subtract the numerators and express the result What is the minimum droop to which the aileron as the numerator of a fraction whose denominator can be rigged?
is the common denominator. When the denomi- FIRST: Reduce the fractions to a common fractions to a common denominator before pro- nators are unlike, it is necessary to first reduce the denominator.
ceeding with the addition or subtraction process. ] = 4Ở 35 5 - ≤ 10 8
EXAMPLES NEXT: Subtract the numerators, and express
1. A certain switch installation requires g-inch the result as in the above examples.
plunger travel before switch actuation occurs. If š-inch travel is required after actuation, what will be the total plunger travel? 35 10 - 40 - 40 8 27 FIRST: Add the numerators.
5+1=6 NEXT: Express the result as the numerator of a fraction whose denominator is the common denominator.
8† 8
2. The total travel of a jackscrew is 18 of an inch. If the travel in one direction from the neutral position is 76 of an inch, what is the travel in the opposite direction?
FIRST: Subtract the numerators.
13 - 7 = 6 NEXT: Express the result as the numerator of a fraction whose denominator is the common denominator.
1б-}-
3. Find the outside diameter of a section of tubing that has a l-inch inside diameter and a combined wall thickness of & inch.
FIRST: Reduce the fractions to a common denominator.
2 5 5 8 8 NEXT: Add the numerators, and express the result as the numerator of a fraction whose denominator is the common denominator.
2 5
4. The tolerance for rigging the aileron droop of an airplane is ⅜ inch plus or minus ⅖ inch.
Finding the Least Common Denominator When the denominators of fractions to be added or subtracted are such that a common denominator cannot be determined readily, the LCD (least common denominator) can be found by the continued division method.
To find the LCD of a group of fractions, write the denominators in a horizontal row. Next, divide the denominators in this row by the smallest integer that will exactly divide two or more of the denominators. Bring down to a new row all the quotients and numbers that were not divisible.
Continue this process until there are no two numbers in the resulting row that are divisible by any integer other than one. Multiply together all the divisors and the remaining terms in the last row to obtain the least common denominator.
EXAMPLE What is the LCD for 3, 18, 36, 13?
FIRST: Write the denominators in a horizontal row and divide this row by the smallest integer that will exactly divide two or more of the numbers.
2 8 20 36 45 4 10 18 45 NEXT: Continue this process until there are no two numbers in the resulting row that are divisible by any integer other 2 8 20 36 45 2 10 18 45 2 9 45 3 2 5 15 5 5 1 3
THEN: Multiply together all the divisors and Also, the 18 and 3 are both exaetly divisible remaining terms in the last row to by 3.
obtain the LCD. 6 LCD = 2X2 X 3X3 X 5X2 = 360 18 Multiplication of Fractions 2 1 The product of two or more fractions is obtained The resulting 6 in the numerator and the 2 in ator of the product and by multiplying the denomi- by multiplying the numerators to form the numer- the denominator are both divisible by 2.
nators to form the denominator of the product.
The resulting fraction is then reduced to its lowest 1 terms. A common denominator need not be found 3×1 3 for this operation, as the new denominator in most 1×1 = 3 cases will be different from that of all the original 2 1
EXAMPLE What is the product of 3 X 22 X ½?
FIRST: Multiply the numerators together.
3 × 12 × 1 = 36 NEXT: Multiply the denominators together.
5 × 22 × 2 = 220 THEN: Reduce the resulting fraction to its lowest terms.
36 9 220 = 55 Cancellation Cancellation is a technique of dividing out or cancelling all common factors that exist between numerators and denominators. 'This aids in locating the ultimate product by eliminating much of the burdensome multiplication.
EXAMPLE What is the product of To X 3?
The product could be found by multiplying 18 X 5 and 10 × 3, then dividing the product of the numerators by the product of the denominators. However, a much easier method of solution is by cancellation. It is apparent that the 10 in the denominator and the 5 in the numerator can both be divided an exact number of times by 5.
18 Iø X = 2 The fraction is thus reduced to its lowest terms, and the final multiplication and division steps are performed with ease when compared with the task of multiplying and dividing the larger fractions.
Division of Common Fractions The division of common fractions is accomplished most conveniently by converting the problem into a multiplication of two common fractions.
To divide one fraction by another fraction, invert the divisor fraction and multiply the numerators together and the denominators together. This is known as the inverted divisor method.
Always keep in mind the order in which the fractions are written. It is important in division that the operations be performed in the order indicated. Also, remember that it is always the divisor that is inverted, never the dividend.
MIXED NUMBERS Mixed numbers can be added, subtracted, multiplied, or divided by changing them to improper fractions and proceeding as when performing the operations with other fractions.
EXAMPLE A piece of tubing 616 inches long is cut from a piece 24½ inches long. Allowing is inch for the cut, what is the length of the remaining piece?
FIRST: Reduce the fractional parts to like fractions and complete the subtraction process.
1 - 16 3 8 16-16-16 16 - I6 = 4
NEXT: Subtract the integer parts.
24 - 6 = 18 THEN: Combine the results obtained in each step.
18 + = 18- inches.
4 4
DECIMALS Decimals are fractions whose denominators are 10 or some multiple of 10, such as 100, 1,000, 10,000, etc. They are indicated by writing one or more digits to the right of a reference mark called a decimal point. Thus:
10 = 6, both read six tenths.
6 = .06, both read six hundredths.
100 1,000 = .006, both read six thousandths.
When writing a decimal, any number of zeros may be written at the right end without changing the value of the decimal. This may be illustrated in the following manner:
5 = 10 5 =≥50 = 100 = 500 = 1,000 = 2 A decimal fraction that is written where there is no whole number as .6, .06, etc., is called a pure decimal. When a whole number and a decimal fraction are written together as 3.6, 12.2, 131.12, etc., the number is known as a mixed decimal.
2.34 Ohms
37.5 Ohms .09 Ohms FIGURE 1-1. A series circuit.
Addition of Decimals When computing decimals, the rule of likeness requires that we add or subtract only like denominations. This rule was discussed previously under addition and subtraction of whole numbers.
To add or subtract decimal expressions, arrange the decimals so that the decimal points align vertically, and add or subtract as with integers.
Place the decimal point in the result directly below the decimal points in the addends or minuend and subtrahend.
EXAMPLES The total resistance of series circuit (figure 1-1) is equal to the sum of the individual resistances. What is the total resistance for the diagram shown in this example?
FIRST: Arrange the decimals in a vertical column so that the decimal points are in alinement.
2.34
37.5
- 115 NEXT: Complete the addition following the technique used in adding whole numbers. Place the decimal point in the result directly below the other decimal points.
2.31
37.5
- 09
39.93 ohms Subtraction of Decimals A series circuit containing two resistors has a total resistance of 37.27 ohms. One of the resistors has a value of 14.88 ohms. What is the value of the remaining resistor?
FIRST: Arrange the decimals in a vertical column so that the decimal points are in alinement.
37.27
- 14.88 NEXT: Perform the subtraction process using the procedure for subtracting whole numbers. Place the decimal point in the result directly below the other decimal points.
37.27 -14.88
22.39 ohms 5
Multiplication of Decimals The multiplication of a decimal by another decimal will always produce an answer smaller than either of the two numbers. When a decimal is multiplied by a whole number or by a mixed decimal, the answer will lie between the two When multiplying a decimal fraction by an integer or another decimal, establishing the position of the decimal point in the product causes the greatest amount of difficulty.
To multiply decimals, ignore the decimal points and multiply the terms as though they were whole numbers. To locate the decimal point in the product, begin at the right of the product and point off toward the left the number of decimal places that will equal the sum of the decimal places in the quantities multiplied.
EXAMPLE Using the formula, Watts = Amperes X Voltage, what is the wattage of an electric heater that uses 9.45 amperes from a 120-volt source?
FIRST: Arrange the terms and multiply. Ignore the decimal point.
9.45 ×120 000 1890 945 113400 NEXT: Locate the decimal point. Begin at the right of the product and point off toward the left the number of places that will equal the sum of the decimal places in the quantities multiplied.
In some problems the number of digits in the product will be less than the sum of the decimal places in the quantities multiplied. Where this occurs, merely add zeros to the left of the product until the number of digits equals the sum of the decimal places in the quantities multiplied.
EXAMPLE Multiply 218 by 203.
FIRST: Arrange the terms and multiply, ignoring the decimal point.
.218 .203 654 4360 44254 NEXT: Locate the decimal point. Add a zero to the left of the product so that the number of places will equal the sum of the decimal places in the quantities mult plied.
.218 X .203 654 4360 . 044254 Division of Decimals When one or both of the terms of a division problem involve decimal expressions, the quotient is found by converting the problem to one involving a whole number.
Two facts relating to division of decimals that must be borne in mind are: (1) When the dividend and divisor are multiplied by the same number, the quotient remains unchanged; and (2) if the divisor is a whole number, the decimal place in the quotient will align vertically with the decimal in the dividend when the problem is expressed in long division form.
To divide decimal expressions, count off to the right of the decimal point in the dividend the same number of places that are located to the right of the decimal point in the divisor. Insert a caret (A) to the right of the last digit counted.
If the number of decimal places in the dividend is less than the number of decimal places in the divisor, add zeros to the dividend, remembering that there must be at least as many decimal places in the dividend as in the divisor. Divide the terms, disregarding the decimal points entirely. Place the decimal point in the quotient so that it alines vertically with the caret mark in the dividend.
EXAMPLE The wing area of a certain airplane is 245 square feet; its span is 40.33 feet. What is the mean chord of its wings?
FIRST: Arrange the terms as in long division and move the decimal point to the right, adding zeros as necessary, and insert a caret.
10.33,/245.00.
NEXT: Divide the terms, disregarding the decimal points entirely. Add adaccuracy.
6 07
40.3311245.00.00 241 98 3 020 0 000 3 0200 2 8221 1979 THEN: Place the decimal point in the quotient so that it alines vertically with the caret mark in the dividend.
6.07 feet
40.33,21500,00 24198 3020 0000 30200 28221 1979 Rounding Off Decimals There is a general tendency to think of all numbers as being precise. Actually the whole realm of measurement involves numbers that are only approximations of precise numbers. For example, measurements of length, area, and volume are at best approximations. The degree of accuracy of these measurements depends on the refinement of the measuring instruments.
Occasionally it is necessary to round a number to some value that is practical to use. For example, a measurement is computed to be 29.4948 inches.
It is impractical, if not impossible, to measure this accurately with a steel rule which is accurate only to s1 of an inch.
To use this measurement, we can use the process of "rounding." A decimal expression is "rounded offi by retaining the digits for a certain number of places and discarding the rest. The retained number is an approximation of the computed or exact number. The degree of accuracy desired determines the number of digits to be retained.
When the digit immediately to the right of the last retained digit is a 5, or greater than 5, increase the last retained digit by 1. When the digit immediately to the right of the last retained digit is less than 5, leave the last retained digit unchanged.
EXAMPLE Round 29.4948 to the nearest tenth.
FIRST: Determine the number of digits to retain. In this case one-tenths being the first place to the right of the decimal point.
29.4948 NEXT: Change the value of the last retained digit, if required. In this case, since 9 is greater than 5, the final decimal is expressed thus:
29.4948 becomes 29.5 inches.
Converting Decimals to Common Fractions To change a decimal fraction to a common fraction, count the number of digits to the right of the decimal point. Express the number as the numerator of a fraction whose denominator is l followed by the number of zeros that will equal the number of digits to the right of the decimal
EXAMPLE Express .375 as a common fraction.
FIRST: Count the number of digits to the right of the decimal point.
375 NEXT: Express the number as the numerator of a fraction whose denominator is 1 followed by the number of zeros that will equal the number of digits to the right of the decimal point.
375 375 = 1000 7
Converting Common Fractions to Decimals To convert a common fraction, whether proper or improper, to a decimal, divide the numerator by the denominator. Add zeros to the right to permit carrying the quotient to the desired accu- 5 16 FIGURE 1-2. Locating the hole center.
Many times a dimension appearing in a maintenance manual or on a blueprint is expressed in decimal fractions. In order to use the dimension, it must be converted to some equivalent approximation applicable to the available measuring device. From the mechanic's standpoint, the steel rule will be the device most frequently used.
To change a decimal to the nearest equivalent fraction having a desired denominator, multiply the decimal by the desired denominator. The result will be the numerator of the desired fraction.
EXAMPLE When accurate holes of uniform diameter are required, they are first drilled an inch undersize and reamed to the desired diameter. What size drill would be used before reaming a hole to .763?
FIRST: Multiply the decimal by the desired denominator of 61.
3052 1578 48832 NEXT: Round the product to a whole number and express it as the numerator of the desired denominator.
18.332 - 19 THEN: To determine the drill size, subtract ay inch from the finished hole size.
19 48 61-71-61 = pinch dru.
EXAMPLE Find the distance the center of the hole (figure I-2) is from the plate edges when the center of the hole is in the center of the plate. Express the length and width of the plate in decimal forms, and then divide each by 2. Express the final result to the nearest 32nd.
FIRST: Change the mixed numbers to improper fractions.
16 - 16° 5 = 29 NEXT: Convert the improper fractions to decimal expressions.
87 29
- = 5.4375; = 3.625 16 8 THEN: Divide the decimal expressions by 2 to find the center of the plate.
5.4375 = 2.7188; 3.625
- = 1.813 2 2 FINALLY: Express the final results to the nearest 32nd.
32' 1.813 = 126 23
2.7188 = 2 32
PERCENTAGE There are many problems that arise every day involving the percent expression. The greatest number of percentage problems involve some kind of comparison of a part to the whole. Such comparisons become percentage problems when the ratio fraction is expressed as a percent.
A fraction having the specific power of 100 for the denominator is given the name percent. When writing these fractions, the percent symbol (%) is substituted for the denominator. Any common fraction or decimal can be expressed as a percent.
The fraction 4 can thus be expressed as .20 or as 20 percent or simply as 20%. Note that the percent is the same as the decimal fraction except that the decimal point has been moved two places to the right and deleted after "percent" or the symbol "%" has been added.
Expressing a Decimal as a Percent decimal point two places to the right (add a zero if necessary) and affix the percent symbol.
EXAMPLE Express .90 as a percent.
FIRST: Move the decimal point two places to the right.
90.
NEXT: Affix the percent symbol to the right after dropping the decimal point.
90% Expressing a Percent as a Decimal Sometimes it may be necessary to express a percent as a decimal. Keeping in mind that a percent is simply a decimal with the decimal point moved two places to the right, all that is necessary to express a percent as a decimal is to move the decimal point two places to the left.
Expressing a Common Fraction as a Percent FIRST: Write the part number, 10.75, as the numerator of a fraction whose denominator is the whole number, 12.
10.75 12 NEXT: Convert the fraction to its decimal equivalent.
10.75 ÷ 12 = .8958 THEN: Express the decimal as a percent.
.8958 = 89.58% efficient.
Finding a Percent of a Given Number The technique used in determining a percent of a given number is based on the fundamental process of multiplication. It is necessary to state the desired percent as a decimal or common fraction and multiply the given number by the percent expressed as a decimal or other fraction.
The technique involved in expressing a common fraction as a percent is essentially the same as that for a decimal fraction. The one difference is the procedure necessary to convert the fraction to a decimal.
EXAMPLE Express & as a percent.
FIRST: Convert the fraction to a decimal.
= 5 ÷ 8 = .625 8 NEXT: Move the decimal point two places to the right and affix the percent symbol.
625 = 62.5% Finding What Percent One Number is of Another Determining what percent one number is of another is done by writing the part number as the numerator of a fraction and the whole number as the denominator of that fraction, and expressing this fraction as a percentage.
EXAMPLE The cruising speed of an airplane at an altitude of 7,500 feet is 290 knots. What is the cruising speed at 9,000 feet if it has increased 6 percent?
FIRST: State the desired percent as a decimal.
6% = .06 NEXT: Multiply the given number by the decimal expression.
290 X.06 = 17.40 THEN: Add the new product to the given number. This is the new cruising speed.
290 + 17.4 = 307.4 knots.
Finding a Number When a Percent of It is Known To determine a number when a percent of it is known, express the percent as a decimal and divide the known number by the decimal expression.
EXAMPLE EXAMPLE delivering 10.75 horsepower. What is the motor A motor rated as 12 horsepower is found to be i the motor total restine. Find the top rene.
efficiency expressed in percent? circuit.
FIRST: Express the percent as a decimal. THEN: Perform the indicated mathematical manipulations and reduce to lowest 52% = .52 terms.
NEXT: Divide the known number by the (800 × 7.2) 5760 decimal expression. R = 10,080 - 10,080 =→ 0 4:7 80 ÷ .52 = 153.8 ohms total resistance. What is the ratio in gallons?
FIRST: Express the ratio in fractional form.
RATIO 800 gal.
An important application of the common frac- R = 10,080 lb.
parison of one number to another number. Com- tion is that of ratio. A ratio represents the com- NEXT: Express both quantities in the same parison by the use of ratios has widespread appli- unit (gallons).
cation in the field of aviation. A ratio is used to R = 800 gal.
express the comparison of the volume of a cylinder 10,080 when the piston is at bottom center to the volume 7.2 of a cylinder when the piston is at top center.
This is referred to as the compression ratio. The THEN: Perform the indicated mathematical aspect ratio of an aircraft wing is a comparison manipulations and reduce to lowest of the wing span to the wing chord. The relation- terms.
ship of maximum speed, wing area, wing span, 800 800 loaded weight, and horsepower of different makes R = 10,080 - 1,400 = → or +7 and models of aircraft may be compared through the use of ratios. 7.2 A ratio is the quotient of one number divided by another number, expressed in like terms. A ratio, therefore, is the fractional part that one number is of another. A ratio may be expressed as a fraction, or it may be written using the colon (:) as the symbol for expressing ratio; thus the ratio ½ can be written 7:8.
2. If the cruising speed of an airplane is 200 knots and its maximum speed is 250 knots, what is the ratio of cruising speed to maximum speed?
FIRST: Express the cruising speed as the numerator of a fraction whose denominator is the maximum speed.
Finding the Ratio of Two Quantitites To find a ratio, the first term is divided by the second term. Both quantities of both terms must be expressed in the same units, and reduced to their lowest terms.
EXAMPLES
1. What is the weight ratio of a fuel load of 800 gallons to one of 10,080 pounds? Assume that the fuel weighs 7.2 pounds per gallon.
FIRST: Express the fuel load in gallons as the numerator of a fraction whose denominator is the fuel load in pounds.
R = 10,080 lb. 800 gal.
NEXT: Express both quantities in the same unit (pounds).
R = (800 × 7.2) 1b. 10,080 lb.
R = 250 200 NEXT: Reduce the resulting fraction to its lowest terms.
R = 250 200 THEN: Express the result as a ratio of one.
4 R = 3 or 8:1 (Read 8/10ths to one) Finding the Quantity of the First Term Now consider the situation when the ratio and the quantity that corresponds to the second term are given, and it is required to find the quantity that corresponds to the first term. To solve this type problem, multiply the term that corresponds to the second term by the fraction that represents 10
EXAMPLE The given ratio is 5/7 and the quantity that corresponds to the second term is 35. Find the quantity that corresponds to the first term.
FIRST: Express the problem as the product of the second term times the ratio.
35 × 5/7 = NEXT: Perform the indicated operation.
38 × 5/7 - 25 The first term is 25. The proof of this can be demonstrated by showing that the ratio of 25 to 35 is 5:7, reduced to lowest terms.
25/35 = 5/7 Finding the Quantity of the Second Term To solve a problem of this type, the ratio of the two quantities and the quantity that corresponds to the first term must be known. The solution is obtained by dividing the known number by the fraction that represents the ratio.
EXAMPLE The ratio of two quantities is 2/3; the quantity that corresponds to the first term is 100. Find the quantity that corresponds to the second term.
FIRST: Express the problem as the quotient of the first term divided by the ratio.
100 ÷ 2/3 = NEXT: Perform the indicated operation.
100 ÷ 2/3 = 50 100 × 3/2 = 150 The second term is 150. Again, this can be proved by expressing 100 as a ratio of 150.
100/150 = 2/3
PROPORTION A proportion is a statement of equality between two or more ratios. Thus, = g: or 3:4 = 6:8.
This is read 3 is to 4 as 6 is to 8. The first and last terms of the proportion are called the extremes.
The second and third terms are called the means.
In any proportion, the product of the extremes is equal to the product of the means. In the proportion 2:3 = 4:6 the product of the extremes, 2 X 6, is 12; the product of the means, 3 X 4, also is 12. An inspection of any proportion will show this to be true. This rule simplifies the solution of many practical problems.
An airplane flying a distance of 300 miles used 24 gallons of gasoline. How many gallons will it need to travel 750 miles?
300:750 = 24:x
(300) (x) = (750) (24) 300x = 18,000 × = 60 Sixty gallons of gasoline will be required to travel a distance of 750 miles.
POSITIVE AND NEGATIVE NUMBERS Positive and negative numbers are numbers that have directional value from a given starting point or from zero. Numbers above or to one side, usually right, of zero are designated as positive (+); those below or to the opposite side, usually left, of zero are designated as negative (-).
Figure 1-3 is representative of signed numbers on a horizontal scale.
-5 3-21 +1+2+3+4+5 FIGURE 1-3. A scale of signed numbers.
The sum of positive numbers is positive.
The sum of negative numbers is negative.
Addition To add a positive and a negative number, find the difference in their actual values and give this difference the sign (tor-) of the larger number.
EXAMPLE The weight of an aircraft is 2,000 pounds. A radio rack weighing 3 pounds and a transceiver weighing 10 pounds are removed from the aircraft.
What is the new weight? For weight and balance purposes, all weight removed from an aircraft is given a minus sign, and all weight added is given a plus sign.
FIRST: Add the values for the removed weights.
10+3=13 NEXT: Prefix the sign for removed weights.
- 13 FIRST: Change the sign of the number to be subtracted. +20 becomes - 20.
NEXT: Combine the two terms, following the rule for adding like signs.
(-6) + (-20) = -26 degrees.
Multiplication The product of two positive numbers is positive (+). The product of two negative numbers is positive (+). The product of a positive and a negative number is negative (-).
EXAMPLES 3X6=18 - 3X6= - 18 -3X -6=18 3X -6= -18 THEN: Add the sum of the removed weights to the total weight, following the rule for unlike signs.
+ 2000-13 = + 1987 pounds.
Division The quotient of two positive numbers is positive.
The quotient of two negative numbers is positive.
The quotient of a positive and negative number is negative.
EXAMPLES Subtraction To subtract positive and negative numbers, change the sign of the subtrahend (the number to be subtracted from another) and proceed as in addition.
EXAMPLE What is the temperature difference between a temperature reading of +20 at 5,000 feet and a reading of -6 at 25,000 feet? Follow the rule, "a change in temperature is equal to the first reading, subtracted from the second reading."
6÷3=2 -6÷3=-2 -6÷-3=2 6÷-3= -2
POWERS AND ROOTS Power When one number, the base, is used as a factor two or more times, the result is a power of the base. A positive integral exponent, written as a small number just to the right and slightly above the base number, indicates the number of times the base is used as a factor. Thus, 4 squared, or 12
4' means 4 X 4, which is 16. The 4 is the base, the 2 is the exponent, and the 16 is the power.
Roots A root of a number is one of two or more equal numbers that, when multiplied together, will produce the number. Such a number is called an equal factor. Thus, two equal factors that will produce 9 when multiplied together are 3 and 3.
Therefore, the square root of 9 equals 3. This may be written V9 = 3. The symbol V" is called a radical sign. Another method of indicating the square root of a number is to use a fractional exponent such as 912 = 3. If the root to be taken is other than a square root, it may be shown in a similar manner; that is, the cube root of 9 may be written 91. For example, the cube root of 8 equals 2 and may be written 18 = 2, or 81/3 = 2;
the fourth root of 256 equals 4 and may be written V256 = 4, or 256%4 = 4.
Computation of Square Root It is comparatively easy to determine the square root of such numbers as 4, 9, 16, and 144. The numbers are the perfect squares of small numbers.
Unfortunately, all numbers are not perfect squares; neither are they small. The square of a number is the product of that number multiplied by itself. Extracting the square root of a number is the reverse process of squaring a number, and is essentially a special division process. A description of this process follows and is presented in example form.
EXAMPLE Find the square root of 213.16 FIRST: Starting at the decimal point, and marking off in both directions from the decimal point, separate the number into periods of two figures each.
The last period at the left end need not have two figures; all others must have two figures. A zero may be added to the right end so that the period will have two figures.
V213.16 NEXT: Select the largest number that can be squared in the first period. Place the selected number above the radical sign, and place the square of this 13 number under the first period and subtract.
V213.16 THEN: Bring down the next pair.
(1) Multiply the root by 2 and place the product to the left of the remainder as the trial divisor.
(2) Determine the number of times the trial divisor will go into that portion of the remainder that is one digit more than the trial divisor. Write this number to the right of the digit in the trial divisor to form the final divisor and also to the right of the digit in the root.
(3) Multiply this number times the completed divisor. If the resulting product is larger than the remainder, reduce the number by one, both in the root and in the final divisor, and repeat the multiplication process.
(4) Subtract the product formed from the remainder and bring down the next pair to form a new remainder.
(5) To complete the solution of extracting the square root, simply repeat the procedure set forth in this step for each period of numbers remaining. It is unnecessary to carry the root beyond the number of digits possessed by the original number.
14. 6 V213.16 24 113 96 286 17 16 17 16
Two will divide into 11, 5 times.
However, 5 X 25 is greater than 113, so the 5 must be reduced to a 4.
The decimal is placed in the root so that the number of digits in the whole number portion of the root is equal to the sum of the periods, or pairs, in the whole number portion of the number from which the root was extracted.
(3) The value of a number does not change when it is both multiplied and divided by the same factor (5 X 10 ÷ 10 = 5).
Moving the decimal point of a number to the left is the same as dividing the number by 10 for each place the decimal point moves. Conversely, moving the decimal point to the right is the same as multiplying the number by 10 for each place the decimal point moves.
Powers of Ten
POWER OF
EXPANSION VALUE The difficulty of performing mathematical problems with very large (or very small) numbers and Positive Exponent 10° 10 × 10 × 10 × 10 × 10 x10 1,000,000 the counting and writing of many decimal places 105 10 × 10 × 10 × 10 × 10 100,000 10° 10 × 10 × 10 × 10 10,000 are both an annoyance and a source of error. 10° 10 × 20 × 10 1,000 The problems of representation and calculation 10ª 10 × 10 100 are simplified by the use of "powers of ten." (See 10' 10° 10 10 figure 1-1.). This system, sometimes referred to as The velocity of light, 30, 000, 000, 000 centimeters per second, simplifies to 3 × 10" centi- "Engineer's Shorthand, requires an understand« meters per second.
ing of the principles of the exponent. These are Negative Exponent summarized as follows: 10~ = 1a ł0=01 10°=10 10 × 10 ł00 = 00:
10 = 10° 10 × 10 × 10 1,000 = 0.001
(1) The positive exponent (or power) of a number is a shorthand method of indi- 10% = To 10 × 10 × 10 × 10 10.000 = 0.0001 cating how many times the number is 10→ = 10 10 × 10 × 10 x 10 × 10 100.000 = 0.00001 multiplied by itself. For example, 23 (read as 2-cubed or 2 to the third 10 ° = 10° 10 × 10 × 10 × 10 × 10 x 10 1,000,000 = 0,000001 power) means 2 is to be multiplied by The mass of an clectron, 0.000, 000, 000, 000, 000, 000, 000, 000, 000, 911 gram, become itself 3 times: 2 X 2 X 2 = 8. A number with a negative exponent may be defined as its inverse or reciprocal (1 divided by the number) with the same exponent made positive. For example, 2-3 (read as 2 to the minus 3 power) is the same as FIGURE 1-4. Powers of ten and their equivalents.
The procedure for use of powers of ten may be summarized as follows:
(2) 2x2X2-8
(2) Auy number, except zero, to the zero power is equal to 1. When a number is written without an exponent, the value of the exponent is 1. When an exponent has no sign (+ or -) preceding it, the exponent is positive.
(1). Move the decimal point to the place desired. Count the number of places the decimal point is moved.
(2) Multiply the altered number by 10 to a power equal to the number of places the decimal point was moved.
(3) The exponent of 10 is negative if the decimal point is moved to the right, and it is positive if the decimal point is moved to the left. An aid for remembering the sign to be used is: L, 14