NOTATION 92. For a thorough understanding of arithmetic it is necessary to be familiar with the system of notation used in expressing values and quantities in figures. Ten characters (figures) are used, nine of which have a positive or integral value. These are represented by the figures 1 2 3 4 5 6 7 8 9. The tenth figure, 0 (read cipher, zero, or naught), represents nothing and has no integral value. In other words, the figure 3 stands for three individual units; 5, for five individual units; 7, for seven; 9, for nine; while the cipher is used to visualize nothing. These nine integral units, with their accompanying cipher, are given a distinct value according to their position in relation to a fixed line represented by the decimal point. Thus, one in the units column - the first column to the left of the line unit. means one whole Move this 1 to the next column, one place to the left; fill in the space from which it has been taken with a cipher to show that nothing is there; it then represents the value of "ten" and is so read. Move it one more column to the left and it represents ten times ten, or one hundred, and so on. Writing the 1 in the second, or tens column, and the figure 3 in the first, or units column, we show 10 units (in the tens column) and 3 units (in the units column), that is, 13, read thirteen. 93. It will readily be seen that the removal of a figure one column to the left multiplies its value by ten. It will also be apparent that bringing it back one place to the right divides its value by ten. This is the governing principle of notation and may be applied on either side of the decimal line. The figure 1 starting to the left of the line, in the units column, and moving one place to the right becomes of 1; moved another place to the right it becomes of o, which is 1o; moved another place to the right it becomes o of 100, or 1000, and so on without limit. If a student has difficulty in learning the value of a figure to the right of a decimal point (which is merely the line of division between whole numbers and their fractional parts represented by tenths), it is suggested that he take an ordinary sheet of writing paper, turn it sidewise, write the names of the various places in the columns thus formed, draw a heavy line to represent the decimal line, write the decimal notation to the right of this line, and then place figures in such columns as may appeal to him, calling them by the names of the values written in the columns. Exercises of this kind are very valuable for students whose minds are so constituted that they have difficulty in reading decimals. Difficulty in reading decimals should not be ascribed to arithmetical weakness, but rather to inability to use the imagination in repreзenting figure pictures. 94. Figures to the left of the decimal line represent whole numbers and are called Integers. 95. Figures to the right of the decimal line represent parts (tenths, hundredths, etc.), and are called Decimals. ་་ COMMON PROCESSES ADDITION INTEGERS 96. Addition is the process of combining several numbers into one quantity that shall equal the value of all. (a) Only numbers representing like values or like quantities, or parts of like values or like quantities, can be added, thus, 5 cows can be added to 2 cows and the result will be 7 cows. 5 horses added to 5 cows would produce 10 things, but they would be neither horses nor cows just 10 animals. (b) Figures not used to measure value or quantity can be added as a mere matter of counting. Furthermore, unlike things may be added, if first reduced to common terms. The term "animal" is common to both horses and cows. 97. The name Addend is applied to any of the individual quantities or values that are added. 98. The name Sum is applied to the total value of the addends. It is the result of addition. ILLUSTRATED SOLUTION 99. Problem: $324.23 $89.96+$742.05 + $23 + $1.95+$796.45 ? = $324 23 89 96 742 05 23 195 796 45 $1977 64 Arrange the addends in a column, placing decimal point under decimal point so as to form the decimal line. Add either up or down. Beginning with cents (first right-hand column) reading up and combining values as we go, we have 5, 10, 15, 21, 24 cents. This is 2 tens-cents and 4 units-cents. Write the 4 units-cents under the first column. Carry 2 tens-cents to the next column. as before the result is 26. Write 6, carry 2. the next, or unit-dollars column, we have 27. Write the 7, The sum of the next column is 27. Write the 7, carry 2. The total of the next column is 19. As this is the last column write both figures. Adding carry 2. 100. To Check the Work: Add each column separately, beginning with the first right-hand column. Write the results as partial sums, one under the other, each one place to the left of its predecessor, thus: or we may begin on the left and work in the opposite direction, adding, either up or down, thus: |