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In Arithmetic we can add together two or more abstract numbers and express them more shortly as a single number, thus : 2 + 3 + 5 = 10; but in Algebra we can only add together and express more shortly terms which are alike, thus: 2a + 3a + 5a = 10a. Terms which are unlike cannot be added together; thus a + b + c cannot be expressed in a shorter form.

1. 
   Only like terms can be added.
   
2. 
   Add together all the like terms that are positive and all the like terms that are negative; subtract the smaller of these sums from the larger, and prefix the sign of the larger sum.
   

Conversely, since +7 added to 20 gives 27, we might infer that —7 subtracted from 20 would give 27, and this would be correct. Hence we can also infer a general rule for subtraction, viz.: — Change the signs of all the terms in the express-ion which has to lie subtracted, and then proceed as in addition.

Set down as in addition, and change the signs in the lower line, thus:

By adding, we get 5a + 6i as the difference required.

In working subtraction sums, the signs in the lower lines should be changed mentally. The above sum would then appear thus:

8a +2b

3a - 4b

5a+6b.

The actual process of working this sum after setting it down would be as follows: — Begin with the a's, minus 3 and plus 8; the plus is the larger by 5; therefore set down 5a, omitting the plus sign because 5a is the first term. Again, plus 4 and plus 2 give plus 6; therefore set down plus 6b.

35. As in addition, like terms must be arranged under like terms. Take another example. From 5a³ + 5a² — 7a + 3a⁴—5 take 5a² — 6a + 7 — 2a⁴ + 2a³. Arrange in order thus:

3a⁴ + 5a³ + 5a² - 7a – 5

-2a⁴ + 2a³ +5a² - 6a+ 7

5a⁴ + 3a³ — a — 12.
The working of this question was as follows: — Begin with a⁴ plus 2 and plus 3 give plus 5a⁴. Then, for a³, minus 2 and plus 5, the plus is the larger by 3; therefore set down + 3a³. Then a² minus 5 and plus 5 give 0; therefore set nothing down. Then a plus 6 and minus 7, the minus is the larger by 1; therefore set down —a, the 1 being understood before the a. Lastly, minus 7 and minus 5 give —12.
36. Notice that in Algebra we do not consider which expression is the larger in a subtraction sum. The answer may be either a positive or a negative quantity; and so in Algebra we may subtract either a larger quantity from a smaller, or a smaller quantity from a larger. Also the letters used in an algebraical expression may have any value, so that we cannot always tell which, is the larger of two expressions. We usually, therefore, pay no attention in Algebra to the magnitudes of the quantities we use.
37. Since subtraction and addition are inverse opera­tions, we can prove the accuracy of our work in an addition sum by subtracting one or more of the expres­sions added together from the sum; and we can prove the accuracy of a subtraction sum by adding the expres­sion subtracted to the remainder. To take an example from Arithmetic: 7+5+3=15; to prove that this is correct, we subtract 3 from 15, thus: 15— 3 = 12; then subtract 5 from 12, thus: 12 — 5 = 7; we have now come back to 7, which is the number we started with; so we infer that our addition was correct. In subtraction, 14—5 = 9 and 5 + 9 = 14. This will be evident to the student from his knowledge of Arithmetic.
MULTIPLICATION
38. In Arithmetic we say 2×3 = 6. No notice is taken of signs; but, if this be expressed fully and cor­rectly, we should say + 2× +3 = +6. Therefore, when two terms with plus signs are multiplied together the product is plus.

Suppose +2 × -3 or -2 × +3. Evidently the pro­duct will not be the same in either of these cases as in +2 × +3. Therefore we assume that +2 × -3 = -6 and -2 × +3= -6.

Again, suppose -2 × -3. This is different from the last two cases, and we assume that -2 × -3 = +6. Therefore, when two terms with minus signs are multi­plied together the product is plus.

From these results we can infer the rule of signs.

Rule of signs. — Like signs produce plus; unlike produce minus.
39. The application of the rule of signs is very im­portant when we come to deal with indices or powers, and roots of quantities. For example:

(+a)² =+a × +a =+a² = a².

(-a)² =-a × -a =+a² = a².

(+a)³ =+a × +a ×+a =+a³ = a³.

(-a)³ =-a × -a ×-a = -a³.

We see that a plus quantity raised to any power produces a plus result; a minus quantity raised to an even power produces a plus result, e.g., (-a)⁶ = a⁶; but a minus quantity raised to an odd power pro­duces a minus result,

e.g., (-a)⁷ = -a⁷.

Again, with roots √(a²) = either +a or -a, since + a × +a = a², and -a × - a = a² also.

So that the square root of a positive or plus quantity is either plus or minus; that is, every positive quantity which is an exact square has two roots, these roots being of opposite sign — the one plus and the other minus.

Since like signs produce plus, we cannot find the square root of any negative or minus quantity, e.g., √(-a²) is impossible quantity, for - a ×- a = +a², and

Again, ³√(+a³) = +a³, since +a × +a × +a = +a³; and ³√(-a³) = -a³, since -a × -a × -a = -a³.

We see, therefore, that apparently there is only one real or possible cube root of a given quantity, but this given quantity may be either plus or minus.

Similarly, for higher powers; if we are asked to find the 4th, 6th, 8th, or any even root of a, given quantity, we can only do so when the given quantity is plus, and then we can find two real roots, one of each sign. But, if we are asked to find the 5th, 7th, 9th, or any odd root of a given quantity, we may be able to do so whatever the sign of the quantity is, but we can only find one real root, and the sign of this root will be the same as the sign of the given quantity.

Hence we infer that different powers of the same form may be multiplied by writing the quantity with an index equal to the sum of the indices of the multipliers. In the above example, 4 + 2 = 6; therefore b⁴×b² =b⁶.

Similarly, b³×b⁵ =b⁸.

Also, since a= a¹, a² × a= a³, a⁴ × a= a⁵; and so on.

Also we have seen that when two different terms are multiplied together the product may be ex­pressed by writing the two terms side by side. Thus: a × b = ab; c² ×d⁴ =c²d⁴.
42. In Arithmetic the student knows that, if several numbers have to be multiplied together, the numbers may be taken in any order. For example:

2 ×3 ×4 = 2 ×4 ×3 = 3 ×4 ×2 = 4 ×3 ×2, &c., for each product is equal to 24. So also in Algebra the terms in any product may be taken in any order. So that

If, therefore, in Algebra we have to multiply together two or more simple factors, we may place the numerical factors all together, and we may gather together any factors which are powers of the same quantity, and apply the rule for the multiplication of indices. For example:

(1) There are as many lines as there are terms in the multiplier.

(2) There are as -many terms in each line as there are
terms in the multiplicand

(3) With regard to signs, a plus sign in the multiplier

(4) It is advisable to arrange both multiplicand and multiplier in descending powers of some letter, because by so doing we shall find that the products produced in the working will be easier to arrange in columns.

Since the dimension of every term in a product is equal to the sum of the dimensions of its factors, it follows that, if we multiply together two homogeneous expressions, we shall obtain a homogeneous product.

(1) The first and last terms in the product are obtained by multiplying together the two first terms, and then the two last terms.

(2) The coefficient of the middle term in the product is obtained by adding together, algebraically, the two last terms; e.g.,

(a + 6)(a + 4) = a² + 10a+24.

Similar rules will hold if, instead of a number, we use any other kind of term for the second term in each multiplier; e.g., (a + 2b)(a + 3b) = a² +5ab+6b².

2. 
   Base Two Numerals
   

During the latter part of the seventeenth century a great German philosopher and mathematician Gottfried Wilhelm von Leibnitz (1646—1716), was doing research on the simplest numeration system. He developed a numeration system using only the symbols 1 and 0. This system is called a base two or binary numeration system.

Leibnitz actually built a mechanical calculating machine which until recently was standing useless in a museum in Germany. Actu­ally he made his calculating machine some 3 centuries before they were made by modern machine makers.

The binary numeration system introduced by Leibnitz is used only in some of the most complicated electronic computers. The numeral 0 corresponds to off and the numeral 1 corresponds to on for the elec­trical circuit of the computer.

Base two numerals indicate groups of ones, twos, fours, eights, and so on. The place value of each digit in 1101two is shown by the above words (on or off) and also by powers of 2 in base ten notation as shown below.

The numeral 1101two means (1x2³)⁴ + (1x2²) + (0x2) + (1x1) = (1x8) + (1x4) + (0x2) + (lxl) = 8 + 4 + + 0 + 1 = 13. Therefore 1101_two = 13

A base ten numeral can be changed to a base two numeral by di­viding by powers of two.

From the above you know that the binary system of numeration is used extensively in high-speed electronic computers. The correspon­dence between the two digits used in the binary system and the two positions (on and off) of a mechanical switch used in an electric circuit accounts for this extensive use.

The binary system is the simplest⁵ place-value, power-position system of numeration. In every such numeration system there must be symbols for the numbers zero and one. We are using 0 and 1 because we are well familiar with them.

The binary numeration system has the advantage of having only two digit symbols but it also has a disadvantage of using many mo­re digits to name the same numeral in base two than in base ten. See for example:

476 = 111011100 two

It is interesting to note that any base two numeral looks like a numeral in any other base. The sum of 10110 and 1001 appears the same in any numeration system, but the meaning is quite different. Compare these numerals:

10110 two 10110ten 10110seven

+ 1001 two + 1001ten + 1001 seven

3. 
   CLOSURE PROPERTY
   

In this lesson we shall be concerned with the closure property.

If we add two natural numbers, the sum will also be a natural number. For example, 5 is a natural number and 3 is a natural number. The sum of these two numbers, 8, is also a natural num­ber. Following are other examples in which two natural numbers are being added and the sum is another natural number. 19+4 = 23 and only 23; 6+6=12 and only 12; 1429+357=1786 and only 1786. In fact, if you add any two natural numbers, the sum is again a na­tural number. Because this is true, we say that the set of natural numbers is closed under addition.

Notice that in each of the above equations we were able to name the sum. That is, the sum of 5 and 3 exists, or there is a number which is the sum of 19 and 4. In fact, the sum of any two numbers exists. This is called the existence property.

Notice also that if you are to add 5 and 3, you will get 8 and only 8 and not some other number. Since there is one and only one sum for 19+4, we say that the sum is unique. This is called the uni­queness property.

Both uniqueness and existence are implied in the definition of closure.

Now, let us state the closure property of addition.

If a and b are numbers of a given set, then a+ b is also a num­ber of that same set. For example, if a and b are any two natural numbers, then a + b exists, it is unique, and it is again a natural number.

If we use the operation of subtraction instead of the operation of addition, we shall not be able to make the statement we made above. If we are to subtract natural numbers, the resullt is sometimes a na­tural number, and sometimes not. 11—6 = 5 and 5 is a natural num­ber, while 9—9 = 0 and 0 is not a natural number.

Consider the equation 4—7 = n. We shall not be able to solve it if we must have a natural number as an answer. Therefore, the set of natural numbers is not closed under subtraction.

What about the operation of multiplication? Find the product of several pairs of natural numbers. Given two natural numbers, is there always a natural number which is the product of the two numbers?

Every pair of natural numbers has a unique product which is aga­in a natural number. Thus the set of natural numbers is closed under multiplication.

In general, the closure property may be defined as follows: if x and y are any elements, not necessarily the same, of set A (A capi­tal) and * (asterisk) denotes an operation *, then set A is closed under the operation asterisk if (x*y) is an element of set A.

To summarize, we shall say that there are two operations, addi­tion and multiplication, for which the set of natural numbers is clo­sed. Given any two natural numbers x and y, x + y and x x y are again natural numbers. This implies that the sum and the product of two natural numbers exists. It so happens that with the set of natu­ral numbers (but not with every mathematical system) the results of the operations of addition and multiplication are unique.

It should be pointed out that it is practically impossible to find the sum or the product of every possible pair of natural numbers. Hence, we have to accept the closure property without proof, that is, as an axiom.

4. 
   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.

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