Ratios, Percentages, Powers, and Notation

3-8 its width (or chord). A typical aspect ratio for a commercial airliner might be 7:1 (or 7 to 1). Air-fuel ratio is the ratio of the weight of the air to the weight of fuel in the mixture being fed into the cylinders of a reciprocating engine. For example, a typical air-fuel ratio might be 14.3:1 (or 14.3 to 1). Glide ratio is the ratio of the forward distance traveled to the vertical distance descended when an aircraft is operating without power. For example, if an aircraft descends 1,000 feet while it travels through the air for two linear miles (10,560 feet), it has a glide ratio of 10,560:1,000 which can be reduced to 10.56: 1 (or 10.56 to 1).

Gear ratio is the number of teeth each gear represents when two gears are used in an aircraft component. In Figure 3-7, the pinion gear has 8 teeth and a spur gear has 28 teeth. The gear ratio is 8:28. Using 7 as the LCD, 8:28 becomes 2:7. Speed ratio is when two gears are used in an aircraft component; the rotational speed of each gear is represented as a speed ratio. As the number of teeth in a gear decreases, the rotational speed of that gear increases, and vice-versa. Therefore, the speed ratio of two gears is the inverse (or opposite) of the gear ratio. If two gears have a gear ratio of 2:9, then their speed ratio is 9:2.

Example: A pinion gear with 10 teeth is driving a spur gear with 40 teeth. The spur gear is rotating at 160 rpm. Calculate the speed of the pinion gear. Teeth in Pinion Gear = Speed of Spur Gear Teeth in Spur Gear Speed of Pinion Gear 10 teeth = 160 rpm 40 teeth SP (speed of pinion gear) To solve for SP, multiply 40 × 160, then divide by 10. The speed of the pinion gear is 640 rpm. Example: 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.

Ratio = 200 250 Next, reduce the resulting fraction to its simplest form. Ratio = 200 250 = 4 5 Therefore, the ratio of cruising speed to maximum speed is 4:5. Another common use of ratios is to convert any given ratio to an equivalent ratio with a denominator of 1. Example: Express the ratio 9:5 as a ratio with a denominator of 1. R = 9 5 = ? 1 Since 9 ÷ 5 = 1.8, then 9 5 = 1.8 1 Therefore, 9:5 is the same ratio as 1.8:1. In other words, 9 to 5 is the same ratio as 1.8 to 1.

Proportion

A proportion is a statement of equality between two or more ratios. For example, 3 4 = 6 8 or 3:4 = 6:8 This proportion is read as, “3 is to 4 as 6 is to 8.” Extremes and Means The first and last terms of the proportion (the 3 and 8 in this example) are called the extremes. The second and third terms (the 4 and 6 in this example) 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 × 6, is 12; the product of the means, 3 × 4, is also 12. An inspection of any proportion shows this to be true.

Solving Proportions Normally when solving a proportion, three quantities are known, and the fourth is 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 100 First, multiply 65 × 100: 65 × 100 = 6500 Next, divide by 80: 6500 ÷ 80 = 81.25 Therefore, X = 81.25. Example: An airplane flying 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: Miles Gallons 300 24 = 750 G 3-9 Fraction Decimal MM 1/64 1/32 3/64 1/16 5/64 3/32 7/64 1/8 9/64 5/32 11/64 3/16 13/64 7/32 15/64 1/4 17/64 9/32 19/64 5/16 21/64 11/32 23/64 3/8 25/64 13/32 27/64 7/16 29/64 15/32 31/64 1/2 33/64 17/32 35/64 39341 37/64 19/32 39/64 5/8 41/64 21/32 43/64 11/16 45/64 23/32 47/64 3/4 49/64 25/32 51/64 13/16 53/64 27/32 55/64 7/8 57/64 29/32 59/64 15/16 61/64 31/32 63/64 1 0.015 0.031 0.046 0.062 0.078 0.093 0.109 0.125 0.140 0.156 0.171 0.187 0.203 0.218 0.234 0.25 0.265 0.281 0.296 0.312 0.328 0.343 0.359 0.375 0.390 0.406 0.421 0.437 0.453 0.468 0.484 0.5 0.515 0.531 0.546 0.562 0.578 0.593 0.609 0.625 0.640 0.656 0.671 0.687 0.703 0.718 0.734 0.75 0.765 0.781 0.796 0.812 0.828 0.843 0.859 0.875 0.890 0.906 0.921 0.937 0.953 0.968 0.984 1 0.396 0.793 1.190 1.587 1.984 2.381 2.778 3.175 3.571 3.968 4.365 4.762 5.159 5.556 5.953 6.35 6.746 7.143 7.540 7.937 8.334 8.731 9.128 9.525 9.921 10.318 10.715 11.112 11.509 11.906 12.303 12.7 13.096 13.493 13.890 14.287 14.684 15.081 15.478 15.875 16.271 16.668 17.065 17.462 17.859 18.256 18.653 19.05 19.446 19.843 20.240 20.637 21.034 21.431 21.828 22.225 22.621 23.018 23.415 23.812 24.209 24.606 25.003 25.4 Fraction Decimal MM 1 1/64 1 1/32 1 3/64 1 1/16 1 5/64 1 3/32 1 7/64 1 1/8 1 9/64 1 5/32 1 11/64 1 3/16 1 13/64 1 7/32 1 15/64 1 1/4 1 17/64 1 9/32 1 19/64 1 5/16 1 21/64 1 11/32 1 23/64 1 3/8 1 25/64 1 13/32 1 27/64 1 7/16 1 29/64 1 15/32 1 31/64 1 1/2 1 33/64 1 17/32 1 35/64 1 9/16 1 37/64 1 19/32 1 39/64 1 5/8 1 41/64 1 21/32 1 43/64 1 11/16 1 45/64 1 23/32 1 47/64 1 3/4 1 49/64 1 25/32 1 51/64 1 13/16 1 53/64 1 27/32 1 55/64 1 7/8 1 57/64 1 29/32 1 59/64 1 15/16 1 61/64 1 31/32 1 63/64 2 1.015 1.031 1.046 1.062 1.078 1.093 1.109 1.125 1.140 1.156 1.171 1.187 1.203 1.218 1.234 1.25 1.265 1.281 1.296 1.312 1.328 1.343 1.359 1.375 1.390 1.406 1.421 1.437 1.453 1.468 1.484 1.5 1.515 1.531 1.546 1.562 1.578 1.593 1.609 1.625 1.640 1.656 1.671 1.687 1.703 1.718 1.734 1.75 1.765 1.781 1.796 1.812 1.828 1.843 1.859 1.875 1.890 1.906 1.921 1.937 1.953 1.968 1.984 2 25.796 26.193 26.590 26.987 27.384 27.781 28.178 28.575 28.971 29.368 29.765 30.162 30.559 30.956 31.353 31.75 32.146 32.543 32.940 33.337 33.734 34.131 34.528 34.925 35.321 35.718 36.115 36.512 36.909 37.306 37.703 38.1 38.496 38.893 39.290 39.687 40.084 40.481 40.878 41.275 41.671 42.068 42.465 42.862 43.259 43.656 44.053 44.45 44.846 45.243 45.640 46.037 46.434 46.831 47.228 47.625 48.021 48.418 48.815 49.212 49.609 50.006 50.403 50.8 Fraction Decimal MM 2 1/64 2 1/32 2 3/64 2 1/16 2 5/64 2 3/32 2 7/64 2 1/8 2 9/64 2 5/32 2 11/64 2 3/16 2 13/64 2 7/32 2 15/64 2 1/4 2 17/64 2 9/32 2 19/64 2 5/16 2 21/64 2 11/32 2 23/64 2 3/8 2 25/64 2 13/32 2 27/64 2 7/16 2 29/64 2 15/32 2 31/64 2 1/2 2 33/64 2 17/32 2 35/64 2 9/16 2 37/64 2 19/32 2 39/64 2 5/8 2 41/64 2 21/32 2 43/64 2 11/16 2 45/64 2 23/32 2 47/64 2 3/4 2 49/64 2 25/32 2 51/64 2 13/16 2 53/64 2 27/32 2 55/64 2 7/8 2 57/64 2 29/32 2 59/64 2 15/16 2 61/64 2 31/32 2 63/64 3 2.015 2.031 2.046 2.062 2.078 2.093 2.109 2.125 2.140 2.156 2.171 2.187 2.203 2.218 2.234 2.25 2.265 2.281 2.296 2.312 2.328 2.343 2.359 2.375 2.390 2.406 2.421 2.437 2.453 2.468 2.484 2.5 2.515 2.531 2.546 2.562 2.578 2.593 2.609 2.625 2.640 2.656 2.671 2.687 2.703 2.718 2.734 2.75 2.765 2.781 2.796 2.812 2.828 2.843 2.859 2.875 2.890 2.906 2.921 2.937 2.953 2.968 2.984 3 51.196 51.593 51.990 52.387 52.784 53.181 53.578 53.975 54.371 54.768 55.165 55.562 55.959 56.356 56.753 57.15 57.546 57.943 58.340 58.737 59.134 59.531 59.928 60.325 60.721 61.118 61.515 61.912 62.309 62.706 63.103 63.5 63.896 64.293 64.690 65.087 65.484 65.881 66.278 66.675 67.071 67.468 67.865 68.262 68.659 69.056 69.453 69.85 70.246 70.643 71.040 71.437 71.834 72.231 72.628 73.025 73.421 73.818 74.215 74.612 75.009 75.406 75.803 76.2 3-10 2:7 the denominator), and then convert the decimal number to a percentage by multiplying by 100 as shown earlier.

Example: Express the fraction 5⁄8 as a percentage. 5 8 = 5 ÷ 8 = 0.625 = 62.5% Finding a Percentage of a Given Number This is the most common type of percentage calculation. Here are two methods to solve percentage problems: using algebra or using proportions. Each method is shown next 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? Algebraic 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 = 15 100 To solve for N: N × 100 = 80 × 15 N × 100 = 1,200 N = 1,200 ÷ 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?

Algebraic Method: N% of 12 rated horsepower = 10.75 actual horsepower N% × 12 = 10.75 N% = 10.75 ÷ 12 N% = 0.8958 N = 89.58 Therefore, the motor efficiency is 89.58%. Proportion Method: 10.75 12 = N 100 Solve for G: (750 × 24) ÷ 300 = 60 Therefore, to fly 750 miles, 60 gallons of gasoline is 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 zeroes if necessary) and then affix the percent symbol. Example: Express the following decimal numbers as a percent: 0.90 = 90% 0.5 = 50% 1.25 = 125% 0.335 = 33.5% Expressing a Percentage as a Decimal Number Sometimes it may be necessary to express a percentage 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.

For example: Express the following percentages as decimal numbers: 90% = 0.90 50% = 0.50 5% = 0.05 150% = 1.5 Expressing a Fraction as a Percentage To express a fraction as a percentage, first change the fraction to a decimal number (by dividing the numerator by 3-11 To solve for N: N × 12 = 10.75 × 100 N × 12 = 1,075 N = 1,075 ÷ 12 N = 89.58 or N = (1,075 × 100) ÷ 12 N = 89.58 Therefore, the motor efficiency is 89.58%. Finding a Number When a Percentage of it is Known Example: Eighty ohms represents 52% of a microphone’s total resistance. Find the total resistance of this microphone. Algebraic Method: 52% of N = 80 ohms 52% × N = 80 N = 80 ÷ 0.52 N = 153.846 The total resistance of the microphone is 153.846 ohms.

Proportion Method: = 52 100 To solve for N: N × 52 = 80 × 100 N × 52 = 8,000 N = 8,000 ÷ 52 N = 153.846 ohms or N = (80 × 100) ÷ 52 N = 153.846 ohms Positive & Negative Numbers (Signed Numbers) Positive numbers are numbers that are greater than zero. Negative numbers are numbers less than zero. [Figure 3-8] Signed numbers are also called integers. Addition of Positive & Negative Numbers The sum (addition) of two positive numbers is positive. The sum (addition) of two negative numbers is negative. 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 $100 (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 weighing 10 pounds 80 N 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 = 1,987 Therefore, the new weight is 1,987 pounds. Subtraction of Positive & Negative Numbers To subtract positive and negative numbers, first change the “–” (subtraction symbol) to a “+” (addition symbol), 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 Denver was 6° below zero (−6°). An airplane is cruising at 15,000 feet above Denver. The temperature at 15,000 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) = −26 The temperature is −26°, or 26° below zero at 15,000 feet above the city. Multiplication of Positive & Negative Numbers The product of two positive numbers is always positive. The product of two negative numbers is always positive. The product of a positive and a negative number is always negative. Examples: 3 × 6 = 18 −3 × 6 = −18 −3 × −6 = 18 3 × −6 = −18 Division of Positive & Negative Numbers The quotient of two positive numbers is always positive. The quotient of two negative numbers is always positive. The quotient of a positive and negative number is always negative.

Examples: 6 ÷ 3 = 2 −6 ÷ 3 = −2 −6 ÷ −3 = 2 6 ÷ −3 = −2

Powers

The power (or exponent) of a number is a shorthand method of indicating how many times a number, called the base, is multiplied by itself. For example, 3 4 is read as “3 to the power of 4.” That is, 3 multiplied by itself 4 times. The 3 is the base and 4 is the power. Examples: 23 = 2 × 2 × 2 = 8 3-12 −5 −4 −3 −2 −1 0 +1 +2 +3 +4 +5 Read “two to the third power equals 8.” 105 = 10 × 10 × 10 × 10 × 10 = 100,000 Read “ten to the fifth power equals 100,000.” Special Powers Squared When a number has a power of 2, it is commonly referred to as “squared.” For example, 72 is read as “seven squared” or “seven to the second power.” To remember this, think about how a square has two dimensions: length and width.

Cubed When a number has a power of 3, it is commonly referred to as “cubed.” For example, 7 3 is read as “seven cubed” or “seven to the third power.” To remember this, think about how a cube has three dimensions: length, width, and depth. Power of Zero Any non-zero number raised to the zero power always equals 1. Example: 70 = 1 1810 = 1 (–24)0 = 1 Negative Powers A number with a negative power equals its reciprocal with the same power made positive. Example: The number 2 -3 is read as “2 to the negative 3rd power,” and is calculated by: 2-3 = 1 23 = 1 2 × 2 × 2 = 1 8 When using a calculator to raise a negative number to a power, always place parentheses around the negative number (before raising it to a power) so that the entire number gets raised to the power.

Law of Exponents When multiplying numbers with powers, the powers can be added as long as the bases are the same. Example: 32 × 34 = (3 × 3) × (3 × 3 × 3 × 3) = 3 × 3 × 3 × 3 × 3 × 3 = 36 or 32 × 34 = 3(2+4) = 36 When dividing numbers with powers, the powers can be subtracted as long as the bases are the same. Example: 104 ÷ 102 =10 × 10 × 10 × 10 10 × 10 =10 × 10 × 10 × 10 10 × 10 =10 × 10 =102 or 104 ÷ 102 = 10(4 – 2) = 102 Powers of Ten Because we use the decimal system of numbers, powers of ten are frequently seen in everyday applications. For example, scientific notation uses powers of ten. Also, many aircraft drawings are scaled to powers of ten. Figure 3-9 gives more information on the powers of ten and their values.

Roots A root is a number that when multiplied by itself a specified number of times produces a given number. The two most common roots are the square root and the cube root. For more examples of roots, see Figure 3-10. 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 collections 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 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.

3-13

Powers

of Ten Expansion Value Positive Exponents 106 105 104 103 102 101 100 1,000,000 100,000 10,000 1,000 100 10 1 10 x 10 x 10 x 10 x 10 x 10 10 x 10 x 10 x 10 x 10 10 x 10 x 10 x 10 10 x 10 x 10 10 x 10 10 Negative Exponents 10-1 10-2 10-3 10-4 10-5 10-6 1/10=0.1 1/100=0.01 1/1,000=0.001 1/10,000=0.0001 1/100,000=0.00001 1/1,000,000=0.000001 1/10 1/(10 x 10) 1/(10 x 10 x 10) 1/(10 x 10 x 10 x 10) 1/(10 x 10 x 10 x 10 x 10) 1/(10 x 10 x 10 x 10 x 10 x 10) 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 produces 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 = 273 = 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 found in Figure 3-10 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 (N2) • N cubed (N3) • 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: 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 × r2. Remember that the radius is equal to one-half of the diameter.

Example: A flight deck instrument gauge has a round face that is 3 inches in diameter. What is the area of the face of the gauge? From Figure 3-10 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 × r2 = 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, 106). The number 2,350,000 is expressed in scientific notation as 2.35 × 106. It is important that the decimal point is always placed to the right of the first digit. Notice that very large numbers always have 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 miles per second (mps). This can be expressed as 1.86 × 105 mps in scientific notation. The mass of an electron is approximately 0.000,000,000,000,000,000,000,000,000,911 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.) 3-14 Number Square Cube Square Root Cube Root Circumference Area 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 441 484 529 576 625 676 729 784 841 900 961 1,024 1,089 1,156 1,225 1,296 1,369 1,444 1,521 1,600 1,681 1,764 1,849 1,936 2,025 2,116 2,209 2,304 2,401 2,500 1 8 27 64 125 216 343 512 729 1,000 1,331 1,728 2,197 2,744 3,375 4,096 4,913 5,832 6,859 8,000 9,261 10,648 12,167 13,824 15,625 17,576 19,683 21,952 24,389 27,000 29,791 32,768 35,937 39,304 42,875 46,656 50,653 54,872 59,319 64,000 68,921 74,088 79,507 85,184 91,125 97,336 103,823 110,592 117,649 125,000 1.000 1.414 1.732 2.000 2.236 2.449 2.646 2.828 3.000 3.162 3.317 3.464 3.606 3.742 3.873 4.000 4.123 4.243 4.359 4.472 4.583 4.690 4.796 4.899 5.000 5.099 5.196 5.292 5.385 5.477 5.568 5.657 5.745 5.831 5.916 6.000 6.083 6.164 6.245 6.325 6.403 6.481 6.557 6.633 6.708 6.782 6.856 6.928 7.000 7.071 1.000 1.260 1.442 1.587 1.710 1.817 1.913 2.000 2.080 2.154 2.224 2.289 2.351 2.410 2.466 2.520 2.571 2.621 2.668 2.714 2.759 2.802 2.844 2.885 2.924 2.963 3.000 3.037 3.072 3.107 3.141 3.175 3.208 3.240 3.271 3.302 3.332 3.362 3.391 3.420 3.448 3.476 3.503 3.530 3.557 3.583 3.609 3.634 3.659 3.684 3.142 6.283 9.425 12.566 15.708 18.850 21.991 25.133 28.274 31.416 34.558 37.699 40.841 43.982 47.124 50.265 53.407 56.549 59.690 62.832 65.973 69.115 72.257 75.398 78.540 81.681 84.823 87.965 91.106 94.248 97.389 100.531 103.672 106.814 109.956 113.097 116.239 119.380 122.522 125.664 128.805 131.947 135.088 138.230 141.372 144.513 147.655 150.796 153.938 157.080 0.785 3.142 7.069 12.566 19.635 28.274 38.484 50.265 63.617 78.540 95.033 113.01 132.73 153.94 176.71 201.06 226.98 254.47 283.53 314.16 346.36 380.13 415.48 452.39 490.87 530.93 572.55 615.75 660.52 706.86 754.77 804.25 855.30 907.92 962.11 1017.88 1075.21 1134.11 1194.59 1256.64 1320.25 1385.44 1452.20 1520.53 1590.43 1661.90 1734.94 1809.56 1885.74 1963.49 Number (N) N Squared (N2) N Cubed (N3) Circumference of a circle with diameter = N Area of a circle with diameter = N Square Root of N (√N) Cube Root of N (√N) 3 3-15 Number Square Cube Square Root Cube Root Circumference Area 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 2,601 2,704 2,809 2,916 3,025 3,136 3,249 3,364 3,481 3,600 3,721 3,844 3,969 4,096 4,225 4,356 4,489 4,624 4,761 4,900 5,041 5,184 5,329 5,476 5,625 5,776 5,929 6,084 6,241 6,400 6,561 6,724 6,889 7,056 7,225 7,396 7,569 7,744 7,921 8,100 8,281 8,464 8,649 8,836 9,025 9,216 9,409 9,604 9,801 10,000 132,651 140,608 148,877 157,464 166,375 175,616 185,193 195,112 205,379 216,000 226,981 238,328 250,047 262,144 274,625 287,496 300,763 314,432 328,509 343,000 357,911 373,248 389,017 405,224 421,875 438,976 456,533 474,552 493,039 512,000 531,441 551,368 571,787 592,704 614,125 636,056 658,503 681,472 704,969 729,000 753,571 778,688 804,357 830,584 857,375 884,736 912,673 941,192 970,299 1,000,000 7.141 7.211 7.280 7.348 7.416 7.483 7.550 7.616 7.681 7.746 7.810 7.874 7.937 8.000 8.062 8.124 8.185 8.246 8.307 8.367 8.426 8.485 8.544 8.602 8.660 8.718 8.775 8.832 8.888 8.944 9.000 9.055 9.110 9.165 9.220 9.274 9.327 9.381 9.434 9.487 9.539 9.592 9.644 9.695 9.747 9.798 9.849 9.900 9.950 10.000 3.708 3.733 3.756 3.780 3.803 3.826 3.849 3.871 3.893 3.915 3.937 3.958 3.979 4.000 4.021 4.041 4.062 4.082 4.102 4.121 4.141 4.160 4.179 4.198 4.217 4.236 4.254 4.273 4.291 4.309 4.327 4.344 4.362 4.380 4.397 4.414 4.431 4.448 4.465 4.481 4.498 4.514 4.531 4.547 4.563 4.579 4.595 4.610 4.626 4.642 160.221 163.363 166.504 169.646 172.787 175.929 179.071 182.212 185.354 188.495 191.637 194.779 197.920 201.062 204.203 207.345 210.487 213.628 216.770 219.911 223.053 226.194 229.336 232.478 235.619 238.761 241.902 245.044 248.186 251.327 254.469 257.610 260.752 263.894 267.035 270.177 273.318 276.460 279.602 282.743 285.885 289.026 292.168 295.309 298.451 301.593 304.734 307.876 311.017 314.159 2042.82 2123.71 2206.18 2290.22 2375.83 2463.01 2551.76 2642.08 2733.97 2827.43 2922.46 3109.07 3117.24 3216.99 3318.30 3421.19 3525.65 3631.68 3739.28 3848.45 3959.19 4071.50 4185.38 4300.84 4417.86 4536.46 4656.62 4778.36 4901.67 5026.54 5152.99 5281.01 5410.60 5541.76 5674.50 5808.80 5944.67 6082.12 6221.13 6361.72 6503.88 6647.60 6792.90 6939.77 7088.21 7238.22 7389.81 7542.96 7697.68 7853.98 Number (N) N Squared (N2) N Cubed (N3) Circumference of a circle with diameter = N Area of a circle with diameter = N Square Root of N (√N) Cube Root of N (√N) 3 3-16 1,244,000,000,000 = 1,244,000,000,000.0 To change to the format of scientific notation, the decimal point must be moved to the position between the first and second digits. In this case, it 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 is 12. Remember that large numbers always have a positive exponent. Therefore, 1,244,000,000,000 = 1.244 × 1012 when written in scientific notation.

Example: Convert 0.000000457 from standard notation 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 is −7. Remember that small numbers (those less than one) 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 scientific notation to standard notation, move the decimal place 7 places to the right. 3.68 × 107 = 36,800,000. Another way to think about the conversion is 3.68 × 107 = 3.68 × 10,000,000 = 36,800,000.

Example: Convert 7.1543 × 10 -10 from scientific notation 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 × 0.0000000001 = 0.00000000071543 When converting, remember that large numbers always have positive powers of ten and small numbers always have negative powers of ten. Refer to Figure 3-11 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 scientific 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 × r2 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. 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) to get the variable onto one side of the equals sign. 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 = 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 3-12, find the total capacitance (CT) of the series circuit containing three capacitors with C1 = 0.1 microfarad C2 = 0.015 microfarad C3 = 0.05 microfarad First, substitute the given values into the formula: