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3. Mathematical Signs. (a.) Characters, or signs, are often used to indicate numerical coperations and relations. Among them are the following:

1st. The sign of Equality ( =) is two parallel lines. It is read EQUALS, or IS EQUAL TO, and signifies that the quantities between which it is placed are equal to each other.

ILLUSTRATION. —“4 pecks 1 bushel,” is read, “Four pecks equal one bushel,or,

Four pecks are equal to one bushel.2d. The sign of Addition ( + ) is a horizontal line crossing a vertical one. It is read and, or Plus, and indicates that the quantities between which it is placed are to be added together.

ILLUSTRATION. —“4 + 7 = 11,” is read, “ Four and seven are eleven,” or, Four plus seven equals eleven.”

3d. The sign of Subtraction (–) is a single horizontal line. It is read minus, or less, and indicates that the number which follows it is to be subtracted.

ILLUSTRATION. –“7 — 4 = 3,” is read, "Seven minus four equals three,” or, "Seven less four equals three."

4th. The sign of Multiplication (x) is a cross formed by two oblique lines. It is read Times, or MULTIPLIED BY, and indicates that the quantities between which it is placed are to be multiplied together.

ILLUSTRATION. -“3 X 4 = 12,” is read, “Three times four are twelve," or, " Three multiplied by four equals twelve."

5th. The sign of Division ( - ) is a horizontal line with a dot above and a dot below it. It is read DIVIDED BY, and indicates that the number which precedes it is to be divided by that which follows it. ILLUSTRATION.

.-"8+ 2 = 4,” is read, "Eight divided by two equals four.” 6th. Division may also be expressed by writing the number to be divided over the one by which it is to be divided, and drawing a line between them.

ILLUSTRATION. — is sometimes read, “Twelve divided by three,” but more commonly “twelve-thirds," in which case it is regarded as a FRAC

See 74.


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(a.) Notation and NUMERATION treat of the various methods of representing and expressing numbers.

(b.) Numbers may be expressed by visible objects or marks, by words, and by letters or other written characters.

(c.) Numbers are usually represented to the eye by characters called FIGURES, though sometimes by LETTERS OF THE ALPHABET.

(d.) The method by figures is called the ARABIC METHOD, because it was introduced into Europe by the Arabs.

(e.) The method by letters is called the ROMAN METHOD, because it was used by the ancient Romans.

5. Primitive Numbers and the Figures.

(a.) The first ten numbers are called the PRIMITIVE NUMBERS because in the DECIMAL System OF NUMBERS, which is universally used, all other numbers are derived from them.

(b.) Their names are: one, two, three, four, five, six, seven, eight, nine, ten.

(c.) The following figures are used in writing numbers : 1, or 1, called one. 6, or 6, called six. 2, or 2, called two. 7, or 7, called seven. 3, or 3, called three.

8, or 8, called eight. 4, or 4, called four. 9, or 9, called nine. 5, or 5, called five.

0, or 0, called zero, or cipher. The number ten is written 10, or 10.

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6. The Derived or Higher Numbers. (a.) 1 + 10 eleven 11.

6 + 10 = sixteen = 16. 2 + 10 twelve 12.

7+ 10 seventeen 17. 3 + 10 thirteen


8 + 10 - eighteen = 18. 4 + 10 = fourteen 14.

9 + 10 = nineteen 19. 5 + 10 fifteen 15.

10 + 10 - twenty (b.) 1 ten ten - 10.

6 tens

- sixty = 60.
2 tens
twenty = 20.

7 tens

seventy = 70. 3 tens = thirty = 30.

8 tens eighty = 80. 4 tens forty


9 tens = ninety - 90. 5 tens - fifty =

10 tens

= one hundred =100. (c.) 20 + 1 = twenty-one 21. 90 +1 ninety-one 91.

20 + 2 twenty-two 22. 90 + 8= ninety-eight = 98. 20 + 3 twenty-three = 23. 90 +9= ninety-nine

99. (d.) Read the following numbers : 1. 27. 3. 26. 5. 37.

7. 70.
2. 58.
4. 95.
6. 51.

8. 36. (e.) Analyze each of the above numbers. MODEL. --Twenty-seven equals two tens and seven units; fifty-eight equals five tens and eight units, etc.

= 50.

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7. The Decimal Point and Decimal Places.

(a.) The dot or point (like a period) at the right of each of the above numbers, is called the DECIMAL Point.

(b.) The first figure at the left of the decimal point is in the UNITS' PLACE, and represents units; the second figure at the left of the point is in the Tens' PLACE, and represents tens; the third figure at the left of the point is in the HUNDREDS' PLACE, and represents hundreds.

(c.) When the right-hand figure of a number is in the units' place, the decimal point is usually omitted. • (d.) The method of writing and reading numbers containing hundreds, tens, and units, may be seen below. 100 1 hundred.

287 = 2 hundred and 87. 200 2 hundred.

509 = 5 hundred and 9. 300 3 hundred.

480 - 4 hundred and 80.

(e.) Read each of the following numbers, and tell its denominations:

MODEL. — 518 five hundred and eighteen five hundreds, one ten, and eight units. 1. 518. 4. 204. 7. 240.

10. 307.
2. 185.
5. 402.
8. 576.

11. 490.
3. 851.
6. 420.
9. 894.

12. 036. (f.) Explain the use of the figures in the above numbers.

MODEL. — In 518, the 8 marks the units' place, and shows that there are 8 units; the 1 marks the tens' place, and shows that there is 1 ten; the 5 marks the hundreds place, and shows that there are 5 hundreds.

(g.) Give the value of each figure in the above numbers.

MODEL.-In 518, the 8 = 8 units; the 1 = 1 ten 10 units; the 5 = 5* hundreds 50 tens = 500 units.

(h.) Give the value of each of the above numbers in tens and units.

MODEL.— 518 = 51 tens and 8 units; 185 = 18 tens and 5 units.

Note.- Such exercises as the foregoing are of great value. The student who masters them will not be likely to find any serious difficulty in reading and understanding larger numbers, and decimal fractions.

(i.) Write each of the following numbers in figures: 1. Five hundred and eighty-nine. 6. Eight hundred and eighty. 2. Eight hundred and ninety-five. 7. Eight hundred and eight. 3. Nine hundred and fifty-eight. 8. Four hundred and sixty-one. 4. Six hundred and seven,

9. Nine hundred and two. 5. Seven hundred and six. 10. Two hundred and ninety.

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8. Higher Denominations and Places. (a.) As 10 units 1 ten, and 10 tens = 1 hundred, so 10 hundreds 1 thousand, 10 thousands 1 ten-thousand, etc.

(b.) The first figure at the left of the decimal point is in the units' place; the second figure is in the tens' place; and the third in the hundreds' place; and, in like manner, the fourth figure is in the thousands' place; the fifth in the ten-thousands' place, etc.; as illustrated in the following table:

15th place o Hundred-trillions.
14th place o Ten-trillions.
13th place © Trillions.
12th place Hundred-billions.
11th place o Ten-billions.

9th place o Hundred-millions. 8th place o Ten-millions.

6th place o Hundred-thousands.
5th place o Ten-thousands.
4th place o Thousands.

3d place o Hundreds.

• Decimal Point. 2d place o Tens. 1st place © Units.

(c.) Name the above places in their order, from the point towards the left, and also from the left towards the point.

(d.) Large numbers can be most easily read by dividing the figures representing them into sets or periods of three figures each, beginning at the right.

(e.) The first period is called the UNITS' PERIOD, the second the THOUSANDS' PERIOD, the third the MILLIONS' PERIOD, etc. as is illustrated below:

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(f.) Name the above periods in their order from right to left, and also from left to right.

(g.) Learn the following table:

1000 units 1 thousand.
1000 thousands 1 million.
1000 millions 1 billion.

1000 billions
1000 trillions

1 trillion.
1 quadrillion.

Note.— The English usually divide numbers into periods of six figures each, thus making a billion equal a million millions, a trillion equal a mil. lion billions, etc. This method is now rarely used in this country.

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