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The study of dimension is kept constantly before the student by the introduction of numerous problems in measurement throughout the book, and is given very full and definite treatment in the chapter on mensuration.

A chapter is devoted to the principles of algebra, including the use of letters to indicate quantities and of the equation in solving problems. Mathematically speaking, this is of the greatest consequence, introducing the child, as it does, to the higher forms of thinking involved in the use of general terms as distinguished from the merely specific.

An arithmetical statement of fact, though a generalization, stands for itself alone; 2 + 3 = 5 remains that and nothing more, while a + b = e stands for an infinite number of possible terms, for all of which it is equally true.

Dr. Oliver Wendell Holmes, in the Autocrat, wittily puts it thus:

"I was just going to say when I was interrupted, that one of the many ways of classifying minds is under the heads of arithmetical and algebraical intellects. All economical and practical wisdom is an extension or variation of the following arithmetical formula: 2+2 = 4. Every philosophical proposition has the more general character of the expression a+b=c. We are mere operatives, empirics, and egotists, until we learn to think in letters instead of figures."

The metric system of measurement is given full and adequate treatment. But it is put by itself in the supplement, to be used or not as school authorities may desire.

Book III is a complete arithmetic, orderly in plan and simple in statement. It is believed that any child of ordinary intelligence can read it alone, and even without a teacher, can obtain from it a fair measure of both the utilitarian and the cultural values of the subject.

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BOOK III

CHAPTER I

THE FUNDAMENTAL PROCESSES

NOTATION AND NUMERATION

1. Write in as many ways as you can the number of the present year.

Read 26,057; MDXCVII.

Writing numbers is called notation.

Reading numbers is called numeration.

A regular plan of writing or reading numbers is called a system of notation or of numeration.

There are two systems of notation and numeration in use in this country: the Arabic and the Roman.

The Arabic system was invented by the ancient Hindus, the people of India. It was introduced into Europe by the Arabs, hence its name.

The Roman system was used by the ancient Romans.

The Arabic system is the one used in all common business transactions. The Roman system is used chiefly in the numbering of certain chapters and pages of books, and on the faces of clocks and watches.

Read 349; CCCXLIX. Write eighty-eight in both Arabic and Roman notation. Which do you think the better system? Why?

THE ARABIC SYSTEM

2. The characters, or figures, of the Arabic system are

1, 2, 3, 4, 5, 6, 7, 8, 9, called digits, and 0, called zero,

or cipher.

Ten ones, or units

Ten tens

Ten hundreds

1,000 thousands

1,000 millions

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= 1 hundred; 10 x 10 = 100.

= 1 thousand; 10 x 100 = 1,000.

= 1 million; 1,000 × 1,000 = 1,000,000. = 1 billion; 1,000 x 1,000,000

= 1,000,000,000.

The use of numbers higher than millions is rare, though the following numbers are found by using 1,000 as a multiplier: billions, trillions, quadrillions, quintillions, sextillions, septillions, octillions, nonillions, and decillions. 1 decillion would contain 33 ciphers.

The Arabic system is called a decimal system (from the Latin word decem, meaning "ten") because it is a system of tens, each place having a value ten times greater or ten times less than the one next to it.

The word digit means "finger." It indicates that the decimal system grew out of the custom of counting on the fingers. Numbers as written by the Arabic system are divided into periods of three figures each. According to its position each figure stands for units, tens, or hundreds of its period. In 3,126,374,201 the value of each figure is as follows:

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