THE NEW ADVANCED ARITHMETIC. Part I. SECTION L DEFINITIONS. 1. Measuring is the process of finding how many times a quantity contains a part of itself which is taken as a standard. Illustrate. 2. That portion of a measured quantity which is used as a standard is called a Unit. When we count a basket of eggs, we measure the quantity of eggs by using one of them as a standard. Such a unit is a natural unit. When we measure a quantity of cloth, we use a portion of itself, called a yard, as a standard. Such a unit is an artificial unit. 3. From the repetitions of the unit in counting or measuring, the successive numbers, one, two, three, etc., arise; thus, Number is that which answers the question "How many?" 4. Arithmetic is the science which treats of number and the methods of employing it in computation. 5. To simplify counting or measuring, units are gathered into equal groups, each of which forms a new unit. Thus, in measuring a basket of eggs, they are grouped into dozens. The quantity may be expressed as a number of single eggs or as a number of dozens. A Decimal System of numbers is a system in which ones are grouped into tens; tens into tens of tens, or hundreds; hundreds into tens of hundreds, or thousands; thousands into ten-thousands, etc. Ones are units of the first rank, or order; tens are units of the second order; hundreds, of the third order, etc. Illustrate these groupings with bundles of splints. The scale in any system of numbers is the number of units in each order required to form one of the next higher. NOTATION. 6. The art of expressing numbers by means of characters is Notation. 7. A system of Notation which will express all numbers must include a set of characters to represent numbers and the laws for using them. 8. The Arabic System of Notation employs ten characters called figures. They are 1 (one), 2 (two), 3 (three), 4 (four), 5 (five), 6 (six), 7 (seven), 8 (eight), 9 (nine), 0 (cipher). 9. Any given figure always expresses the same number of units, but the order or kind of units is expressed by the place in which the figure is written. Ones stand in the first place, tens in the next place to the left, hundreds in the third place, etc.; thus, A figure standing in any place expresses units ten times as large as if standing one place to the right. The cipher is used to fill vacant orders. NUMERATION. 10. The art of reading numbers expressed by figures is Numeration. 11. Three orders form a period. The name of the lowest order in each period is ones; of the next higher is tens; of the third is hundreds. The names of the first twelve periods in their order are as follows: 1. Units. 2. Thousands. 3. Millions. 4. Billions. 5. Trillions. 6. Quadrillions. 7. Quintillions. 8. Sextillions. 9. Septillions. 10. Octillions. 11. Nonillions. 12. Decillions. NOTE. — The meaning of the prefix in the word billion is two; in the word trillion is three; in quadrillion is four, and so on. Observe that the number of any period above millions is two more than the meaning of the prefix in the name of that period. 12. Arrangement of orders and periods. Trillions. Billions. Millions. Thousands. Units. ∞ Hundred-trillions. Ten-trillions. Hundred-billions. ∞ Ten-millions. Millions. -Hundred-thousands. NOTE. right. 8 6 5,4 0 6,3 8 2,1 0 4,5 7 9 Learn the names of the periods in their order from left to 13. To read a number, group the figures into periods, beginning at the right, and separating the periods by commas. Beginning at the left, read the number in each period as if it stood alone; then add the name of the period. NOTE. - The English system of Numeration is in use in England and upon the continent of Europe, except in France. It employs six orders for a period. In studying this system the meaning of the names of the periods is made plain. A million is a thousand thousand. A billion is the square of a million; a trillion, the third power of a million; a quadrillion, the fourth power of a million, etc. 14. Read the following numbers: |