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students who take up the work of Stage II. without having previously used this book for Stage I.

The amount of Plane Trigonometry included in this volume is small, extending only to the solution of triangles and the simpler cases of heights and distances. There is, however, sufficient to fully cover the requirements of Stage II. of the Government Syllabus. The treatment of the higher parts of Algebra and Plane Trigonometry, as well as that of Spherical Trigonometry, is reserved for a second volume.

E А.

LEICESTER, November, 1873.

MATHEMATICS.

FIRST STAGE.

SECTION I.

A RITHMETIC.

CHAPTER 1.

THE FUNDAMENTAL PRINCIPLES AND RULES APPLIED

TO WHOLE NUMBERS AND DECIMAL FRACTIONS.

Notation and Numeration. 1. We learn from elementary books on Arithmetic, that figures have a local as well as an intrinsic value, and that the local value of a figure increases tenfold, or diminishes tenfold, according as its position is changed from right to left, or from left to right. Thus, commencing with the right hand figure of an ordinary number, the respective figures of the number stand for units, tens, hundreds, thousands, &c.; or, beginning with the left hand figure, which, we will suppose, stands for thousands, the respective figures represent thousands, hundreds, tens, units. Let us carry this principle a little further. Take the figures 68754, and suppose that 7 represents 7 units; the question then arises as to the number represented by 68754. Now, as 7 is the units figure, we have evidently, by the above principle, 6 hundreds, 8 tens, 7 units; and further, remembering that the local value of a figure decreases tenfold for every remove to the right, the 5, on our supposition, must represent 5 tenths, and the 4 must represent 4 hundredths. Let us, as is usual in numbers thus represented, mark the units' figure by placing a dot to the right of it. Thus, 357.2605 will then represent 3 hundreds, 5 tens, 7 units, 2 tenths, 6 hundredths, 5 ten-thousandths; and to take one other example, •3065, where the units' figure, though not expressed, is actually 0, will represent 3 tenths, 6 thousandths, 5 ten-thousandths. The dot is called the decimal point, and the digits to the right are called decimals, because they represent portions of the unit obtained by cutting it up into a number of equal parts, which is always some power of 10. It may be remarked, that 10 is called the first power of 10; 100, or 10 x 10, the second power, sometimes written 102; 1000, or 10 x 10 x 10, the third power, written 103, and so on.

To make the subject clear, let us see what the decimals, •237, 2370, :0237 respectively represent. Now, the digits 2, 3, 7, in the first two decimals, are in exactly the same position with regard to the decimal point, and the respective digits in each have the same absolute value; moreover, the cipher affixed to the right of the decimal •2370, has no intrinsic value, and hence the two decimals, .237, and .2370, have the same absolute value. And since the reasoning is the same, no matter how many ciphers are affixed to the right, we get the following important principle :

The value of a decimal is not altered by affixing ciphers to the right.

We will now compare the first and third of our examples, namely, the decimals •237 and .0237. The cipher which is here prefixed to the left, has again no intrinsic value; but it has removed the digits 2, 3, 7, one stage to the right, and has, therefore, diminished their local value tenfold. The effect of prefixing the cipher, is therefore to diminish the absolute value of the decimal tenfold, and as every additional cipher so prefixed has a similar effect, we get another fundamental principle, as follows :

The value of a decimal is diminished tenfold for every cipher prefixed.

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