ART. 9. The Arabic figure, 0, naught, is frequently called cipher or zero. By itself it has no value, but is just as neces sary as any other figure. The other nine figures are called significant figures, because each one signifies or expresses a certain number of units. ART. 10. By these ten figures all numbers can be expressed. Nine is the largest number that can be expressed by one figure. Numbers above nine are represented by combining two or more of these ten figures, thus: ART. 11. As men had ten fingers, it was natural, when they wished to make calculations, to count on the fingers one, two, three, four, five, six, seven, eight, nine, ten, and then to begin again and count another ten; and they would naturally count as many tens as they had fingers. In this way they would count a number of tens, viz.: ART. 12. To express numbers between the tens we pro ceed thus: Twenty-me means 2 tens and one, expressed by 21 ART. 13. After counting to one hundred, men would naturally proceed with the hundreds in the same way that they did with the tens, and they would count as many hundreds as they had fingers. In this way they would get a number of hundreds, viz.: One hundred, expressed by 100. Ten hundred is called one thousand. ART. 14. The particular position a figure occupies with respect to other figures is called its place; thus, in the number 253, counting from the right hand, 3 stands in the first place, 5 in the second place and 2 in the third place, and so on for any number of figures. The figure in the first place is called units, the figure in the second place is called tens, and in the third place hundreds. Thus, 253 would express 2 hundreds 5 tens and 3 units. ART. 15. In Art. 13 the pupil was shown how to write hundreds. If we wish to supply intermediate numbers between 100 and 200, between 200 and 300, etc., we simply write the figure expressing hundreds in the third place, tens in the second place and units in the first place. Thus: Three hundred and fifty-nine would be written 359. 66 567. 935. NOTE. The teacher would do well to pause just here and drill the pupil in writing numbers up to one thousand, until he can write them readily and accurately. "Make haste slowly" is the true secret of teaching Arithmetic. be ART. 16. Sometimes the units or the tens, or both, may wanting. In such cases we supply their places with a cipher, 0. Thus: Two hundred and seven would be written 207. Let the pupil express the following in figures: Eight hundred and three. Two hundred and forty. Six hundred and eighty. Seven hundred and one. Three hundred and seven. Two hundred and thirty. NOTE.-The foregoing examples illustrate the utility of the 0. Although it has no value of itself, it serves a useful purpose in keeping other figures in their proper places. ART. 17. After counting to one thousand, as was explained in Art. 13, men would naturally proceed with the thousands as they did with the hundreds, counting as many thousands as they had fingers, and thus would get one thousand, two thousand, etc., up to ten thousand. After counting to ten-thousands, as just explained, men would proceed with the ten-thousands as they had done with the thousands, and thus would get ONE ten-thousand, Two ten-thousands, etc., up to TEN ten-thousands, now called ONE hundred thousand. Proceeding in the same way, they would finally reach TEN hundred thousand, which is called a MILLION. ART. 18. After counting to MILLIONS, as just explained, men would count ONE million, Two millions, THREE millions, etc., until they reached TEN millions. They would then count ten of these TEN millions, making ONE HUNDRED millions. Counting on, they would reach TEN HUNDRED millions. This last number is now called a BILLION. ART. 19. Counting successively by tens, we have learned the names of the following places of figures, viz.: units, tens, hundreds, thousands, ten-thousands, hundred-thousands, millions, ten-millions, hundred-millions, billions. We have thus formed a scale of units in which each new unit is ten times greater than the preceding one. Thus thousands are ten times greater than hundreds, and hundreds are ten times greater than tens. ART. 20. The figures 1, 2, 3, 4, etc., when they stand alone, or when they occupy the first place, denote simply units or ones, and are called units of the first order. When they occupy the are called units of the second place, they represent tens, and second order. When found in the third place, they stand for hundreds, and are called units of the third order, and so on This may be illustrated by the following table: The 5th order of units is called ten-thousands. And we may extend this table to trillions, quadrillions, etc. ART. 21. Suppose we wish to express in figures the number thirty-two thousand five hundred and nine. In this example the highest order of units is ten-thousands, of which there are 3; the next order is thousands, of which there are 2; the next order is hundreds, of which there are 5; the next order is tens, of which there are 0; and the next order is units, of which there are 9. To write the number we first set down the units of the highest order, viz.: tenthousands; then the thousands, hundreds, tens and units in succession. The number is therefore written thus: 32509. Hence we have the RULE FOR NOTATION Begin at the left hand and write the units of the HIGHEST order mentioned, and set down each order in succession in its proper place, filling the place of each absent order with a cipher. EXERCISES IN NOTATION. ART. 22. Let the pupil express the following numbers in figures 1. Eighty-seven. 2. Three hundred and seventy-six. 3 Five hundred and seven. 4. Six thousand three hundred and seventy-five. Ans 87. |