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Suggestions for the Study of
Arithmetic

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By ERNEST L. CRANDALL

Former Civil Service Examiner

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There are certain “standard errors,” so to speak, that the unsuccessful candidate makes nine times out of ten, and if these are eliminated every one, with a little practice, may put himself in line for 100 per cent.

While the examples may take the form of “problems,” the only processes involved will be simple addition, subtraction, multiplication and division—no fractions or decimals.

In addition there is but one thing to be observed. If your numbers are not all of equal length arrange them so that the last figures are all in the same column. Suppose you have to add 357,856, 7,596, 452 and 29,360. Following are the right and wrong ways to arrange them:

This arrangement is necessary because of the inherent properties of numbers as expressed in figures, under what we call our decimal system, which means simply the practice we have adopted of expressing our numbers in multiples of ten. This arose from the fact that we happen to be born with ten fingers, and our ancestors, like our children, learned to count by means of those very useful “markers.”

In the system of counting every place, or column, counting from the right, has a value ten times greater than the one in the place or column nearest on the right. Thus in the number 36,542 the first figure on the right represents “ones,” the next ten times as much or “tens,” the next ten times as much again or “hundreds,” and so on. We really read this number backward when we name it, for in handling it in any way we have to start with the last figure, representing the “ones.” The number really means two ones, four tens, five hundreds, six thousands and three ten thousands. It is built up this way, really by addition:

Now, this principle underlies the processes called “carrying” and “borrowing.” You wish to add 26 and 37. Adding the 6 ones to the 7 you get 13 ones, or 3 ones and 1 ten. So you “carry” that 1 ten to the column where it belongs, leaving the 3 ones in their proper column. Thus, in your tens column you have 2 tens plus 3 tens plus the 1 ten “carried,” which makes 6 tens; and your result is 63, or 6 tens and 3 ones.

Again, you want to subtract 19 from 38. As you cannot take 9 from 8, you “borrow” one of the 3 tens, making your 8 into 18 and subtract 9 from that, leaving 9. By so doing you have left but 2 tens in your tens column, and so there your subtraction is now from 2, leaving 1. Hence your result is 9 ones and 1 ten, or 19.

Here is an example in subtraction which was once used, and which is as likely to trip one up as any that could be set. Subtract 199,999 from 320,012. The result is as follows:

Now, you cannot take 9 from 2, so you “borrow” one from the left and make your two 12. Then 9 from 12 leaves 3. In borrowing from the left you reduce the 1 in the tens column to 0. As you cannot take 9 from 0, you must again borrow from the left. But what are you to borrow from? In the third, or hundreds column there is only a 0. Hence, before you can borrow from this column you must make this 0 a 10 by borrowing from the fourth, or thousands column (counting your columns always from the right).

But again here you find only a 0, and so before you can make even this “borrow” you must borrow one from the 2 in the ten thousands column. Now see what happens. With the one which you have finally borrowed you have made the 0 left in the second or tens column into a 10, and you take 9 from 10, which leaves 1.

Now, here is where you forget something. When you started out to “borrow” you had to go away over to the 2 in the fifth column; that made your 0 in the fourth column a 10, but you immediately passed this one on to the third column, which left only 9; again you passed it on from the third to the second column, which left only a 9 in the third column. Hence you have now a 9 in the third and in the fourth columns, and your results there will be in each case 9 from 9 leaves 0.

Coming to the fifth you have a 1 instead of a 2, having borrowed 1; and you have to borrow again from the 3 to make your 1 into an 11, obtaining 9 from 11 leaves 2; and your sixth and last figure, being reduced from 3 to 2, your last result is 1 from 2 leaves 1.

This last part is easy, but one out of practice is almost certain to forget that his 0’s in the third and fourth columns became 9’s. If you have any difficulty with subtraction, study out the processes in this example until you understand them and you will never make a mistake again.

Now, as to the shape in which the examples will be given: The plain problems in addition will be unmistakable. You will be told that a concern sold 27,356 barrels of flour in one month, 38,452 the next, etc., and you cannot well run off the track. But you may find both processes involved in one “problem,” and you must then be careful to understand just what is meant by the question, so that you will know what you are expected to do with the figures.

Take this, for example: “A had $3,465 and B $4,895. A gained $1,146 and B lost $602. Which then had the more, and how much?”

Here you must add A’s gain to his principal—that is, the sum he had to start with—and subtract B’s loss from his principal; then subtract the smaller result from the larger, stating which is the “winner.” Thus:

Answer.—A has $318 more.

When it comes to multiplication and division, there is just one “catch,” so it might appear to the untrained mind of some poor candidate, which is made to play a part in nearly every problem. It is safe to say that 90 per cent. of the failures on these two processes turn on this one point. It is a very simple one and really the same in both processes. It arises in the handling of the “naught” or “cipher,” as we used to call it, the “zero”—call it what you like, it is nothing, anyhow. And that’s the point to be remembered.

Here is an example: Multiply 3,125 by 208. Now it seems almost incredible, but I have seen literally hundreds of papers, it seems to me, where this very simple problem was worked out this way:

The Wrong Way.

Or else this:

Another Wrong Way.

The trouble is that when the poor fellow came to multiply by the “naught” he forgot in the first instance that it was nothing, and that the biggest number in the world multiplied by nothing will produce nothing. He knew that something ought to go down there, and so in sheer desperation he wrote down the number he was multiplying.

In the second instance, while he recognized that nothing is nothing, he forgot that all our figuring is done by columns, as we saw in our last lesson; so that when we are multiplying by tens we must put our first figure down in the hundreds column, and so on. By forgetting this he multiplied his number by two hundreds, but put his first figure down in the tens columns, and thus he really multiplied by only 28 instead of 208.

Now, the very simplest way to avoid this sort of mistake is to “go through the motions” of multiplying by the “naught” or “zero.” Thus:

This looks a little clumsy, perhaps, but it is the logical way—to go through the process of saying naught times 5 is naught, naught times 2 is naught, etc., putting down the results in the proper columns. It is the safest way, if you are the least bit weak on the principles of numbers, to do even the process of multiplying by whole hundreds. Thus:

By writing his example in the “short cut” style I have seen many a man make this mistake:

That is, after setting down his two surplus ciphers, when he obtained another in multiplying 5 by 2, he forgot that it was a new one and went right on to the next process. If you are in that position that you must really learn your arithmetic all over again, stick to the logical method of showing every process and learn the “short cuts” afterward.

Now, when the reverse situation arises in division, a similar error is of frequent occurrence. Suppose we are to divide 650,000 by 3,125. This sometimes results:

That is, the figurer, when he came to try to divide 2,500 by 3,125, realizing that it would not “go,” simply “brought down” another figure. He forgot that the real mental process was 3,125 goes into 2,500 no times, or produces “naught,” and that “naught,” or “cipher,” must be set down in the proper tens column. The only safe way, again, is to indicate every process; to “bring down” but one figure at a time and to set down every result, even the “nothings,” in its proper place. That will make our example look like this:

Very simple, but let me “whisper,” if you really master and understand the mysteries of “long division,” you have crossed the Rubicon of education. There is no door in all human learning that need remain forever sealed to a persistent mind that has truly found its way clearly and understandingly through this first great stumbling block. Ask any old-fashioned school teacher to dispute that proposition. And, “whisper” again, there are men counting coupons who can do long division, to be sure, but who could not tell you why it is done as it is, if the price of stocks depended on it.

Punctuation.

Punctuation is a system of marks the purpose of which is to indicate to the eye the relation of words to one another in meaning, and so the relative importance of the component parts of a written composition.

The marks of Punctuation, corresponding, for the most part, to pauses in spoken language, are the comma (,), the period (.), the note of interrogation (?), the note of exclamation (!), the colon (:), the semi-colon (;), the dash (—), parentheses ( ), brackets [ ], quotation marks (“ ”), and the hyphen (-).

Purpose of Punctuation.—To make a written composition clear and intelligent, and to facilitate the task of reading.

Avoid All Unnecessary Remarks.—In modern writings punctuation marks are less frequently used than they were among writers in the early part of the last century. A sentence consisting of a simple subject, a simple predicate, and a simple object, or the relation of whose parts is clearly intelligible without marks, should not be encumbered with any. Take, for instance, the following two sentences:

“The attack was prepared with impenetrable secrecy.”

“On the very morning of the massacre they were in the houses and at the tables of those whose deaths they were plotting.”

Comma.—Three or more words of the same part of speech not connected by conjunctions should be separated from one another by commas.

“He was strong, alert, active.”

“New York City is grand, immense, beautiful.”

Two words contrasted with one another are separated by a comma.

“He is slow, but sure.”

Words in a series of pairs should be separated by a comma. “Young and old, strong and weak, fair and dark, good and bad.”

Explanatory and parenthetical words or phrases (such as “therefore,” “moreover,” “indeed,” “however,” “in fact,” “to some extent,” etc.), inserted into the body of a sentence are usually marked off by commas.

A comma is inserted after the name of a person or thing addressed.

“John, you were mistaken.”

“My country, I am proud of thee.”

Period.—The period (.) is put at the end of every complete sentence that is not exclamatory or interrogative. It is also used as a part of every abbreviation, and after every initial letter standing in place of the full word in a name. “A. M.” (for Master of Arts), “Mr.” (for Mister), “Esq.” (for Esquire), “R. W. Emerson” (for Ralph Waldo Emerson), “Dr.” (for doctor).

Note of Interrogation.—The note of interrogation (?) should follow every direct question: “Are you coming?” “Shall I buy it?” An interrogation point does not, however, follow an indirect question, such as “Let me know what he says.”

Note of Exclamation.—The note of exclamation (!) follows an exclamation, or any series of words denoting an outburst of feeling. “Alas!” “Three cheers!” “Hurrah!”

Colon.—The colon (:) is used to divide from one another the several co-ordinate members of a compound sentence, when they might each of them form an independent sentence, but are ranged side by side in a compound sentence for the sake of better showing how they illustrate one another.

“New York is a wonderful city: The wealthiest in America.”

A quotation or enumeration of details is often preceded by a colon.

“He spoke as follows:” “His last words were:” “Among those present were:”

Semi-Colon.—The semi-colon (;) separates co-ordinate sentences more dependent on one another than are those parted by the colon.

“Where it is prescribed that an act is to be done; or that the adverse party has a specified time to do an act; if service required is doubly the time allowed; except that,” etc.

In sentences containing two sets of subjects and predicates where either clause is very long or contains a subordinate clause, it is well to use a semi-colon.

Parentheses.—Parentheses ( ) are used to enclose words or phrases in a sentence, inserted by way of explanation or comment, but lying outside of the construction of the sentence:

“You see (as I predicted would be the case) I have had a long journey for nothing.”

Dash.—The dash (—) denotes, in most cases, a sudden digression from the general run of the sentence: “I want to tell you—but first let us go into the house.”

Sometimes the dash takes the place of the parentheses, when the clause, though digressive, bears some relation to the context.

Brackets.—Brackets [ ] are used to isolate interpolated words from the passage in which they are used:

“The examiner said that if they [the candidates] were discovered talking with each other he [the examiner] would have them [the candidates] expelled from the room.”

Hyphen.—A hyphen (-) is used, first to connect the part of a word at the end of a line with the remaining letters or syllables of the word beginning the next line; second, to conjoin two or more words into a compound word; as, “a never-to-be-forgotten day;” “long-winded,” etc.

The part of a word to which the hyphen is attached should be an integral part; that is, an entire syllable, and not merely certain letters composing only a part of a syllable.

Quotation Marks.—Quotation marks (“ ”) are used to distinguish a word, phrase, clause, sentence, or passage taken word for word, from any source outside that of the writing into which it is inserted.

A quotation within a quotation is marked off only by a single inverted comma before and after it. But a quotation within the second quotation requires double marks.

A passage quoted, not word for word, but only in substance, is often distinguished by but one quotation mark before and after it.

Capital Letters.—In examinations containing papers the rating of which is determined in part by correctness in the use of capital letters the average candidate is usually at a disadvantage. The following rules, if committed to memory, will enable the candidate to avoid errors which, if made, might cause him to fail in the examination.

The first word of every sentence should begin with a capital letter.

The days of the week, the months of the year, and holidays.

The names of places and countries; as, England, Yonkers, Belmont Park, etc.

The names of States, Mountains, Rivers and Lakes.

All words used to signify the Deity; as, He, Him, His, Thou, Thee, Thine, etc.

The names of persons, the titles of persons, and the titles of books; as, John Brown, Lord Salisbury, Senator Mitchell, “The Marble Faun.”

The first word in every line of poetry.

The pronoun I, and the exclamation O, or Oh.

The first word of a direct quotation should also begin with a capital; as, “To thine own self be true.”