Lessons in Decimals.
The paper on arithmetic in second grade examinations usually contains one, sometimes two, problems in common or decimal fractions. These are no more difficult to solve when one understands the rules governing them, than any simple test in addition, division, etc. In whole numbers, as 57, 563, 4278, the various units increase on a scale of ten to the left (or decrease on the same scale of ten to the right). Thus in the last number we say 8 units, 7 tens, 2 hundreds, and 4 thousands or four thousand two hundred seventy-eight.
Decimals also decrease on a scale of ten to the right (or increase on the same scale of ten to the left). In writing decimals, we first write the decimal point, which is the same mark we use at the close of a sentence and is called a period. Then the first figure to the right is called “tenths” and is written thus .6, meaning six tenths. The second figure stands for hundredths as .06, six hundredths; .006 for six thousandths; .0006 for six ten-thousandths; .00006 for six hundred-thousandths; .000006 for six millionths, etc. When a whole number, previously mentioned, and decimals are written together as 47.328, it is called a mixed number.
The only distinction between reading whole numbers and decimals is made by adding this to the ending of decimals, and the denomination of the right-hand figure must be expressed to give the proper value to decimal parts. For instance, .12, is twelve hundredths; .007, is seven thousandths; .062, is sixty-two thousandths; .201, is two hundred one thousandths; .5562, is five thousand five hundred sixty-two ten-thousandths; .24371, is twenty-four thousand three hundred seventy-one hundred-thousandths; .893254, is eight hundred ninety-three thousand two hundred fifty-four millionths, etc. Remember that in decimals the first figure stands for, tenths; the second, hundredths; the third, thousandths; the fourth, ten-thousandths; the fifth, hundred-thousandths; the sixth, millionths, and that in reading decimals we add the denomination of the right-hand figure. When reading a mixed number the word “and” is used, and then only, to indicate the decimal point. Thus 45.304 should be read forty-five AND three hundred four thousandths.
Addition and subtraction of decimals differ from similar operations of whole numbers only in the placing of the figures. In whole numbers units come under units, tens under tens, etc. To illustrate:
What is the sum of 260, 4398, 305, 2, 29?
The figures are placed thus:
| 260 |
| 4,398 |
| 305 |
| 2 |
| 29 |
| ——— |
| 4,994 |
Now let us take the same figures expressed decimally: .260, .4398, .305, .2, .29.
| .260 |
| .4398 |
| .305 |
| .2 |
| .29 |
| ——— |
| 1.4948 |
In subtraction of whole numbers or decimals the figures are placed as in addition.
Examples—Subtract .204 from .4723.
| .4723 |
| .204 |
| ——– |
| .2683 |
Subtract 5.346 from .937.
| 5.346 |
| .937 |
| ——– |
| 4.409 |
Subtract .753 from 18. (Note that the point or period is placed to the left of “753” indicating decimals, but in connection with the number “18,” a dot is placed to the right as a mark of punctuation merely, thus showing that “18” is a whole number.)
Now from the whole number “18,” which is the minuend because it is the number to be subtracted from, we are to subtract .753, and it is done in this way:
| Minuend | 18.000 |
| Subtrahend | .753 |
| ——— | |
| 17.247 |
The three ciphers are added to the minuend to correspond to the decimal places in the subtrahend. It is not necessary to put the ciphers down, but beginners are apt to get confused if there is nothing there to correspond to the decimals below. Annex as many ciphers to the minuend as there are decimals in the subtrahend, and place in the remainder a decimal point under those of the numbers subtracted.
Multiplication of decimals differs somewhat from the previous operations mentioned for the reason that we do not necessarily place the decimal points directly under each other. The right-hand figure of the multiplier usually goes under the right-hand figure of the multiplicand and the problem is then worked out as in multiplying whole numbers. When the product is obtained we point off as many decimal places in it as there are in both the multiplier and the multiplicand.
Let us take as an example: Multiply 2.648 by 2.35
| Multiplicand | 2.648 |
| Multiplier | 2.35 |
| ———– | |
| 13240 | |
| 7944 | |
| 5296 | |
| ———– | |
| Product | 6.22280 |
It will be seen that there are three decimals in the multiplicand and 2 decimals in the multiplier, hence we point off five decimals in the product.
In the operation of division of decimals the decimal point is not considered until the result is obtained. If the number of decimal places in the dividend is less than the number of decimal places in the divisor ciphers must be annexed or added to make up the deficiency, and the decimal point is then suppressed, thus reducing the operation to the division of two whole numbers. If there is no remainder, the quotient is a whole number, if there is a remainder, add a cipher to the right of it and place a decimal point to the right of the quotient obtained, then continue the division as far as desirable by adding ciphers to the right of the successive remainders, for each of which a new decimal will be obtained in the quotient.
Divide 460 by .5.
| .5) 460 (92 |
| 45 |
| — |
| 10 |
| 10 |
| — |
| 0 |
Fractions are reduced to decimals by annexing ciphers to the numerator and then dividing by the denominator.
For instance—5/8 equals what decimal?
| 8) 5.000 (.625 = 5/8 |
| 4 8 |
| — |
| .20 |
| 16 |
| — |
| .40 |
| 40 |
Lessons by Prof. Jean P. Genthon, C.E., Member Society of Municipal Engineers and Author of “The Assistant Engineer,” “The Chief’s” Text Book on Civil Engineering.
In solving problems the process should be not merely indicated, but all the figures necessary in solving each problem should be given in full. The answers to each problem should be indicated by writing “Ans.” after it.
Arithmetic is the science of numbers.
A Number is the result of the comparison (also called measurement) of a magnitude or quantity with another magnitude or quantity of the same kind supposed to be known.
A Concrete Number is one the nature of the unit of which is known.
Denominate Number.—A concrete number the standard of which is fixed by law or established by long usage.
An Abstract Number is one of which the nature of the unit is unknown.
How to Read Numbers.—The right way to read 101,274, etc., is one hundred one, two hundred seventy-four, etc.
The Decimal Point.—A period, called decimal point, is placed in a mixed number between the integral part and the decimal portion which follows. It should never be omitted.
Roman Numbers.—I stands for 1, V for 5, X for 10, L for 50, C for 100, D for 500 and M for 1,000.
Abbreviations.—A smaller unit, written to the left of a greater one, is subtracted from the latter, as: IV = 4 (IV is marked IIII on clock and watch dials); IX = 9; XC = 90; CD = 400, etc. Sometimes a Roman number is surmounted by a dash or vinculum; it then expresses thousands, as IX = 9,000.