ARITHMETICAL SIGNS OR CHARACTERS. + is the sign of ADDITION; and is called plus, or more. is called minus, or less, and denotes SUBTRACTION. into, and denotes MULTIPLICATION.

S

by, and denotes DIVISION.

is the sign of EQUALITY.

::

are the signs of PROPORTION.

is the Radical Sign, or Sign of EVOLUTION.

denotes the difference between two quantities when it is uncertain which is the greater.

ILLUSTRATIONS.

6+3=9 signifies 6 plus 3 equal 9: that is, 6 added to 3 equal 9. 7-4-3 signifies 7 minus 4 equal 3: that is, 4 taken from 7

leaves 3.

4x3=12 is read 4 into 3 equal 12: that is, 4 multiplied by 3 equal 12.

12÷43 is read 12 by 4 equal 3. But Division is more conveniently expressed in the form of a Fraction:

=

thus, 123; twelve divided by four equal three. As 2:48: 16; As 2 are to 4, so are 8 to 16.

164; the Square Root of 16 equal 4. 3√64 = 4; the Cube Root of 64 equal 4.

A vinculum connects two or more terms which are to be considered as forming one term or quantity. It is signified by a line drawn over them; or by parentheses including them.

A point is often used instead of the cross to denote Multiplication; and in Algebraical Theorems, &c., the multiplication of the quantities denoted by different letters, is understood by placing the letters together without any sign between them. Thus 2.5+7, denotes that 5 and 7 are to be added, and the sum (12) to be multiplied by 2. Also, (ab+c). (n-1) denotes that a and b are to be multiplied, c added to the product, and the whole to be multiplied by one less than the number or quantity represented by n.

THE

TUTOR'S ASSISTANT;

BEING A COMPENDIUM

OF PRACTICAL ARITHMETIC.

Integers, or Whole Numbers.

ARITHMETIC is the science of numbers; or the art of numerical computation. A whole number is a unit, or a collection of units.

Numbers are expressed by ten written characters called figures, or digits: viz. 1, 2, 3, 4, 5, 6, 7, 8, 9, which are significant figures, all declaring their own values by the names; and the cipher, or nought (0) an insignificant figure, indicating no value when it stands alone.

NUMERATION AND NOTATION.

A figure standing alone, or the first on the right of others, denotes only its simple value, as so many units, or ones; the second is so many tens; the third, so many hundreds, &c. increasing continually towards the left in a tenfold proportion.

Numeration is the art of reading numbers expressed in figures; and Notation, the art of expressing numbers by figures.

NOTE. To read any Number. Divide it into periods of six figures each, beginning at the right hand; and each period into semi-periods with a different mark, for the sake of distinction. The first on the right hand is the Units' period, the second the Millions' period, &c. Beginning at the left, observe that the three figures of every complete semi-period must be reckoned as so many hundreds, tens, and units; joining the word thousands when you come to the middle of the period, and the proper name of the period at the end of it.

2. To express any given Number in Figures. Begin at the left, and write the figures which denote (as so many hundreds, tens, and units) the number in that semi-period; and proceed thus with each successive semi-period, till the whole is completed; placing a separating comma in the middle of each period, or immediately after the thou sands, and a semicolon between the periods. But observe, that though every semi-period but the first on the left must have its complete number of three figures, that may be incomplete, and consist of only one or two figures: also, where significant figures are not required in any part of a number, no semi-period must be omitted, but the places must be filled up with ciphers.

Example. Write in figures, seventy thousand four hundred billions, two hundred and ten thousand millions, and ninety-six.

First, write 70 (seventy) with a comma, these being thousands; then 400 (four hundred) with a semicolon, denoting the end of the period; next, write 210 (two hundred and ten) and, because they are thousands, put a comma after them, and then 000 (three ciphers, there being no more millions) followed by a semicolon, to denote the completion of the period; again, put 000 (three more ciphers, denoting the absence of thousands) with a comma after them, and then 096, (ninety-six) which will complete the number : thus, 70,400; 210,000; 000,096.

EXERCISES IN NUMERATION AND NOTATION.

Read, or write in words the following numbers.

The figures in parentheses refer to the Editor's Key to this work. See Advertisement on the first page.

Express in figures the following numbers.

(1) Nine; ninety; ninety-nine; nine hundred ; nine hundred and nine; nine hundred and ninety ; nine hundred and ninety-nine.

(2) One hundred and eight; one hundred and eighty; eight hundred and one; eight hundred and ten; one hundred and sixteen; one hundred and sixty-one; six hundred and eleven.

(3) One hundred and twenty-three; one hundred and thirty-two; two hundred and thirteen; two hundred and thirty-one; three hundred and twelve; three hundred and twenty-one.

(4) Two thousand five hundred and seventy-two.

(5) Seventy-two thousand five hundred and seventy-two. (6) Five hundred and seventy-two thousand five hundred and seventy-two.

(7) Ten thousand nine hundred and ten.

(8) Nine hundred and ten thousand nine hundred and ten. (9) One hundred and nine thousand nine hundred and one. (10) One hundred and ninety thousand and ninety-one. (11) Nine hundred and one thousand and nineteen. (12) One hundred and fourteen millions, one hundred and forty-one thousand four hundred and eleven.

(13) Four hundred and six millions, six hundred and four thousand four hundred and sixty.

(14) Six hundred and forty millions, forty-six thousand and sixty-four.

(15) Seven millions, seventy thousand seven hundred. (16) Seven hundred millions, seven thousand and seventy. (17) Ten millions, one thousand one hundred.

(18) One hundred and one millions, eleven thousand one hundred and ten.

(19) Twelve billions, seventeen thousand and nine millions, and eighty-nine.

(20) Seven thousand five hundred and four trillions, sixty thousand millions, eight hundred thousand.