IV. Something about Mathematical Sentences

In all branches of mathematics you need to write many sentences about numbers. For example, you may be asked to write an arithme­tic sentence that includes two numerals which may name the same number or even different numbers. Suppose that for your sentence you choose the numerals 8 and 11—3 which name the same number. You can denote this by writing the following arithmetic sentence, which is true: 8= 11—3.

Suppose that you choose the numerals 9+6 and 13 for your sen­tence. If you use the equal sign (=) between the numerals you will get the following sentence 9+6=13. But do 9+6 and 13 both name the same number? Is 9+6= 13 a true sentence? Why or why not?

You will remember that the symbol of equality (=) in an arith­metic sentence is used to mean is equal to. Another symbol that is the symbol of non-equality (≠) is used to mean is not equal to. When an equal sign (=) is replaced by a non-equal sign (≠), the opposite meaning is implied. Thus the following sentence (9+6 ≠ 13) is read: nine plus six is not equal to thirteen. Is it a true sentence? Why or why not?

An important feature about a sentence involving numerals is that it is either true or false, but not both.

A mathematical sentence that is either true or false, but not both is called a closed sentence. To decide whether a closed sentence containing an equal sign (=) is true or false, we check to see that both elements, or expressions, of the sentence name the same num­ber. To decide whether a closed sentence containing a non-equal sign (≠) is true or false, we check to see that both elements do not name the same number.

As a matter of fact, there is nothing incorrect or wrong, about writing a false sentence; in fact, in some mathematical proofs it is essential that you write false sentences. The important thing is that you must be able to determine whether arithmetic sentences are true or false.

The following properties of equality will help you to do so.

Reflexive: a = a

Symmetric: If a = b, then b — a.

Transitive: If a = b and b = c, then a = c.

The relation of equality between two numbers satisfies these basic axioms for the numbers a, b, and c.

Using mathematical symbols, we are constantly building a new language. In many respects it is more concise and direct than our everyday language. But if we are going to use this mathematical language correctly we must have a very good understanding of the meaning of each symbol used.

You already know that drawing a short line across the = sign (equality sign) we change it to ≠ sign (non-equality sign). The non-equality symbol (≠) implies either of the two things, namely: is greater then or is less than. In other words, the sign of non-equa­lity (≠) in 3+4 ≠ 6 merely tells us that the numerals 3+4 and 6 name different numbers; it does not tell us which numeral names the greater or the lesser of the two numbers.

If we are interested to know which of the two numerals is grea­ter we use the conventional symbols meaning less than (<) or grea­ter than (>). These are inequality symbols or ordering symbols be­cause they indicate order of numbers. If you want to say that six is less than seven, you will write it in the following way: 6<7. If you want to show that twenty is greater than five, you will write 20>5.

The signs which express equality or inequality (=, ≠, >, <) are called relation symbols because they indicate how two expressions are related.

V. Rational Numbers

In this chapter you will deal with rational numbers. Let us begin like this.

John has read twice as many books as Bill. John has read 7 books. How many books has Bill read?

This problem is easily translated into the equation 2 n = 7, where n represents the number of books that Bill has read. If we are allowed to use only integers, the equation 2 n =7has no solution. This is an indication that the set of integers does not meet all of our needs.

If we attempt to solve the equation 2 n = 7, our work might appear as follows.

2 n =7, 2 n 7 2 77 7

2 = 2, 2 x n = 2, 1x n= 2, n = 2.

The symbol, or fraction, 7/2 means 7 divided by 2. This is not the name of an integer but involves a pair of integers. It is the name for a rational number. A rational number is the quotient of two in­tegers (divisor and zero). The rational numbers can be named by fractions. The following fractions name rational numbers:

1 8 0 3 9

2, 3, 5, 1, 4. a

We might define a rational number as any number named by n

where a and n name integers and n ≠ 0.

Let us dwell on fractions in some greater detail.

Every fraction has a numerator and a denominator. The deno­minator tells you the number of parts of equal size into which some quantity is to be divided. The numerator tells you how many of these parts are to be taken.

2

Fractions representing values less than 1, like 3 (two thirds) for example, are called proper fractions. Fractions which name a number

23

equal to or greater than 1, like 2 or 2, are called improper fractions.

1

There are numerals like 1 2 (one and one second), which name a whole number and a fractional number. Such numerals are called mixed fractions.

Fractions which represent the same fractional number like

12 3 4

2, 4, 6, 8, and so on, are called equivalent fractions.

We have already seen that if we multiply a whole number by I we shall leave the number unchanged. The same is true of fractions since when we multiply both integers named in a fraction by the same number we simply produce another name for the fractional number.

11

For example, l x 2 = 2 We can also use the idea that I can be as

2 3 4

expressed a fraction in various ways: 2, 3, 4, and so on.

 

1 2

Now see what happens when you multiply 2 by 2. You will have

1 12 1 2x1 2

2 = 1 2 = 2 x 2 = 2x2 = 4. As a matter of fact in the above

operation you have changed the fraction to its higher terms.

6 6 2 6: 2 3

Now look at this: 8: 1 = 8: 2 = 8: 2 = 4.

In both of the above operations the number you have chosen for 1 is

2 6

2 In the second example you have used division to change 8 to lower

3

terms, that is to 4.The numerator and the denominator in this fraction are

relatively prime and accordingly we call such a fraction the simplest fraction for the given rational number.

You may conclude that dividing both of the numbers named by the numerator and the denominator by the same number, not 0 or 1 leaves the fractional number unchanged. The process of bringing a fractional number to lower terms is called reducing a fraction.

To reduce a fraction to lowest terms, you are to determine the greatest common factor. The greatest common factor is the largest possible integer by which both numbers named in the fraction are divisible.

From the above you can draw the following conclusion⁶: mathe­matical concepts and principles are just as valid in the case of rational numbers (fractions) as in the case of integers (whole num­bers).

VI. DECIMAL NUMERALS

In our numeration system we use ten numerals called digits. These digits are used over and over again in various combinations..Suppose, you have been given numerals 1, 2, 3 and have been asked to write all possible combinations of these digits. You may write 123, 132, 213 and so on. The position in which each digit is written af­fects its value. How many digits are in the numeral 7086? How many place value positions does it have? The diagram below may prove helpful. A comma separates each group or period. To read 529, 248, 650, 396, you must say: five hundred twenty-nine billion, two hundred forty-eight million, six hundred fifty thousand, three hundred ninety-six.

 

Billions period	Millions period	Thousands period	Ones period
Hundred billions Ten-billions One-billion	Hundred millions Ten-millions One-million	Hundred- thousands Ten-thousands One-thousand	Hundreds Tens Ones
5 2 9,	2 4 8,	6 5 0,	3 9 6

 

But suppose you have been given a numeral 587.9 where 9 has been separated from 587 by a point, but not by a comma. The nume­ral 587 names a whole number. The sign (.) is called a decimal point.

All digits to the left of the decimal point represent whole numbers. All digits to the right of the decimal point represent fractional parts of 1.

The place-value position at the right of the ones place is called tenths. You obtain a tenth by dividing 1 by 10. Such numerals like 687.9 are called decimals.

You read.2 as two tenths. To read.0054 you skip two zeroes and say fifty four ten thousandths.

Decimals like.666..., or.242424..., are called repeating decimals. In a repeating decimal the same numeral or the same set of nume­rals is repeated over and over again indefinitely.

We can express rational numbers as decimal numerals. See how it may be done.

31 4 4 x 4 16

100 = 0.31. 25 = 4 x 25 = 100 = 0.16

The digits to the right of the decimal point name the numerator of the fraction, and the number of such digits indicates the power of 10 which is the denominator. For example,.217 denotes numerator 217 and a denominator of 10³ (ten cubed) or 1000.

In our development of rational numbers we have named them by fractional numerals. We know that rational numerals can just as well be named by decimal numerals. As you might expect, calcula­tions with decimal numerals give the same results as calculations with the corresponding fractional numerals.

Before performing addition with fractional numerals, the fractions must have a common denominator. This is also true of decimal nu­merals.

When multiplying with fractions, we find the product of the numerators and the product of denominators. The same procedure is used in multiplication with decimals.

Division of numbers in decimal form is more difficult to learn because there is no such simple pattern as has been observed for mul­tiplication.

Yet, we can introduce a procedure that reduces all decimal-divi­sion situations to one standard situation, namely the situation where the divisor is an integer. If we do so we shall see that there exists a simple algorithm that will take care of all possible division cases.

In operating with decimal numbers you will see that the arithme­tic of numbers in decimal form is in full agreement with the arith­metic of numbers in fractional form.

You only have to use your knowledge of fractional numbers.

Take addition, for example. Each step of addition in fractional form has a corresponding step in decimal form.

Suppose you are to find the sum of, say,.26 and 2.18. You can change the decimal numerals, if necessary, so that they denote a com­mon denominator. We may write.26 =.260 or 2.18 = 2.180. Then we add the numbers just as we have added integers and denote the common denominator in the sum by proper placement of the decimal point.

We only have to write the decimals so that all the decimal points lie on the same vertical line. This keeps each digit in its proper pla­ce-value position.

Since zero is the identity element of addition it is unnecessary to write.26 as.260, or 2.18 as 2.180 if you are careful to align the de­cimal points, as appropriate.