real number system,ordering,rounding numbers

The Real Numbers

In higher standard we explored the real number system. This is a set of numbers that includes the integers or counting numbers, all the rational numbers (numbers that can be represented as a ratio of two whole numbers, such as 1/3 or 3/7) and the irrational numbers (numbers that cannot be represented as a ratio of two whole numbers, such as π, e, and sqrt(2)).

You learned about combining numbers using basic operations like addition (+), subtraction (-), multiplication (× or *) and division (/). You have also been introduced to several other mathematical functions including: exponentiation (^) -- raising something to a power -- and square root (). You may even remember some of the properties that real numbers obey:

Closure

If you operated on any two real numbers A and B with +, -, ×, or /, you get a real number.

Commutativity

A + B = B + A and A × B = B × A

Associativity

(A + B) + C = A + (B + C) and (A × B) × C = A × (B × C)

Inverse

A + -A = 0, A × (1/A) = 1

Identity

A + 0 = A, A × 1 = A

Distributivity

A × (B + C) = (A × B) + (A × C)

These properties are useful to review and keep in mind because they help you simplify more complex calculations.

Order of Operations

In chemistry, as in the other sciences, we will encounter many problems in which the solution involves a series of calculations. When doing more than one operation we need follow a set of rules regarding which calculations to do first. For example, what is the "right" answer to

3 + 2 × 6 = ?

Should we go "left to right" and just do the + first and get 30, or do we do the × first and get 15? Well, in order to avoid confusion and get the correct answer, mathematicians decided long ago that all calculations should be done in the same order. You may have learned the order of operations as being: Please Excuse My Dear Aunt Sally! where the words stand forParentheses, Exponentiation, Multiplication or Division, Addition or Subtraction.

So what is the correct answer for our problem? The order of operations would say that in the absence of parentheses, you would multiply 2×6 first, then add 3, so the result should be 15.

Rounding Numbers

Another issue we need to deal with when we perform operations is how to state the answer. For example, if we are dividing a 20 centimeter wire into 3 equal pieces, we would divide 20 by 3 to get the length of each piece. If we took the time to work this division out by hand -- ack! -- we would get

20 / 3 = 6.666666666666666666.........

The 6 repeats forever. How do we report this number? We round to some usually pre-determined number of digits or decimal places. By "digits" we mean the total number of numbers both left and right of the decimal point. By "decimal places" we specifically refer to the number of numbers to the right of the decimal point.

For comparison, let's try rounding this number to 2 decimal places -- two numbers to the right of the point. To round, look at the digit after the one of interest -- in this case the third decimal place -- and use the rule:

if the digit is 0, 1, 2, 3 or 4 round down
if the digit is 5, 6, 7, 8 or 9 round up

In our example:

6.666666666666.....
    ^              
the next digit is 6 so we round up,

giving 6.67 as the desired answer. If instead we had been asked to round the number 20/3 to 2 digits the answer would have been 6.7 (two digits, one of which is a "decimal place").

Sometimes rounding is the result of an approximation. If you had 101 or 98 meters of some wire, in each case you would have "about 100 meters."

We will round many of our answers in science because the numbers will often be reporting measurements. Numbers representing measurements are only as accurate as the device used for measuring. For example, we could use a standard meter stick marked off in centimeters to measure the length of a wire as 15 cm. If sometime later we cut the wire in pieces, reporting the size of a piece of the wire to nine or ten decimal places would not make sense. You may encounter instructions in labs or in problem sets or on exams which ask you to round a measurement or result to the "nearest cm" or "nearest mL" or "nearest second". We'll look more at these accuracy issues in the section "Numbers in Science."

One other operation which you may or may not be familiar with is often used in chemistry is absolute value . The "absolute value" of a number is sometimes used instead of the number itself. Absolute value is represented as a pair of upright lines around whatever expression or number is supposed to be treated that way. The absolute value of a number expresses its distance from zero, regardless of sign. So, the absolute value of -20 is 20, and the absolute value of 19 is 19. Basically, all numbers become positive but do not change value. Most calculators will take an absolute value, usually called abs().