Older adding machines added by moving beads (or gears) forward and
subtracted by moving them backward. Modern calculators and computers
subtract a number by making it negative and then adding it. Both
methods come out with the same result.
As I mentioned in my Biography, my day job is software engineering. I set up the instructions for the computer to manipulate values, using symbols to represent the numbers. The first step of programming is understanding basic mathematic operations and how they can produce useful information.
You may think that you will never program any computer; but you have probably used spreadsheets. Even the entry in a spreadsheet to sum a column of numbers is a little computer program. The skills necessary for problem solving, including critical thinking, data manipulation, and how to solve mathematical problems are the same with or without the computer.
This week's lecture covers real numbers and their properties.
The concept of real numbers developed in stages. Most courses in mathematics follow the historic sequence, starting with the natural numbers (also called the “counting numbers”). People were using natural numbers (one, two, and so forth) before the start of writing.
The concept of zero, to complete the set of whole numbers is as old as counting; but until the development of the Arabic numbering system, zero was not considered a true number. Zero is still considered a special number because it can not be used as a divisor.
Negatives of natural numbers, which complete the set of integers, were also used before they were included with the other whole numbers. Of course, ancient merchants subtracted sold items from their inventory, but they viewed subtraction as the reverse of addition; for example, they asked “what number of items must be added to eight to make ten” and demonstrated that the answer matched “what number of items are left when eight are taken away from ten”. Another view of subtraction is adding a negative value. These three views are the same.
10 – 8 = 2
x
+ 8 = 10; What is x?
10 + (-8) = 2
Older adding machines added by moving beads (or gears) forward and
subtracted by moving them backward. Modern calculators and computers
subtract a number by making it negative and then adding it. Both
methods come out with the same result.
Rational numbers are the result of dividing two integers. The term “rational” is related to “ratio” and “rationing”; the alternate meaning, “ability to think” originated in the use of long division as a sanity test.
Euclid developed the concept of irrational numbers by proving that there is no pair of numbers such that (p/q)² = 2, so the square root of 2 is not rational. The proof involved factoring, which is not covered until chapter 6 of our textbook; however you can read an explanation in the University of Toronto Math Question Corner
Computers work with two types of numbers, “integers” and “real”. The “real” numbers are handled in the form of fractions. A fraction cannot be equal to an irrational number, but the computer can come very close. On the computer, the size of a real number determines the number of digits it can hold; even for small real numbers, the accuracy is better than one part in a million. In theory, we can find a rational number within any margin of error, no matter how small, of any irrational number. More formally:
For any real number, R, and any margin of error, e , there is a rational number Q such that
|R-Q| < e .
More about absolute values later)
The number line provides a graphic representation of the set of real numbers, or a specific subset. Some number-line conventions are:
The selected subset is in a contrasting color (such as red),
The end-point marker, "[" or "]" indicates that the end point is included in the subset
The end-point marker, "(" or ")" indicates that the end point is not included in the subset
The end point marker, "<" or ">" indicates that the subset continues to infinity.
One easy method of drawing a number line in Outlook Express is plain text. Set the style (the style selector is just to the left of the "B" on the tool bar) to "formatted" to retain extra spaces. Use the color selector and the bold setting to highlight the portion of the line in the number set.
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -----------------------------(---------------------->
The mathematic operations on a basic calculator include addition, subtraction, multiplication and division. A scientific calculator has a number of other function buttons, such as exponents. The precedence of operations in standard mathematics is the same as the Arithmetic Precedence in Spreadsheets (from first to last)
Items in parenthesis
Arithmetic Opposite
Exponents
Multiplication and division
Addition and subtraction
Absolute Value
On the calculator, the button to calculate the arithmetic opposite (by reversing the sign) is marked “+/-”. In arithmetic, the same minus sign can be used in both for subtraction and reversing the sign of a number; the use can be determined by the positioning, since a minus sign used for subtraction comes between two numbers. Since the minus sign used for the arithmetic opposite only pertains to one number; it is also called a “unary minus sign”.
As noted above, we can perform subtraction by reversing the sign and adding. In the same way, we can divide by inverting the divisor (the number to divide by) and multiplying. On the Microsoft calculator, the arithmetic inversion key is marked “1/x”.
Division by a fraction is always started by first inverting the divisor (which can be done by interchanging the numerator with the denominator). Before computers and calculators were available, many people inverted the divisor because multiplication with paper and pencil is easier than long division.
The absolute value of a number |number| is its distance from zero. Notice on the number line that +4 and -4 are at the same distance (4) from the zero point so they have the same absolute value. The absolute value is always zero or positive, the usual definition for real numbers is:
|x| = x if x>0
-x if x<0
0 if x=0The third case "x=0" is often included with the first to make
x if x≥0
on the other hand, it could also be lumped in with the second case
x if x≤0
and so it makes sense to consider zero separately.
Remember that the
absolute value has the lowest priority, so you always perform the
calculations within the vertical bars before you calculate the
absolute value. For example,
|7 + (-4)| = |3| = 3.
One of the simplest tools of problem solving is retaining the unit of measure while calculating. As a sample problem, suppose we want to know the speed, in miles per hour, of a golf cart that goes 35 feet per second. We know that there are 60 seconds in an a minute and 60 minutes in an hour. We also know that there are 5280 feet in a mile.
35 feet x 60 seconds x 60 minutes x 1 mile second minute hour 5280 feet so 35 feet x 60 seconds x 60 minutes x 1 mile second minute hour 5280 feet so the speed is 35*60*60/5280 miles/hour or about 23.9 mph.
A hint in formatting: if you set the style of the text to formatted, the text and numbers are easier to align.
This method may seem silly and cumbersome, but confusing feet with meters in calculating altitude has caused at least one costly crash.
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Credits:
Soroban picture from http://www.giocomania.org/Legno/soroban/soroban.htm
A classic article described the invisible steel plate that slides into place whenever the eyes see a mathematical formula and prevents the information on the page from reaching the brain. Have you noticed this mental block in yourself or your colleagues?
Try using your excel spreadsheet to evaluate “(A1 + B1) * (C1 + D1)” and the same expression without the parenthesis. Do the results match what you would expect from our textbook?
How would you answer question 107 on page 51 of our textbook?
Start up your Microsoft calculator and
change the mode to “Standard”.
Type in the
calculation “4 + 5 · 6” and hit
“=”; notice the result.
Change the mode to
“Scientific”.
Type in the calculation “4 + 5 ·
6” and hit “=”; notice the result.
How do the
results relate to the mode?