropping out of high school. To that end, there was a very clever advertising campaign, signs for which were posted on the jet ways going to/from the planes. Next to a picture of a pregnant girl was the text, "Make sure your daughter learns how to add and subtract before she learns how to multiply!"
Regarding my earlier comment about tests and "Never say Never", I'm reminded of the orientation night I spent at my son's high school the week after he started his freshman (first) year. As a group, the parents spent 10 minutes with each teacher with whom their child was taking a class. The math teacher said, "Before you tell me that you learned all this stuff when you were in school, but never ever found a use for it, consider the following" and he projected one of Gary Larson's Far Side cartoons. The picture showed a man waiting for admission at the Pearly Gates of Heaven. St. Peter said that although the records showed the man had lived a decent life, before he could be admitted, he had to answer one question. "Two trains are going in opposite directions …" Many of us have been faced with such questions in school math class, so you just never know when you might need something you learned a long time ago. [In my case, 16 years after I was exposed to differential calculus for the first time, I actually found my one and only use for it, in a computer program I was working on that monitored and controlled hydroelectric dams. Who'd have thought?]
There's an old joke about asking a lawyer a question like, "What answer do you get when you add 10 and 15?" The lawyer replies, "What answer do you want?"
I'll end with one of my pet peeves: People insist on misusing the term number when they really mean digit. A number is made up of one or more digits. A number cannot contain another number! Now that you've been reminded about this, watch out for the number-vs.-digit police (who sometimes double as the fewer-vs.-less police as well).