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When adding several whole numbers, such as 4,314, 122, 93,132, and 10, align them into columns according to place value and then add. 4, 122 93,
- 10 97,578 This is the sum of the four whole numbers.
Subtraction of Whole Numbers Subtraction is the process in which the value of one number is taken from the value of another. The answer is called the difference. When subtracting two whole numbers, such as 3,461 from 97,564, align them into columns according to place value and then subtract.
97, –3, 94,103 This is the difference of the two whole numbers.
Multiplication of Whole Numbers Multiplication is the process of repeated addition. For example, 4 × 3 is the same as 4 + 4 + 4. The result is called the product.
Example: How many hydraulic system filters are in the supply room if there are 35 cartons and each carton contains 18 filters?
Place Value
Ten ThousandsThousandsHundreds
TensOnes
3 5 2 6 9 1 2 7 4 9
35 shown as 269 shown as 12,749 shown as
Mathematics in Aviation
Maintenance
Mathematics is woven into many areas of everyday life. Performing mathematical calculations with success requires an understanding of the correct methods and procedures, and practice and review of these principles. Mathematics may be thought of as a set of tools. The aviation mechanic will need these tools to success- fully complete the maintenance, repair, installation, or certification of aircraft equipment.
Many examples of using mathematical principles by the aviation mechanic are available. Tolerances in turbine engine components are critical, making it necessary to measure within a ten-thousandth of an inch. Because of these close tolerances, it is important that the aviation mechanic be able to make accurate measurements and mathematical calculations. An avia- tion mechanic working on aircraft fuel systems will also use mathematical principles to calculate volumes and capacities of fuel tanks. The use of fractions and surface area calculations are required to perform sheet metal repair on aircraft structures.
Whole Numbers
Whole numbers are the numbers 0, 1, 2, 3, 4, 5, and so on.
Addition of Whole Numbers
Addition is the process in which the value of one number is added to the value of another. The result is called the sum. When working with whole numbers, it is important to understand the principle of the place value. The place value in a whole number is the value of the position of the digit within the number. For example, in the number 512, the 5 is in the hundreds column, the 1 is in the tens column, and the 2 is in the ones column. The place values of three whole numbers are shown in Figure 1-1.
Figure 1-1. Example of place values of whole numbers.
× 35
Therefore, there are 630 filters in the supply room.
Division of Whole Numbers
Division is the process of finding how many times one number (called the divisor) is contained in another number (called the dividend). The result is the quotient, and any amount left over is called the remainder.
Example: 218 landing gear bolts need to be divided between 7 aircraft. How many bolts will each aircraft receive?
The solution is 31 bolts per aircraft with a remainder of 1 bolt left over.
Fractions
A fraction is a number written in the form N⁄D where N is called the numerator and D is called the denominator. The fraction bar between the numerator and denomina- tor shows that division is taking place.
Some examples of fractions are:
The denominator of a fraction cannot be a zero. For example, the fraction 2 ⁄ 0 is not allowed. An improper fraction is a fraction in which the numerator is equal to or larger than the denominator. For example, 4 ⁄ 4 or (^15) ⁄ 8 are examples of improper fractions.
Finding the Least Common Denominator
To add or subtract fractions, they must have a common denominator. In math, the least common denominator (LCD) is commonly used. One way to find the LCD is to list the multiples of each denominator and then choose the smallest one that they have in common.
Example: Add 1 ⁄ 5 + 1 ⁄ 10 by finding the least common denominator.
Multiples of 5 are: 5, 10, 15, 20, 25, and on. Multiples of 10 are: 10, 20, 30, 40, and on. Notice that 10, 20, and 30 are in both lists, but 10 is the smallest or least common denominator (LCD). The advantage of find- ing the LCD is that the final answer is more likely to be in lowest terms.
A common denominator can also be found for any group of fractions by multiplying all of the denomina- tors together. This number will not always be the LCD, but it can still be used to add or subtract fractions.
Example: Add 2 ⁄ 3 + 3 ⁄ 5 + 4 ⁄ 7 by finding a common denominator.
A common denominator can be found by multiplying the denominators 3 × 5 × 7 to get 105.
Addition of Fractions In order to add fractions, the denominators must be the same number. This is referred to as having “common denominators.”
Example: Add 1 ⁄ 7 to 3 ⁄ 7
If the fractions do not have the same denominator, then one or all of the denominators must be changed so that every fraction has a common denominator.
Example: Find the total thickness of a panel made from (^3) ⁄ 32 -inch thick aluminum, which has a paint coating that is 1 ⁄64-inch thick. To add these fractions, determine a common denominator. The least common denominator for this example is 1, so only the first fraction must be changed since the denominator of the second fraction is already 64.
Therefore, 7 ⁄ 64 is the total thickness.
Subtraction of Fractions In order to subtract fractions, they must have a com- mon denominator.
Example: Subtract 2 ⁄ 17 from 10 ⁄ 17
7 64
6 + 1 64
3 32
1 64
3 × 2 32 × 2
1 64
6 64
1 64
divisor
quotient dividend
23 35 47 10570 10563 10560 193105 1 10588
2 18 ⁄ 16. (Because, 3 1 ⁄ 8 = 3 2 ⁄ 16 = 2 + 1 + 2 ⁄ 16 = 2 + 16 ⁄ 16
Therefore, the grip length of the bolt is 1^13 ⁄ 16 inches.
(Note: The value for the overall length of the bolt was given in the example, but it was not needed to solve the problem. This type of information is sometimes referred to as a “distracter” because it distracts from the information needed to solve the problem.)
The Decimal Number System
The Origin and Definition The number system that we use every day is called the decimal system. The prefix in the word decimal is a Latin root for the word “ten.” The decimal system probably had its origin in the fact that we have ten fingers (or digits). The decimal system has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. It is a base 10 system and has been in use for over 5,000 years. A decimal is a number with a decimal point. For example, 0.515, .10, and 462.625 are all decimal numbers. Like whole numbers, decimal numbers also have place value. The place values are based on powers of 10, as shown in Figure 1-4.
The fraction 6 ⁄ 16 is not in lowest terms because the numerator (6) and the denominator (16) have a com- mon factor of 2. To reduce 6 ⁄16, divide the numerator and the denominator by 2. The final reduced fraction is 3 ⁄ 8 as shown below.
Therefore, the travel in the opposite direction is 3 ⁄ 8 inch.
Mixed Numbers
A mixed number is a combination of a whole number and a fraction.
Addition of Mixed Numbers
To add mixed numbers, add the whole numbers together. Then add the fractions together by finding a common denominator. The final step is to add the sum of the whole numbers to the sum of the fractions for the final result.
Example: The cargo area behind the rear seat of a small airplane can handle solids that are 4 3 ⁄ 4 feet long. If the rear seats are removed, then 2 1 ⁄ 3 feet is added to the cargo area. What is the total length of the cargo area when the rear seats are removed?
Subtraction of Mixed Numbers
To subtract mixed numbers, find a common denomi- nator for the fractions. Subtract the fractions from each other (it may be necessary to borrow from the larger whole number when subtracting the fractions). Subtract the whole numbers from each other. The final step is to combine the final whole number with the final fraction.
Example: What is the length of the grip of the bolt shown in Figure 1-3? The overall length of the bolt is 3 1 ⁄ 2 inches, the shank length is 3 1 ⁄ 8 inches, and the threaded portion is 1 5 ⁄ 16 inches long. To find the grip, subtract the length of the threaded portion from the length of the shank.
3 1 ⁄ 8 inches – 1 5 ⁄ 16 inches = grip length
To subtract, start with the fractions. Borrowing will be necessary because 5 ⁄ 16 is larger than 1 ⁄ 8 (or 2 ⁄16). From the whole number 3, borrow 1, which is actually 16 ⁄16. After borrowing, the first mixed number will now be
1 (^712)
4 3 2 (4 + 2) + 6 4
1 3
3 4
1 3
13 (^612) 9 12
4 12
feet of cargo room.
Figure 1-3. Bolt dimensions.
6 ÷ 2
16 ÷ 2
Place Value
MillionsHundred ThousandsTen ThousandsThousandsHundredsTensOnesTenthsHundredthsThousandthsTen Thousandths 1 6 2 3 0 5 1 0 0 5 3 1 3 2 4
1,623,
Figure 1-4. Place values.
2.34 Ohms
37.5 Ohms (^) M
.09 Ohms
Addition of Decimal Numbers
To add decimal numbers, they must first be arranged so that the decimal points are aligned vertically and according to place value. That is, adding tenths with tenths, ones with ones, hundreds with hundreds, and so forth.
Example: Find the total resistance for the circuit dia- gram shown in Figure 1-5. The total resistance of a series circuit is equal to the sum of the individual resis- tances. To find the total resistance, RT, the individual resistances are added together.
RT = 2.34 + 37.5 +.
Arrange the resistance values in a vertical column so that the decimal points are aligned and then add.
+.
Therefore, the total resistance, RT = 39.93 ohms.
Subtraction of Decimal Numbers
To subtract decimal numbers, they must first be arranged so that the decimal points are aligned verti- cally and according to place value. That is, subtracting tenths from tenths, ones from ones, hundreds from hundreds, and so forth.
Example: A series circuit containing two resistors has a total resistance (RT ) of 37.272 ohms. One of the resistors (R1) has a value of 14.88 ohms. What is the value of the other resistor (R 2 )?
R 2 = RT – R 1 = 37.272 – 14.
Arrange the decimal numbers in a vertical column so that the decimal points are aligned and then subtract.
Therefore, the second resistor, R2 = 22.392 ohms.
Multiplication of Decimal Numbers
To multiply decimal numbers, vertical alignment of the decimal point is not required. Instead, align the numbers to the right in the same way as whole numbers are multiplied (with no regard to the decimal points or
place values) and then multiply. The last step is to place the decimal point in the correct place in the answer. To do this, “count” the number of decimal places in each of the numbers, add the total, and then “give” that number of decimal places to the result.
Example: To multiply 0.2 × 6.03, arrange the numbers vertically and align them to the right. Multiply the numbers, ignoring the decimal points for now.
(ignore the decimal points, for now)
After multiplying the numbers, count the total number of decimal places in both numbers. For this example, 6.03 has 2 decimal places and 0.2 has 1 decimal place. Together there are a total of 3 decimal places. The decimal point for the answer will be placed 3 decimal places from the right. Therefore, the answer is 1.206.
Example: Using the formula Watts = Amperes × Volt- age, what is the wattage of an electric drill that uses 9.45 amperes from a 120 volt source? Align the num- bers to the right and multiply.
After multiplying the numbers, count the total number of decimal places in both numbers. For this example, 9.45 has 2 decimal places and 120 has no decimal place. Together there are 2 decimal places. The decimal point for the answer will be placed 2 decimal places from
×
2 decimal places × 1 decimal place 3 decimal places
Figure 1-5. Circuit diagram.
Example:
Calculator tip: Numerator (top number) ÷ Denomina- tor (bottom number) = the decimal equivalent of the fraction.
Some fractions when converted to decimals produce a repeating decimal.
Example:
Other examples of repeating decimals: .212121… =.
.6666… = .7 or. .254254… =.
Decimal Equivalent Chart Figure 1-6 (on the next page) is a fraction to decimal to millimeter equivalency chart. Measurements starting at 1 ⁄ 64 inch and up to 23 inches have been converted to decimal numbers and to millimeters.
Ratio
A ratio is the comparison of two numbers or quantities. A ratio may be expressed in three ways: as a fraction, with a colon, or with the word “to.” For example, a gear ratio of 5:7 can be expressed as any of the following: (^5) ⁄ 7 or 5:7 or 5 to 7
to a fraction. An aviation mechanic frequently uses a steel rule that is calibrated in units of 1 ⁄ 64 of an inch. To change a decimal to the nearest equivalent common fraction, multiply the decimal by 64. The product of the decimal and 64 will be the numerator of the fraction and 64 will be the denominator. Reduce the fraction, if needed.
Example: The width of a hex head bolt is 0. inches. Convert the decimal 0.3123 to a common fraction to decide which socket would be the best fit for the bolt head. First, multiply the 0.3123 decimal by 64:
0.3123 × 64 = 19.
Next, round the product to the nearest whole number: 19.98722 ≈ 20.
Use this whole number (20) as the numerator and 64
as the denominator: 20 ⁄ 64.
Now, reduce 20 ⁄ 64 to 5 ⁄ 16.
Therefore, the correct socket would be the 5 ⁄ 16 inch socket ( 20 ⁄ 64 reduced).
Example: When accurate holes of uniform diameter are required for aircraft structures, they are first drilled approximately 1 ⁄ 64 inch undersized and then reamed to the final desired diameter. What size drill bit should be selected for the undersized hole if the final hole is reamed to a diameter of 0.763 inches? First, multiply the 0.763 decimal by 64.
0.763 × 64 = 48.
Next, round the product to the nearest whole number: 48.832 ≈ 49.
Use this number (49) as the numerator and 64 as the denominator: 49 ⁄ 64 is the closest fraction to the final reaming diameter of 0.763 inches. To determine the drill size for the initial undersized hole, subtract 1 ⁄ 64 inch from the finished hole size.
Therefore, a 3 ⁄4-inch drill bit should be used for the initial undersized holes.
Converting Fractions to Decimals
To convert any fraction to a decimal, simply divide the top number (numerator) by the bottom number (denominator). Every fraction will have an approxi- mate decimal equivalent.
= (^) 1 ÷ 2 = 2 1.0 Therefore, 2 =.
0
=3 ÷ 8 Therefore, 8 =.
3 =^ 1 ÷ 3=^ = .3 or. This decimal can be represented with bar, or can be rounded. (A bar indicates that the number(s) beneath it are repeated to infinity.)
fraction decimal mm fraction decimal mm fraction decimal mm
For example: Express the following percentages as decimal numbers:
90% =. 50% =. 5% =. 150% = 1.
Expressing a Fraction as a Percentage To express a fraction as a percentage, first change the fraction to a decimal number (by dividing the numera- tor by the denominator), and then convert the decimal number to a percentage as shown earlier.
Example: Express the fraction 5 ⁄ 8 as a percentage. (^5) = 5 ÷ 8 = 0.625 = 62.5% 8
Finding a Percentage of a Given Number This is the most common type of percentage calcula- tion. Here are two methods to solve percentage prob- lems: using algebra or using proportions. Each method is shown below to find a percent of a given number.
Example: In a shipment of 80 wingtip lights, 15% of the lights were defective. How many of the lights were defective?
Algebra Method: 15% of 80 lights = N (number of defective lights) 0.15 × 80 = N 12 = N
Therefore, 12 defective lights were in the shipment.
Proportion Method: N (^) = 80
To solve for N: N × 100 = 80 × 15 N × 100 = 1200 N = 1200 ÷ 100 N = 12 or N = (80 × 15) ÷ 100 N = 12
Finding What Percentage One Number Is of Another Example: A small engine rated at 12 horsepower is found to be delivering only 10.75 horsepower. What is the motor efficiency expressed as a percent?
Solving Proportions
Normally when solving a proportion, three quantities will be known, and the fourth will be unknown. To solve for the unknown, multiply the two numbers along the diagonal and then divide by the third number.
Example: Solve for X in the proportion given below.
65 80
X
First, multiply 65 × 100: 65 × 100 = 6500 Next, divide by 80: 6500 ÷ 80 = 81. Therefore, X = 81.25.
Example: An airplane flying a distance of 300 miles used 24 gallons of gasoline. How many gallons will it need to travel 750 miles?
The ratio here is: “miles to gallons;” therefore, the proportion is set up as:
300 24
Miles Gallons
G
Solve for G: (750 × 24) ÷ 300 = 60
Therefore, to fly 750 miles, 60 gallons of gasoline will be required.
Percentage
Percentage means “parts out of one hundred.” The percentage sign is “%”. Ninety percent is expressed as 90% (= 90 parts out of 100). The decimal 0.90 equals (^90) ⁄100, or 90 out of 100, or 90%.
Expressing a Decimal Number as a Percentage
To express a decimal number in percent, move the decimal point two places to the right (adding zeros if necessary) and then affix the percent symbol.
Example: Express the following decimal numbers as a percent:
.90 = 90% .5 = 50% 1.25 = 125% .335 = 33.5%
Expressing a Percentage as a Decimal Number
Sometimes it may be necessary to express a percent- age as a decimal number. To express a percentage as a decimal number, move the decimal point two places to the left and drop the % symbol.
Addition of Positive and Negative Numbers The sum (addition) of two positive numbers is positive. The sum (addition) of two negative numbers is nega- tive. The sum of a positive and a negative number can be positive or negative, depending on the values of the numbers. A good way to visualize a negative number is to think in terms of debt. If you are in debt by $ (or, −100) and you add $45 to your account, you are now only $55 in debt (or −55).
Therefore: −100 + 45 = −55.
Example: The weight of an aircraft is 2,000 pounds. A radio rack weighing 3 pounds and a transceiver weigh- ing 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.
2,000 + −3 + −10 = 2,000 + −13 = 1987 Therefore, the new weight is 1,987 pounds.
Subtraction of Positive and Negative Numbers To subtract positive and negative numbers, first change the “–” (subtraction symbol) to a “+” (addition sym- bol), and change the sign of the second number to its opposite (that is, change a positive number to a negative number or vice versa). Finally, add the two numbers together.
Example: The daytime temperature in the city of Den- ver was 6° below zero (−6°). An airplane is cruising at 15,000 feet above Denver. The temperature at 15, feet is 20° colder than in the city of Denver. What is the temperature at 15,000 feet?
Subtract 20 from −6: −6 – 20 = −6 + −20 = −
The temperature is −26°, or 26° below zero at 15, feet above the city.
Multiplication of Positive and Negative Numbers The product of two positive numbers is always positive. The product of two negative numbers is always posi- tive. The product of a positive and a negative number is always negative.
Algebra Method: N% of 12 rated horsepower = 10.75 actual horsepower N% × 12 = 10. N% = 10.75 ÷ 12 N% =. N = 89. Therefore, the motor efficiency is 89.58%.
Proportion Method:
10.75 = 12
N
To solve for N: N × 12 = 10.75 × 100 N × 12 = 1075 N = 1075 ÷ 12 N = 89. or N = (1075 × 100) ÷ 12 N = 89. Therefore, the motor efficiency is 89.58%.
Finding a Number When a Percentage of It Is Known
Example: Eighty ohms represents 52% of a micro- phone’s total resistance. Find the total resistance of this microphone.
Algebraic Method: 52% of N = 80 ohms 52% × N = 80 N = 80 ÷. N = 153. The total resistance of the microphone is 153. ohms.
Proportion Method:
(^80) = N
Solve for N: N × 52 = 80 × 100 N × 52 = 8, N = 8,000 ÷ 52 N = 153.846 ohms or N = (80 × 100) ÷ 52 N = 153.846 ohms
Positive and Negative Numbers
(Signed Numbers)
Positive numbers are numbers that are greater than zero. Negative numbers are numbers less than zero. [Figure 1-8] Signed numbers are also called integers.
Figure 1-8. A scale of signed numbers.
The two most common roots are the square root and the cube root. For more examples of roots, see the chart in Figure 1-10, Functions of Numbers (on page 1-14).
Square Roots
The square root of 25, written as √25, equals 5. That is, when the number 5 is squared (multiplied by itself ), it produces the number 25. The symbol √ is called a radical sign. Finding the square root of a number is the most common application of roots. The collection of numbers whose square roots are whole numbers are called perfect squares. The first ten perfect squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. The square root of each of these numbers is 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, respectively.
For example, √36 = 6 and √81 = 9
To find the square root of a number that is not a perfect square, use either a calculator or the estimation method. A longhand method does exist for finding square roots, but with the advent of calculators and because of its lengthy explanation, it is no longer included in this handbook. The estimation method uses the knowledge of perfect squares to approximate the square root of a number.
Example: Find the square root of 31. Since 31 falls between the two perfect roots 25 and 36, we know that √31 must be between √25 and √36. Therefore,√ 31 must be greater than 5 and less than 6 because √25 = 5 and √36 = 6. If you estimate the square root of 31 at 5.5, you are close to the correct answer. The square root of 31 is actually 5.568.
Cube Roots
The cube root of 125, written as
3 √125, equals 5. That is, when the number 5 is cubed (5 multiplied by itself then multiplying the product (25) by 5 again), it pro- duces the number 125. It is common to confuse the “cube” of a number with the “cube root” of a number. For clarification, the cube of 27 = 27^3 = 27 × 27 × 27 = 19,683. However, the cube root of 27 =
3 √27 = 3.
Fractional Powers
Another way to write a root is to use a fraction as the power (or exponent) instead of the radical sign. The square root of a number is written with a 1 ⁄ 2 as the exponent instead of a radical sign. The cube root of a number is written with an exponent of 1 ⁄ 3 and the fourth root with an exponent of 1 ⁄ 4 and so on.
Example: √31 = 31
(^1) ⁄ 2 3 √125 = 125
(^1) ⁄ 3 4 √16 = 16
(^1) ⁄ 4
Functions of Numbers Chart
The Functions of Numbers chart [Figure 1-10] is included in this chapter for convenience in making computations. Each column in the chart is listed below, with new concepts explained.
- Number, (N)
- N squared, (N 2 )
- N cubed, (N^3 )
- Square root of N, (√N)
- Cube root of N, (
3 √N )
- Circumference of a circle with diameter = N. Circumference is the linear measurement of the distance around a circle. The circumference is calculated by multiplying the diameter of the circle by 3.1416 (3.1416 is the number referred to as pi, which has the symbol π). If the diameter of a circle is 10 inches, then the circumference would be 31.416 inches because 10 × 3.1416 = 31.4160.
- Area of a circle with diameter = N. Area of a circle is the number of square units of measurement contained in the circle with a diameter of N. The area of a circle equals π multiplied by the radius squared. This is calculated by the formula: A = π × r^2. Remember that the radius is equal to one-half of the diameter. Example: A cockpit instrument gauge has a round face that is 3 inches in diameter. What is the area of the face of the gauge? From Figure 1-10 for N = 3, the answer is 7.0686 square inches. This is calculated by: If the diameter of the gauge is 3 inches, then the radius = D⁄ 2 = 3 ⁄ 2 = 1.5 inches. Area = π × r^2 = 3.1416 × 1.5 2 = 3.1416 × 2.25 = 7.0686 square inches.
Scientific Notation
Scientific notation is used as a type of shorthand to express very large or very small numbers. It is a way to write numbers so that they do not take up as much space on the page. The format of a number written in scientific notation has two parts. The first part is a number greater than or equal to 1 and less than 10 (for example, 2.35). The second part is a power of 10 (for example, 10^6 ). The number 2,350,000 is expressed in scientific notation as 2.35 × 10^6. It is important that the decimal point is always placed to the right of the first digit. Notice that very large numbers always have
Number Square Cube Square Root Cube Root Circumference Area
When converting, remember that large numbers always have positive powers of ten and small numbers always have negative powers of ten. Refer to Figure 1- to determine which direction to move the decimal point.
Addition, Subtraction, Multiplication, and Division of Scientific Numbers To add, subtract, multiply, or divide numbers in sci- entific notation, change the scientific notation number back to standard notation. Then add, subtract, multiply or divide the standard notation numbers. After the computation, change the final standard notation number back to scientific notation.
Algebra
Algebra is the branch of mathematics that uses letters or symbols to represent variables in formulas and equations.
For example, in the equation D = V × T, where Distance = Velocity × Time, the variables are: D, V, and T.
Equations Algebraic equations are frequently used in aviation to show the relationship between two or more variables. Equations normally have an equals sign (=) in the expression.
Example: The formula A = π × r^2 shows the relationship between the area of a circle (A) and the length of the radius (r) of the circle. The area of a circle is equal to π (3.1416) times the radius squared. Therefore, the larger the radius, the larger the area of the circle.
Algebraic Rules When solving for a variable in an equation, you can add, subtract, multiply or divide the terms in the equation, you do the same to both sides of the equals sign.
a positive power of 10 and very small numbers always have a negative power of 10.
Example: The velocity of the speed of light is over 186,000,000 mph. This can be expressed as 1.86 × 10 8 mph in scientific notation. The mass of an electron is approximately 0.000,000,000,000,000,000,000, ,000,911 grams. This can be expressed in scientific notation as 9.11 × 10 -28^ grams.
Converting Numbers from Standard Notation to Scientific Notation
Example: Convert 1,244,000,000,000 to scientific notation as follows. First, note that the decimal point is to the right of the last zero. (Even though it is not usually written, it is assumed to be there.)
1,244,000,000,000 = 1,244,000,000,
To change to the format of scientific notation, the deci- mal point must be moved to the position between the first and second digits, which in this case is between the 1 and the 2. Since the decimal point must be moved 12 places to the left to get there, the power of 10 will be 12. Remember that large numbers always have a positive exponent. Therefore, 1,244,000,000,000 = 1.244 × 10^12 when written in scientific notation.
Example: Convert 0.000000457 from standard nota- tion to scientific notation. To change to the format of scientific notation, the decimal point must be moved to the position between the first and second numbers, which in this case is between the 4 and the 5. Since the decimal point must be moved 7 places to the right to get there, the power of 10 will be −7. Remember that small numbers (those less than one) will have a negative exponent. Therefore, 0.000000457 = 4.57 × 10 -7^ when written in scientific notation.
Converting Numbers from Scientific Notation to Standard Notation
Example: Convert 3.68 × 10^7 from scientific notation to standard notation, as follows. To convert from sci- entific notation to standard notation, move the decimal place 7 places to the right. 3.68 × 10 7 = 36800000 = 36,800,000. Another way to think about the conversion is 3.68 × 10^7 = 3.68 × 10,000,000 = 36,800,000.
Example: Convert 7.1543 × 10-10^ from scientific nota- tion to standard notation. Move the decimal place 10 places to the left: 7.1543 × 10 -10^ =.00000000071543. Another way to think about the conversion is 7.1543 × 10 -10^ = 7.1543 × .0000000001 =.
Figure 1-11. Converting between scientific and standard notation.
Conversion
Large numbers with positive powers of 10
Small numbers with negative powers of 10 From standard notation to scientific notation
Move decimal place to the left
Move decimal place to the right
From scientific notation to standard notation
Move decimal place to the right
Move decimal place to the left
Examples: Solve the following equations for the value N.
3N = 21 To solve for N, divide both sides by 3. 3N ÷ 3 = 21 ÷ 3 N = 7
N + 17 = 59 To solve for N, subtract 17 from both sides. N + 17 – 17 = 59 – 17 N = 42
N – 22 = 100 To solve for N, add 22 to both sides. N – 22 + 22 = 100 + 22 N = 122
N 5 = 50 To solve for N, multiply both sides by 5. N 5 × 5 = 50 × 5 N = 250
Solving for a Variable
Another application of algebra is to solve an equation for a given variable.
Example: Using the formula given in Figure 1-12, find the total capacitance (CT) of the series circuit contain- ing three capacitors with
C 1 = .1 microfarad C 2 = .015 microfarad C 3 = .05 microfarad
First, substitute the given values into the formula:
Figure 1-12. Total capacitance in a series circuit.
C^1 T =^
1 1 (^1) + (^1) + 1 =^ =10 + 66.66 + 20 C 1 C 2 C 3
(^1) + 1 + 1 0.1 0.015 0.
Therefore, C (^) T = 1 ⁄96.66 = .01034 microfarad. The microfarad (10 -6^ farad) is a unit of measurement of capacitance. This will be discussed in greater length beginning on page 10-51 in chapter 10, Electricity.
Use of Parentheses In algebraic equations, parentheses are used to group numbers or symbols together. The use of parentheses helps us to identify the order in which we should apply mathematical operations. The operations inside the parentheses are always performed first in algebraic equations.
Example: Solve the algebraic equation N = (4 + 3) 2. First, perform the operation inside the parentheses. That is, 4 + 3 = 7. Then complete the exponent calcula- tion N = (7) 2 = 7 × 7 = 49.
When using more complex equations, which may com- bine several terms and use multiple operations, group- ing the terms together helps organize the equation. Parentheses, ( ), are most commonly used in grouping, but you may also see brackets, [ ]. When a term or expression is inside one of these grouping symbols, it means that any operation indicated to be done on the group is done to the entire term or expression.
Example: Solve the equation N = 2 × [(9 ÷ 3) + (4 + 3)^2 ]. Start with the operations inside the parentheses ( ), then perform the operations inside the brackets [ ].
N = 2 × [(9 ÷ 3) + (4 + 3)^2 ] N = 2 × [3 + (7) 2 ] First, complete the operations inside the parentheses ( ). N = 2 × [3 + 49] N = 2 × [52] Second, complete the operations inside the brackets [ ]. N = 104
Order of Operation In algebra, rules have been set for the order in which operations are evaluated. These same universally accepted rules are also used when programming algebraic equations in calculators. When solving the following equation, the order of operation is given below:
N = (62 – 54)^2 + 6^2 – 4 + 3 × [8 + (10 ÷ 2)] + √25 + (42 × 2) ÷ 4 + 3 ⁄ 4
- Parentheses. First, do everything in parentheses, ( ). Starting from the innermost parentheses. If the expression has a set of brackets, [ ], treat these exactly like parentheses. If you are working with a fraction, treat the top as if it were in parentheses and the denominator as if it were in parentheses, even if there are none shown. From the equation
Since the length and the width of a square are the same value, the formula for the area of a square can also be written as:
Area = Side × Side = S^2
Example: What is the area of a square access plate whose side measures 25 inches? First, determine the known value and substitute it in the formula.
A = L × W = 25 inches × 25 inches = 625 square inches
Triangle
A triangle is a three-sided figure. The sum of the three angles in a triangle is always equal to 180°. Triangles are often classified by their sides. An equilateral tri- angle has 3 sides of equal length. An isosceles triangle has 2 sides of equal length. A scalene triangle has three sides of differing length. Triangles can also be clas- sified by their angles: An acute triangle has all three angles less than 90°. A right triangle has one right angle (a 90° angle). An obtuse triangle has one angle greater than 90°. Each of these types of triangles is shown in Figure 1-15.
The formula for the area of a triangle is
Area = 1 ⁄ 2 × (Base × Height) = 1 ⁄ 2 × (B × H)
Example: Find the area of the obtuse triangle shown in Figure 1-16. First, substitute the known values in the area formula.
A = 1 ⁄ 2 × (B × H) = 1 ⁄ 2 × (2'6" × 3'2")
Next, convert all dimensions to inches:
2'6" = (2 × 12") + 6" = (24 + 6) = 30 inches 3'2" = (3 × 12") + 2" = (36 + 2) = 38 inches
Now, solve the formula for the unknown value:
A = 1 ⁄ 2 × (30 inches × 38 inches) = 570 square inches
Parallelogram
A parallelogram is a four-sided figure with two pairs of parallel sides. [Figure 1-17] Parallelograms do not necessarily have four right angles. The formula for the area of a parallelogram is:
Area = Length × Height = L × H
Trapezoid A trapezoid is a four-sided figure with one pair of parallel sides. [Figure 1-18] The formula for the area of a trapezoid is:
Area = 1 ⁄ 2 (Base 1 + Base 2 ) × Height
Example: What is the area of a trapezoid in Figure 1- whose bases are 14 inches and 10 inches, and whose height (or altitude) is 6 inches? First, substitute the known values in the formula.
Figure 1-15. Types of triangles.
Figure 1-16. Obtuse triangle.
Figure 1-17. Parallelogram.
Figure 1-18. Trapezoid.
A = 1 ⁄ 2 (b 1 + b 2 ) × H
= 1 ⁄ 2 (14 inches + 10 inches) × 6 inches
A = 1 ⁄ 2 (24 inches) × 6 inches
= 12 inches × 6 inches = 72 square inches.
Circle
A circle is a closed, curved, plane figure. [Figure 1-20] Every point on the circle is an equal distance from the center of the circle. The diameter is the distance across the circle (through the center). The radius is the distance from the center to the edge of the circle. The diameter is always twice the length of the radius. The circumference, or distance around, a circle is equal to the diameter times π.
Circumference = C = d π
The formula for the area of a circle is:
Area = π × radius^2 = π × r^2
Example: The bore, or “inside diameter,” of a certain aircraft engine cylinder is 5 inches. Find the area of the cross section of the cylinder.
First, substitute the known values in the formula:
A = π × r^2.
The diameter is 5 inches, so the radius is 2.5 inches. (diameter = radius × 2)
A = 3.1416 × (2.5 inches)^2 = 3.1416 × 6.25 square inches = 19.635 square inches
Ellipse An ellipse is a closed, curved, plane figure and is com- monly called an oval. [Figure 1-21] In a radial engine, the articulating rods connect to the hub by pins, which travel in the pattern of an ellipse (i.e., an elliptical or obital path).
Wing Area To describe the shape of a wing [Figure 1-23], several terms are required. To calculate wing area, it will be necessary to know the meaning of the terms “span” and “chord.” The wingspan, S, is the length of the wing from wingtip to wingtip. The chord is the average width
Figure 1-19. Trapezoid, with dimensions.
Figure 1-20. Circle. Figure 1-22. Wing planform.
Figure 1-21. Ellipse.
