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Elementary algebra. Numbers and number systems. Negative numbers. Rational, irrational and real numbers. Fields.

Introduction. The subject matter with which elementary algebra is concerned is algebraic expressions and algebraic equations and the various laws or rules for manipulating them. An example of an algebraic equation is the formula for the perimeter, p, of a triangle. Let the lengths of the sides of a triangle be denoted by a, b, and c. The perimeter is then given by the formula

p = a + b + c

Here the letters p, a, b, c all represent numbers. The values of a, b, and c can vary and their sum determines p. A distinctive feature of algebra is the use of letters to represent numbers. An algebraic equation, in general, is a statement of equality that involves numbers and symbols that represent numbers. One important way in which equations arise is in so-called “word problems”. Such problems are often “brain-twisters” that can easily be solved by the methods of algebra. A simple example of a word problem is the following:

Problem: A line 28 inches long is to be divided into three parts. The second part is to be twice as long as the first part and the third part is to be four times as long as the first part. Find the length of each part.

Solution. Let x = the length of the first part in inches.

Then 2x = the length of the second part in inches

and 4x = the length of the third part in inches.

We can then write down the equality

1) x + 2x + 4x = 28

which is simply an algebraic way of stating that the sum of the lengths of the three parts is 28 inches.

From 1) we obtain

2) 7x = 28

and from 2) we obtain the solution

3) x = 4

Thus the length of the first part is 4 inches, the length of the second part is 8 inches and the length of the third part is 16 inches.

The general process in solving word problems is as follows: The word problem leads to an equation (possibly more than one equation) such as 1) above. We then proceed to attempt to solve this equation. Typically, using our knowledge of algebra, we manipulate the equation into a succession of simpler equivalent forms, trying to put it into a form that will give us the solution.

Numbers and number systems. Negative numbers. Rational, irrational and real numbers.

Rational numbers. The natural numbers 1, 2, 3, 4, ... are the fundamental building blocks from which man has created different number systems. They are real and natural in a sense in which other numbers are not. All other numbers are invented or created. Ancient man used the natural numbers in conjunction with fractions (he invented the concept of a fraction and defined it using natural numbers) to measure quantities. The set of numbers that he used consisted of numbers of the form k + m/n where k, m, and n are natural numbers. This set of numbers corresponds to what we now call the positive rational numbers. Many centuries later someone came up with the ingenious idea of giving “nothing” a symbol and a name and appending it to this set of numbers. This gave us the set of positive rational numbers with “0" added in. This set of positive rational numbers and 0 are what we are accustomed to in arithmetic.

Irrational numbers. The ancient Greeks came up with a very interesting discovery. They discovered that there are some quantities that cannot be expressed exactly by numbers of the form k + m/n (i.e. by rational numbers). They proved, for example, that in a square with sides of unit length there is no rational number that gives the length of its diagonal exactly. Thus we now know that there are some numbers such as and π that are not equal to any rational number. We call the set of all such numbers the irrational numbers.

Negative numbers. The number system used in algebra is an extended number system that includes negative numbers. We can envision negative numbers as corresponding to points left of zero on a number scale. They find meaning in examples such as temperatures less than zero on a thermometer, dates before Christ, or elevations below sea level. Negative numbers can also be viewed as debts as opposed to assets. The reason for enlarging the number system to include negative numbers was to create a number system that was closed with respect to the subtraction operation (allowing differences such as 3 - 5 to be viewed as numbers whereas they are illegal in the number system of arithmetic). Doing this was essential to making the algebraic method work. The methods employed in algebra would not work on a number system consisting of the positive real numbers only. They require a system that is closed with respect to all operations of addition, subtraction, multiplication and division.

The number system used in algebra, consisting of both positive and negative rational and irrational numbers, constitutes what is known as the real number system.

The real number system. The real number system consists of positive and negative rational and irrational numbers. The set of all rational numbers plus the set of all irrational numbers gives the set of all real numbers. The real numbers correspond to points on a line called the real-number axis (see Fig. 1), with a one-to-one correspondence between the real numbers and points on the line.

Def. Rational number. A rational number is any number that can be expressed as the quotient of two integers (including both positive and negative integers). Thus the set of all rational numbers corresponds to the set of all quotients p/q where p and q are integers (positive or negative). Since any quotient of two integers p/q can be expressed as a mixed number k + m/n, a rational number can be equivalently defined as any number that can be represented as a mixed number k + m/n, where k, m, n are integers.

Examples. 5½, -3⅔, 16⅜ are all rational numbers.

Def. Absolute (or numerical ) value. The absolute (or numerical) value of a number is the nonnegative number that remains when the sign is removed. The absolute value of a is denoted by |a|.

Examples.

|-3| = 3, |5| = 5, |0| = 0.

Defining a new number system that encompassed negative numbers required defining meanings for the sum, difference, product and quotient when negative numbers were involved. For example, what is the definition of the sum a + b when a or b or both are negative? What is the definition of the product ab when a or b or both are negative?

Rules for addition, subtraction, multiplication and division. In the number system assumed in algebra (consisting of both positive and negative real numbers) the following rules obtain for the operations of addition, subtraction, multiplication and division:

Addition.

1. To add two numbers with like signs: 1) add their absolute values and 2) prefix this sum with the common sign.

Examples:

3 + 4 = 7

(-3) + (-4) = -7

2. To add two numbers with unlike signs: 1) find the difference between their absolute values and 2) prefix the number obtained with the sign of the number with the greater absolute value.

Examples:

17 + (-8) = 9

(-6) + 4 = -2

(-18) + 15 = -3

Subtraction.

To subtract one number b from another number a: change the sign of b and add to a i.e. a - b = a + (-b).

Examples:

12 - (7) = 12 + (-7) = 5

(-9) - (4) = -9 + (-4) = -13

2 - (-8) = 2 + 8 = 10

Multiplication.

1. To multiply two numbers having like signs: 1) multiply their absolute values and 2) prefix this product obtained with a plus sign (or no sign).

Examples:

(5)(3) = 15

(-5)(-3) = 15

2. To multiply two numbers having unlike signs: 1) multiply their absolute values and 2) prefix this product with a minus sign.

Examples:

(-3)(6) = -18

(2)(-6) = -12

Division.

1. To divide two numbers having like signs: 1) divide their absolute values and 2) prefix the number obtained a plus sign (or no sign).

Note. Division by zero is not defined.

Examples:

(-6)/(-3) = 2

6/3 = 2

2. To divide two numbers having unlike signs: 1) divide their absolute values and 2) prefix the number obtained with a minus sign.

Examples:

-12/4 = -3

12/(-4) = -3

Where did these rules come from? What is the basis for them? How were they arrived at? What assumptions underlie them? Why is a negative number times a positive number a negative number? Why is a negative number times a negative number a positive number? We now attempt to answer these questions.

A number of laws hold when working with the positive numbers (i.e. the associative laws and commutative laws for both addition and multiplication and the left and right distributive laws). The algebraic method used in algebra utilizes these laws. In order to utilize the algebraic method on a system involving negative numbers, these laws must continue to be valid for the new, expanded number system. That is a requirement for any new system that we might wish to build that includes negative numbers.

In order for the algebraic method to work, the new expanded number system that we wish to create had to have a certain abstract structure. That abstract structure that it needed to possess was the structure known as a field. A field has the following properties:

Field. A set S of elements a, b, c, ... on which are defined two binary operations called addition and multiplication with the following abstract properties:

Addition.

1. Closed under addition. For all elements a,b in S

a,b in S implies a + b in S

2. Associative Law holds under addition. For all elements a,b,c in S

a + (b + c) = (a + b) + c

3. Identity element called 0 exists under addition.

a + 0 = 0 + a = a for all elements a in S

4. Inverse exists for every element under addition. For every element a in S there exists an additive inverse element -a in S such that

a + (-a) = (-a ) + a = 0 for all a in S

5. Commutative Law holds under addition. For all elements a,b in S

a + b = b + a

Multiplication.

6. Closed under multiplication. For all elements a,b in S

a,b in S implies ab in S

7. Associative Law holds under multiplication. For all elements a,b,c in S

a(bc) = (ab)c

8. Identity element called 1 (i.e. unit element) exists under multiplication.

1a = a1 = a for all elements a in S

9. Inverse exists for every nonzero element under multiplication. For every element a in S there
exists a multiplicative inverse element a^{-1} in S such that

aa^{-1} = a^{-1}a = 1 for all a in S (except element 0 is excluded)

10. Commutative Law holds under multiplication. For all elements a,b in S

ab = ba

11. Left Distributive Law holds -- multiplication over addition

a(b+c) = ab + ac

for all elements a,b,c in S

12. Right Distributive Law holds -- multiplication over addition

(a+b)c = ac + bc

for all elements a,b,c in S

13. No proper divisors of zero (i.e. there are no nonzero members a and b for

which ab = 0).

14. Cancellation Law under multiplication holds.

ax = bx implies a = b

Properties 1-11 are the defining properties for a field. Properties 12-14 are logical consequences of the first 11 properties. Thus any field has all of the above 14 properties.

Once we have postulated the existence of an additive inverse for a positive number in the form of a negative number, the rules for operating on our newly postulated signed numbers are forced upon us i.e. they must be as they have been defined above in order for the associative, commutative and distributive laws to be obeyed. Stated differently, if we assume that the new number system that we are creating, a number system including both positive and negative numbers, possesses the properties of a field then the above rules for adding, subtracting, multiplying and dividing numbers can be derived directly. Thus the above rules for adding, subtracting, multiplying and dividing numbers of like and unlike signs are not just arbitrary definitions. These rules are forced from the assumption that our new system obeys the postulates of a field. Hence a negative number times a negative number is a positive number because it has to be if our new number system is to be a field. There is no choice. Likewise, a negative number times a positive number is a negative number because it has to be in order for the expanded number system to be a field. For more insight into this question see Basis for operations involving negative numbers.

References

Hawks, Luby, Touton. First-Year Algebra

Murray R. Spiegel. College Algebra

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