ARITHMETICAL MARKS AND SIGNS. =The sign of equality and is pronounced, equal to ; +The sign of Addition, and is pronounced, added to; -The sign of Subtraction, and is pronounced, subtract ed by. EXAMPLES. 12+7=19, twelve added to seven will be equal to nineteen. 23–8=15, twenty-three subtracted by eight, equal fifteen. *The sign of Multiplication, and is pronounced, Multiplied into; :The sign of Division, and is pronounced, divided by. EXAMPLES. 8x7=56, eight multiplied into seven equal fifty-six. 36-4=9, thirty-six divided by four, equal nine. Division is also implied by the signs 3)6(2 and 2, six divided by three, equal two. : : : : The sign of Proportion, and is pronounced, is to, so is, to. EXAMPLE. 6:9::8:12, as 6 is to 9 so is 8 to 12. ✓ or signifies the Square Root : thus 81 is read, the 1 square root of 81 ; or 812 is read 81 in the square root. 3 or denotes the Cube Root, &c. 3 means that 3 is squared, or to be multiplied, by itself. 2 3 4 3 means that 3 is to be cubed. 48 shows that 48 must be raised to the 4th power. 19+3x9=198 means that 19 added to 3, and the sum multiplied by 9, equal 198 12–2x3 =3 shows that 12 less the product of 2 multi2 plied by 3, and divided by 2 equal 3. ARITHMETIC. ARITHMETIC is the art and science of numbers and has for its operation four fundamental rules, viz. Addition, Subtraction, Multiplication, and Division. To understand these, it is necessary to have a perfect knowledge of our method of Numeration or Notation. NOTATION 1 one, 2 two, TEACHES to express numbers by words or characters. When performed by means of characters or figures, ten are employed. Nine of these are of intrinsic value and are called digits, or significant figures, being written and named thus : 4 four, 7 seven, 8 eight, 9 nine. The tenth figure, namely, 0, is called naught cr cipher, and denotes a want of value wherever it is found. Besides the simple value of the digits, as noted above, they have each a local one, which depends on the following principle. In a combination of figures, reckoning from riglit to left, the figure in the first place represents its simple value ; that in the second place ten times its simple value; that in the third place an hundred times its simple value; and so on ; each figure acquiring anew a tenfold value for every higher place it occupies. Hence our system of arithmetic is called decimal. The names of places are denominated according to their order. The first is the place of units; the second of tens ; the third of hundreds; the fourth of thousands ; the fifth of ten thousands; the sixth of hundred thousands; the seventh of millions; and so on. Thus in the number 8888888; 8 in the first place signifies only eight; 8 in the second place eight tens or eighty ; 8 in the third place eight hundred ; 8 in the fourth place eight thousand; 8 in the fifth place eighty thousand ; 8 in the sixth place eight hundred thousand; 8 in the seventh place eight millions. The whole number is read thus, eight millions, eight hundred and eighty-eight thousand, eight hundred and eighty-eight. Though a cipher has no value of itself, yet it occupies a place; and when set on the right hand of other figures it increases their value in the same tenfold proportion : Thus in the number 8080; the ciphers in the first and third places denote, that, though no simple unit or hundreds are reckoned, yet the place of units and that of hundreds are to be kept up to assist in reckoning the tens and thousands. The above number (8080) is read eight thousand and eighty, which, without the two ciphers, would be read eighty-eight. Large numbers are divided into periods and half periods, each half period consisting of three figures. The name of the first period is units; of the second millions; of the third billions ; of the fourth trillions; and also, the first part of any period is so many units of it; and the latter part so many thousands of it.* NUMERATION TABLE. coHundreds of Millions. A Thousands. APPLICATION. To express in figures Numbers which exceed Nine. Rule.Write down ciphers to so many places as are named in the given number; then, beginning at the left, observe at each place what significant figure is named, and, taking away the cipher, write the significant figure in its place; and thus proceed with each place till you come to the place of units. EXAMPLES. One hundred and one millions, fourteen thousand and fourteen. To read NUMBERS. RULE.First numerate, from the right to the left hand, each figure, in its proper place, by saying, units, tens, hundreds, &c., as in the Numeration Table. Then, to the simple value of each figure, join the name of its place, beginning at the left hand, and reading to the right. EXAMPLES. 64, 396, Note.—The pupil should be accustomed, in each Example, in the following Rules, to read correctly not only every answer, but every line of numbers in his sum. ADDITION. ADDITION in Arithmetic is the uniting or joining together of two or more numbers. Simple Addition is the collecting of several numbers, of the same denomination into one sum ; as, 4 yards and 6 yards, expressed in one sum, are 10 yards. Addition and Subtraction Table. When you would add two numbers, seek one of them in the left hand column, and the other in the top line ; and in the common angle of meeting, or at the right hand of the first, and under the second, you will find the sum; as, 6 and 9 are 15; and so of any others. When you would subtract, seek, in the left hand column, the number to be subtracted from the greater ; then run your eye along, in the same line, towards the right hand, till you find the number from which the other is to be taken ; and exactly over this last, in the top line, you will find the difference; as, 6 from 15, and there remain 9; and so of any others. SIMPLE ADDITION. Rulb.Write the numbers, units under units, tens under tens, &c. and draw a line under the whole. Add up the unit column, and if the sum be less than ten, write |