Numerical and trigonometric fundamentals Numbers and symbols

Numerical and trigonometric fundamentals Numbers and symbols

It is generally believed that our present number system began with the use of the natural numbers, 1, 2, 3, 4, etc. These whole numbers, known as the positive integers, were used primarily for counting. However, as time went on, it became apparent that whole numbers could not be used for defining certain mathematical quantities. For example, a period in time might be between 3 and 4 days or the area of a field might be between 2 and 3 acres (or whatever unit of measure was used at the time). So the positive fractions were introduced, for example, , . These two groups of numbers, the positive integers and the positive fractions, constitute what we call the positive rational numbers. Thus, 711 is an integer or whole number,  is a positive fraction and  is a rational number.

A rational number is any number that can be written in the form a/b,           where a and b represent any integers. Thus ,  and 1 are all rational numbers. The number 1 can be represented by the quotient  = 1, in fact any number divided by itself must always be equal to 1.

The natural numbers are positive integers, but suppose we wish to subtract a larger natural number from a smaller natural number. For example, 10 subtracted from 7, we obviously obtain a number that is less than zero, that is, 7 – 10 = –3.       So, our idea of numbers must be enlarged to include numbers less than zero, called negative numbers. The number zero (0) is unique: it is not a natural number because all natural numbers represent positive integer values (i.e., numbers above zero) and quite clearly from what has been said, it is not a negative number either. It sits uniquely on its own and must be added to our number collection.

So, to the natural numbers (positive integers) we have added negative integers, the concept of zero, positive rational numbers and negative natural numbers. What about numbers like? This is not a rational number because it cannot be represented by the quotient of two integers. So, yet another class of numbers needs to be included, the irrational or non–rational numbers. Together all the above kinds of numbers constitute the broad class of numbers known as real numbers.

They include positive and negative terminating and non–terminating decimals     (e.g., ±  = ±0.1111..., 0.48299999, ±2.5, 1.73205..., etc.). The real numbers are so called to distinguish them from others such as imaginary or complex numbers, the latter may be made up of both real and imaginary number parts. Complex numbers will not be considered during our study of technician mathematics.

Although we have mentioned negative numbers, we have not considered their arithmetic manipulation. All positive and negative numbers are referred to as signed numbers and they obey the arithmetic laws of sign. Before we consider these laws, let us first consider what we mean by signed numbers.

Conventional representation of signed numbers is shown below, with zero at the midpoint. Positive numbers are conventionally shown to the right of zero and negative numbers to the left.

–12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –10 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 +11 +12

The number of units a point is from zero, regardless of its direction, is called the absolute value of the number corresponding to the point on the above number system when points are drawn to scale. Thus the absolute value of a positive number, or of zero, is the number itself. While the absolute value of a negative number is the number with its sign changed. For example, the absolute value of + 10 is 10 and the absolute value of –10 is also 10. Now the absolute value of any number n is represented by the symbol . Thus  means the absolute value of +24. Which is larger, or ?

I hope you said 1–141 because its absolute value is 14, while that of 1+31 is 3 and of course 14 is larger than 3. We are now ready to consider the laws of signs.

The laws of signs, you are probably already familiar with the use of these laws, but may not have seen them written formally before, here they are:

First law: To add two numbers with like signs, add their absolute values and prefix their common sign to the result.

This law works for ordinary arithmetic numbers and simply defines what we have always done in arithmetic addition.

For example: 3 + 4 = 7 or in full (+3) + (+4) = +7

After the introduction of the negative numbers, the unsigned arithmetic numbers became the positive numbers, as illustrated above. So, now all numbers may be considered either positive or negative, and the laws of signs apply to them all.

Does the above law apply to the addition of two negative numbers? From ordinary arithmetic we know that (–7) + (–5) = –12. This again obeys the first law of signs, because we add their absolute value and prefix their common sign.

Second law: To add two signed numbers with unlike signs, subtract the smaller absolute value from the larger and prefix the sign of the number with the larger absolute value to the results.

So, following this rule, we get for example:

5 + (–3) = 2; –12 + 9 = –3; 6 + (–11) = – 5.

The numbers written without signs are, of course, positive numbers. Notice that brackets have been removed when not necessary.

Third law: To subtract one signed number from another, change the sign of the number to be subtracted and follow the rules for addition.

For example, if we subtract 5 from –3, we get –3 – (+5) = –3 + (–5) = –8.

Now what about the multiplication and division of negative and positive numbers, so as not to labour the point the rules for these operations are combined in our fourth and final law.

Fourth law: To multiply (or divide) one signed number by another,  multiply (or divide) their absolute values; then, if the numbers have like signs, prefix the plus sign to the result; if they have unlike sign prefix the minus sign to the result.

Therefore, applying this rule to the multiplication of two positive numbers, for example, 3 X 4 = 12, 12 X 8 = 96 and so on, which of course is simple arithmetic. Now applying the rule to the multiplication of mixed sign numbers we get, for example, –3 X 4 = –12, 12 X (–8) = –96, and so on. We can show, equally well, that the above rule yields similar results for division.

Example 4.1

Apply the fourth law to the following arithmetic problems and determine the arithmetic result.

(a) (–4) (–3) (–7) =?         (b)   14/–2 =?

(c) 5(–6) (–2) =?               (d)   –22/–11 =?

(a) In this example we apply the fourth law twice. (–4) (–3) = 12 (like signs) and so 12(–7) = –84.

(b) 14/(–2) applying the third law for unlike signs immediately gives –7, the correct result.

(c)  Again applying the third law twice. 5(–6) = –30 (unlike signs) and (–30) (–2), = 60.

(d) (–22)/(–11) applying the third law for like sign gives 2, the correct result.

We earlier introduced the concept of symbols to represent numbers when we defined rational numbers where the letters a and b were used to represent any integer. Look at the symbols below, do they represent the same number?

IX; 9; nine;

I hope you answered yes, since each expression is a perfectly valid way of representing the positive integer 9. In algebra, we use letters to represent Arabic numerals, such numbers, are called general numbers or literal numbers, as distinguished from explicit numbers like 1, 2, 3, etc. Thus, a literal number is simply a number represented by a letter, instead of a numeral. Literal numbers are used to state algebraic rules, laws and formulae, these statements being made in mathematical sentences called equations.

If a, is a positive integer and b is 1, what is a/b? I hope you were able to see that   a/b = a. Any number divided by 1 is always itself. Thus a/1 = a, c/1 = c, 45.6/1 = 45.6 and so on.

Suppose a is again any positive integer, but b is 0. What is the value of alb? What we are asking is what is the value of any positive integer divided by zero? Well the answer is that we really do not know! The value of the quotient alb, if b = 0, is not defined in mathematics. This is because there is no such quotient that meets the conditions required of quotients. For example, you know that to check the accuracy of a division problem, you can multiply the quotient by the divisor to get the dividend. Where for example, if 21/7 = 3 then 7 is the divisor, 21 is the dividend and 3 is the quotient and so 3 X 7 = 21, as expected. So, if 17/0 were equal to 17, then 17 X 0 should again equal 17 but it does not! Or if 17/0 were equal to zero, then 0 X 0 should equal 17 but again it does not. Any number multiplied by zero is always zero. Therefore, division of any number by zero (as well as zero divided by zero) is excluded from mathematics. If b = 0, or if both a and b are zero, then alb is meaningless.

When multiplying literal numbers together we try to avoid the multiplication sign (X), this is because it can be easily mistaken for the letter x. Thus instead of writing a X b for the product of two general numbers, we write a^.b (the dot notation for multiplication) or more usually just ab to indicate the product of two general numbers a and b.

Example 4.2

If we let the letter n stand for any real number, what does each of the following expressions equal?

(a) n/n =?         (b) n–0=?         (c) n– 1 =?   (d) n+0=?

(e) n–0 =?        (f) n–n=?          (g) n/0=?

(a) n/n = 1      any number divided by itself is equal to 1.

(b) n ^. 0 = 0    any number multiplied by zero is equal to zero.

(c) n ^. 1 = n    any number multiplied or divided by 1 is itself.

(d) n + 0 = n   addition of zero to any number will not alter that number.

(e) n – 0 = n   subtraction of zero from any number will not alter that number.

(f) n – n = 0    subtraction of any number from itself will always equal zero.

(g) n/0             division by zero is not defined in mathematics.