The place of a figure denotes its value, according to the scale of the Millions. Hundred thousands. Ten thousands. Thousands. ⚫ Hundreds. ⚫ Tens. C Units. 2 Twa 23 Twenty-three. 235 Two hundred thirty-five. 2354 Two thousand three hundred fifty-four. 23546 235460 2 3 5 4 6 0 8 2 3 5 4 6081 2 3 5 4 6 0819 2 3 5 4 6 08197 Two hundred thirty-five million four hundred sixty thousand eight hundred and nineteen. Two billions three hundred fifty-four millions six hundred and eight thousand one hundred ninety-seven. NOTE.-These denominations are sometimes divided into periods of three each from the right, the first period being of units, the second being of thousands, the third being of millions, the fourth being of billions, the fifth being of trillions, the sixth being of quadrillions, etc. A figure being moved one place to the left, in the above table, increases its value ten-fold, and ten of one place equal one of the next place on its left. Although a cipher expresses no number, it is used to locate figures of other orders in their proper places; thus, to express six thousand and eighty-one (6081), the cipher locates the figure 6 in its proper place. Read the following numbers : 302-4516—2007—180960-81108- 9716439 430000 5000006 340657 9977700-18001800-24697-684321—5402835. 17647-2854 The Notation of Decimal Fractions or Decimals, is in harmony with that of Integers, as shown in the following The figures in a decimal expression are the numerator of a decimal fraction; its denominator is understood to be 1 followed by as many ciphers as there are decimal places in the expression; thus, the decimal fraction, .0356, when written as a common fraction, is 10000: 356 The denominator of a decimal fraction is always equal to 1 unit, though it may be understood as tenths, hundredths, etc. Whole numbers, or integers, and decimals, can be written together in one line of figures, by using the decimal point to separate them, because there is the same scale of increasing or decreasing value with both; thus, 2.2 is read, two and two tenths, and signifies 2 units and of another unit; 431.56 is read, four hundred and thirty-one and fifty-six hundredths of one. Etc. ADD., SUB., MULT., DIV. OF DECIMALS. 173 The Addition, Subtraction, Multiplication, and Division of Mixed Decimals, or Numbers expressing integers and decimals, are the same as in whole numbers; thus, NOTE.-In operations in Addition and Subtraction, the figures of each denomination of one number must be added to or subtracted from the figures of the same denominations in the other numbers; thereby, the decimal points will be under each other, and before the tenths placed in the Sum and Remainder. In Multiplication, the Product must have as many decimal places as there are decimal places in both the Multiplicand and Multiplier. If necessary, ciphers must be prefixed to the product to make its number of decimal places equal. In Division, the number of decimal places in the Quotient must equal the decimal places in the Dividend diminished by the number of decimal places in the Divisor. If necessary to produce this result, ciphers must be prefixed to the Quotient, or affixed to the Dividend, and when both Dividend and Divisor contain the same number of decimal places, the Quotient is a whole number, with or without a Remainder, as the case may be. Annexing a Cipher on the right of a decimal expression does not alter its value, but multiplies both terms of the fraction by 10, and reduces it to the next lower denomination; thus, .2==.20=20% or =.200=1000. A decimal, or a mixed number of integers and decimals, can be multiplied by 10, 100, 1000, etc., by removing the decimal point as many places to the right as there are ciphers in the multiplier. Multiply .678 by 10, by 100, by 1000; 178.9 by 10; 19.320 by 10; 16.0054 × 100=? 54.37898 × 10000 ? 732.98 × 1000= ? Prefixing a Cipher in a decimal expression, di vides the decimal by 10, by removing each of its fig. ures one place farther from the decimal point; thus, .5=0, .05=180, .005=1000. A decimal, or a mixed number of integers and decimals, can be divided by 10, 100, 1000, etc., by removing the decimal point as many places to the left as there are ciphers in the divisor. Divide 13.41 by 10, by 100, by 1000; 476.9 by 100, by 10000; .17 by 10; .176 by 1000; 438.15÷10= ? 167.009÷10=? ÷÷1000= ? = .75 of a metre how many decimetres and centi metres? 28.03 expresses how many metres, decimetres, and centimetres? Divide 28.03 by such use of the decimal point, so that it will express 2 grammes 8 decigrammes and 3 milligrammes. Express 3650 cents in dollars and cents, by the use of the decimal point. Read 45.75 in United States currency, in French currency, in decimal measures of weight, decimal measures of capacity. Read 196.071. A Common Fraction can be changed to an equivalent Decimal Fraction, by dividing its numerator with ciphers annexed, by its denominator, and making as many decimal places in the quotient as there are ciphers annexed; thus, =10÷2=5. Reduce to its equivalent decimal. = = ANALYSIS 1ST.-3=30 tenths÷8 gives 3 tenths and 6 tenths remaining; 6 tenths 60 hundredths÷8 gives 7 hundredths and 4 hundredths remaining; 4 hundredths 40 thousandths ÷8 gives 5 thousandths and no remainder; hence =.375. ANALYSIS 2ND.-=:888 +375=375.375. of $1 equals how many cents? .5 of $1 equals how many cents? What is the decimal expression of ? .8 of $1 equals how many dimes? or .75 of $1 equals how much money? of a metre equals how many decimetres and centimetres ? QUESTIONS, QUERIES, AND PUZZLES FOR PUPILS AT HOME. 1. A hound is in pursuit of a fox that is 10 rods ahead of him, and while the hound runs 10 rods the fox runs 1 rod, (i. e. while the hound runs a certain distance, the fox runs one-tenth of that distance.) Will the hound overtake the fox? The conditions remaining the same, what is the greatest distance they can run? 2. A hound is in pursuit of a fox that is 10 rods ahead of him, and while the fox runs 1 rod the hound runs 10 rods, How far will the hound run before he overtakes the fox? 3. Place four 5's in such a position that they shall equal 6. 4. A boy was sent to a spring with a 5 and a 3 quart measure to procure exactly 4 quarts of water. How did he measure it? 5. What is the difference between twice 25, and twice 5 and 20 ? 6. A man had 9 pigs and put them in 4 pens, with an odd number of pigs in each pen. How did he divide them? 7. Two men have an 8 gallon cask full of wine, which they desire to divide equally between them. How can they effect this division, provided the only measures they have are a 5 gallon cask and a 3 gallon cask? 8. Place four 2's in such a manner that they shall exactly equal 23. 9. Place 9 apples in 10 rows so that each row shall contain 3 apples. 10. A squirrel finding 9 ears of corn in a box, took from it daily, 3 ears; how many days was he in removing the corn from the box? 11. If from six you take IX, and from IX you take ten, and if nifty from forty be taken, there will then just half a dozen ren air. |