Positive and negative numbers

Explanations

It is winter. Yesterday, at noon, the thermometer showed 3ºC, today it shows 2ºC. Is the air warmer today than it was yesterday? This question can only be answered, if the statements also tell you whether the temperature was below or above the freezing point. In order to distinguish now between warmer than and colder than the freezing temperature, you add - and + signs to your statements. If the above statements had read: -3º and +2º, you would have known that the temperature rose by 5ºC. If the temperature yesterday was +3ºC and is today -2ºC, you would know that the air became colder by 5ºC, etc.

If you compute your intake and outlay on a day at 40 and 30 Baht, respectively, you do not yet know all. Again you must use positive (intake) and negative (outlay) numbers.

Positive and negative numbers are relative numbers.

You cannot use them when counting apples, pears, people, etc. They have no meaning with quantities like weights. They are required during comparisons and calculations with quantities, which must be ordered in two, mutually opposite directions from zero. Such numbers relate to income, expenditure, have and have not. If equal numbers of one kind and the other are combined, like 30 Baht intake and 30 Baht outlay, the result is 0. If somebody answers the question: "How many apples are here?" by saying +3, it does not make more sense than if he had answered 3, because the + sign is not necessary in this case. If somebody replies 3ºC to the question: "How many degrees do we have today?", the answer is false. It should be +3ºC or -3ºC, or, if he does not want to use the abbreviations, he must tell it is "3ºC warm"or "3ºC cold". The attached signs + and -, or attached words "warm" or "cold", are necessary parts of such statements. When the sign is obvious, it can be omitted, for example, in Bangkok where the temperature never drops below zero.

Under these conditions, the signs - and + are not arithmetic signs. The statement: "The thermometer shows +3ºC" does not involve addition. These leading signs are a part of the number. In what follows, we must distinguish between leading and arithmetic signs.

The absolute value of a relative number (|a|) is obtained by omission of the sign. You write 

The absolute value of +3 and -3 is 3.

Two relative numbers are equal, when they have the same absolute value and the same sign, for example, -3 and -3 or +4 and +4. They are said to be equal, but opposite, when they have the same absolute value, but opposite signs, for example, -3 and +3 or +4 and -4.

Addition of relative numbers

Task: You have 300 Baht and owe 100 Baht. How much do you actually own?
Answer: 200 Baht.

We will now translate this task and its solution into algebraic language. In order to avoid confusion between leading and arithmetic signs, we will enclose relative numbers in brackets and find: 

If we combine expenditures: 

If two credits are to be added, you take their sum and own the result. If two debits are to be added, you take their absolute values and you owe the sum. You have the Rule:

Numbers with equal leading signs are added by adding their absolute values; the sum is given their common leading sign.

Examples: 

Addition of numbers with different leading signs: 

If "debit" and "credit" are to be added, you must subtract the larger number from the smaller one. If "debit" is larger than "credit", the answer is "debit", otherwise it is "credit".

Numbers with different leading signs are added by subtraction
of the smaller one from the larger one
and giving the difference the sign of the larger one.

You arrive at the same rules as in Calculations with sums and differences, but here the rules have a different meaning.

Examples: 

In words, the first example is "Someone owns 6 Baht and owes 7 Baht, whence he owes 1 Baht".

For common fractions, unless their denominators are the same, you give them first the the same denominator; it is then readily seen which fraction has the greater value.

(+¼) + (-½)= (¼) + (-²/₄) = (-¹/₄).

Task: You bank account holds 1000 Baht. You enter your credits and debits with + and - signs, respectively. What do you own after writing down the entries +130 Baht, -540 Baht, - 90 Baht, + 160 Baht?

(+1000) + (+130) + (-540) + (-90) + (+160) = (+660)

Positive numbers mean credit, negative numbers debit. You own 660 Baht.

Graphical representation

Numbers can be related to points on a straight line. The origin O can be selected anywhere on the line.

The coordinate of the point A is +3, that of the point B is -2, etc. You write them either below or above the point. The distance between points, corresponding to two consecutive integers, can be 1 cm, 1 dm, etc., because the choice of the scale is arbitrary.

You must always select a scale and cannot change it during a study.

You can start from O or from any other point. The point B in the next figure is 3 cm from A, so is the point B₁ from A₁. The two distances AB and A₁B₁ on the number line are equal, if they have the same length and direction.

In this figure, B lies 3 cm from A, B₁ -3 cm from A₁. These two distances are opposite, but equal.

Task: You take first -5 steps from O to A, then +6 steps from to B. With how many steps from 0 could you have reached B?

You are +1 step from where you started, that is, you would just take +1 step to reach B.

Addition has been explained earlier as forward counting along the integers and in the graphical representation it is forward motion in the direction of the number to be added.

If you add two equal, but opposite numbers, for example (+3) and (-3), the result is 0. Three steps to the right and three steps to the left return you to your starting point.

(+a) + (-a) = 0.

Interpret the walks of the next figures and confirm the addition rules for numbers with the same and different leading signs.

Examples: 

Subtraction of relative numbers

We have explained subtraction as backward counting along the line of numbers. In the graphical representation, it becomes motion in the opposite direction. Here too the motion starts at the end of the sum from which it is subtracted.

For example, if you want to move in the opposite direction of the subtrahend (+2) (Task 13), it is the same as if you move in the same direction of (-2), that is, you add (-2). If you are to move in the opposite direction of (-2) (Task 14), it is the same as if you move (+2), that is, you add (+2).

You see that you can always reduce subtraction to addition and have the Rule:

Relative numbers are subtracted
by adding them with the opposite sign.

Examples: Our mode of writing has been somewhat difficult. We will now introduce a simplification:

Omit the brackets around relative numbers,
write a instead of (+a), that is, omit the plus sign.

In the new convention, the task (+5) - (+3) = (+2) becomes 5 - 3 = 2. Since every subtraction of a relative number becomes addition of the opposite, but same number, double leading signs are avoided.

Obviously, it is easy to change over from the abbreviated to the explicit mode of representation: 

this is an alternative. The result of the task is unchanged by the mode of representation. As a rule, you encounter

3 - 4 + 5 as the sum (+3) + (-4) + (+5)

and refer to every expression with terms, connected by + and - signs, as an algebraic sum or just as a sum.

You might think now that it is a disadvantage, when you cannot note with a calculation like 5 - 4 = 1 whether you are dealing with absolute or relative numbers. However, once yo become familiar with it, you will recognize the big advantage of the simpler mode and its significance. In fact, calculations are now the same for all kinds of numbers.

The abbreviated mode of representation makes
calculations independent of the type of number.

If a task like 6 - 8 = -2 is posed in the field of numbers or divisible quantities, then -2 becomes a subtraction, which cannot be performed, because 2 must still be subtracted; if the same task occurs in the field of relative numbers, -2 becomes a negative number. Here we assume that the laws of computation apply to all ranges of numbers; later on, we shall show that this is true.

Historical remarks

Tasks in the abbreviated mode of representation, say 6 - 8, have led to the introduction of relative numbers. Here too, the Indians took the decisive step. They were the first to introduce negative numbers; however, they did not give them a negative sign, but placed a point above it. They wrote

= -1, = -2, etc.

Since the Arabs did not adopt this idea, this achievement of the Indians remained unknown for a long time. There elapsed almost 700 years between the first use of negative numbers and when they were gradually accepted in the occident. Fibonacci admitted negative roots of equations. The preacher and mathematician Michael Stifel (1487 - 1567) in Jena, Germany, computed with negative numbers, but still called them absurd numbers; also the Italian mathematician Facio Cardano (1444-1524) and the Dutch inspector of dykes Stevin (1548 - 1620) made use of them, but more carefully, because many mathematicians still rejected them and declared them to be impossible. Especially the proof of the rules of multiplication caused considerable difficulties. Stifel wrote in his treatise Arithmetic Integra (1544), while dealing with these rules (cf. Trofke: History of Elementary Mathematics): "It appears that we must renounce a proof of this rule during multiplication with the signs + and -. It must be ascribed to the impotence of the human brain that we cannot conceive why it is true. Nevertheless, you should not doubt the truth of the given multiplication rule, because it is supported by many examples." Only the work of the French philosopher and mathematician Descartes as well as that of the Swiss mathematician Euler slowly silenced the opposition. Positive and negative numbers are today fundamental ingredients of all mathematics.

Multiplication of relative numbers

You know that

2·3 = 2 + 2 + 2.

The factor 3 indicates how often the factor 2 enters into the sum. Therefore 2·3 represents forward counting from 0 by 3 group units of size 2.

We have written here 3 after the multiplication point. This is not always done and some people would write 3·2. In what follows, we will stick to this rule. A change of this rule would not affect this work, but some operations become so clearer.

Similarly, we define: 

If also the second factor is a relative number, say +3, the task becomes (+2)·(+3) or (-2)·(+3), and we must investigate what this means.

The sequence of the factors, here (+2) or (-2), indicates a definite direction on the number line :

In order to reach agreement with the earlier definitions, we now define: In the case of a positive factor, advance in the same direction as is indicated by the sequence of the factors. For (+2)·(+3), the summation is in the direction of the (+2) sequence, for (-2)·(+3), it is in the direction of the (-2) sequence:

The leading signs + and - always determine two mutually opposite directions. If in the case of positive multiplication we move along the number line from the left to the right, then we must move in the opposite direction for negative multiplication.

In the task (+2)·(-3), the direction of the (+2) sequence is from left to right and we must move in the opposite direction, that is, move from the origin 3 groups to the left. In the task (-2)·(-3), the direction of the factor sequence is from the right to the left, that is, one must move from the origin to the right.

The four possible combinations of factors with general numbers are: 

The product of two numbers with equal leading signs is positive,
with different leading signs negative.

The multiplication rule of Calculations with sums and differences, by which equal signs yield + and different signs - reappears here with a different meaning.

Examples: 

The extension to more than two factors causes no problems. For example, the task (-2)·(-3)·(-4) is taken in two stages: (-2)·(-3) = (+6) and (+6)·(-4) = (-24).

Division of relative numbers

The quotient of two relative numbers is the number which on multiplication by the divisor yields the dividend. If you transfer this definition to relative numbers, then (-a)/(+b) becomes a negative number, because only a negative number can on multiplication by (+b) yield the negative dividend -a.

(-a)/(+b) = (-a/b).

The same reasoning yields the four cases: 

The quotient of two numbers with equal leading signs is positive,
with different leading signs negative.

Examples: 

Equation: -x = 4.

Either you multiply both sides of the equation by -1 to obtain (-x)·(-1)=4·(-1), that is, x = -4, or you divide both sides by -1 to obtain -x/-1 = 4/-1, that is, x = -4.

Note that -a/b = a/-b = - a/b, -1/(a - b) = 1/(b - a), etc.

Permanence principle

While planning rules for new numbers, for example, for relative numbers, you aim, in general, to specify new definitions, so that the old laws of Arithmetic are preserved and no contradictions can arise. This principle has been observed in the development of Arithmetic, but it was first given a conscious and exact form by the German historical mathematician Hankel (1839 - 1873) and became, following him, known as the Permanence Principle or the Principle of the conservation of formal laws.

In order to establish that the old laws are also conserved for relative numbers under the given definitions, consider the two basic laws: The sequence of terms in a sum and that of factors is arbitrary. It is readily shown that they remain valid for relative numbers. We will prove this statement by means of two examples:

You arrive at the same result of the terms in a sum and the factors in a product, although their sequence has been changed.

If the basic laws hold, is this also true for derived rules? Since, by definition, multiplication is repeated addition, the factors 1 and 0 are not taken into consideration. Now you can arrange the rules so that they remain in agreement with the other rules. 

Thus, as you decompose a number into factors, you must take into consideration the Rule:

Every number can be interpreted as a product,
one factor of which is +1 or -1.

For division, we state as before: You may not divide by zero!

Tasks: 

Exercises: Evaluate the expressions:
 

Answers

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