Mathematics

The General handbook calls mathematics a set of tools. Turbine engine parts are measured to within a ten-thousandth of an inch, fuel system work means calculating tank volumes and capacities, and sheet metal repairs call for fractions and surface areas (FAA-H-8083-30B, p. 3-1). The ACS subject for mathematics covers areas and volumes, geometric terms, ratio, proportion and percentage, algebra, number notation, rounding, powers, measurement systems, signed numbers and basic arithmetic (FAA-S-ACS-1, AM.I.H). This lesson works through each one, with the handbook's methods and the mistakes that most often lead to a wrong answer.

Fractions, decimals and rounding

The four basic operations produce a sum (addition), a difference (subtraction), a product (multiplication) and a quotient, with any amount left over called the remainder (division) (FAA-H-8083-30B, pp. 3-1 to 3-2). A fraction is written N/D, where N is the numerator and D the denominator; the denominator can never be zero, and a fraction whose numerator is equal to or larger than its denominator is called improper (FAA-H-8083-30B, p. 3-2).

• Adding and subtracting needs a common denominator, ideally the least common denominator. A 1/8-inch spacer on a 3/16-inch shim is 2/16 + 3/16 = 5/16 inch thick (FAA-H-8083-30B, p. 3-2).
• Multiplying needs no common denominator: multiply the numerators, then the denominators. Dividing means inverting the second fraction and multiplying (FAA-H-8083-30B, p. 3-3).
• Reducing divides out common factors until the fraction is in lowest terms, and subtracting mixed numbers may mean borrowing 1 from the whole number (FAA-H-8083-30B, pp. 3-3 to 3-4).

When multiplying decimals, ignore the decimal points, multiply, then give the answer as many decimal places as the two numbers have together. When dividing by a decimal, move the decimal point in both numbers until the divisor is a whole number (FAA-H-8083-30B, pp. 3-5 to 3-6). To change a fraction to a decimal, divide the numerator by the denominator. To find the nearest fraction on a steel rule graduated in 64ths, multiply the decimal by 64, round to a whole number and put it over 64: 0.656 × 64 = 41.98, about 42, and 42/64 reduces to 21/32 inch (FAA-H-8083-30B, pp. 3-6 to 3-7).

To round, look at the digit just to the right of the last one you keep. If it is 5 or greater, round up by 1; if it is less than 5, leave the last kept digit alone (FAA-H-8083-30B, p. 3-6). So 1.3462 rounded to the nearest hundredth is 1.35. Some calculations round in one direction only: ballast weight, for example, is always rounded up to the next whole pound (FAA-H-8083-30B, p. 6-23).

Signed numbers and the order of operations

Positive numbers are greater than zero and negative numbers are less than zero; signed numbers are also called integers (FAA-H-8083-30B, p. 3-11). The rules are short but they decide many answers:

• Two negatives added give a negative. A positive and a negative can give either sign, depending on which is larger in size: −100 + 45 = −55.
• To subtract, change the minus to a plus and change the sign of the second number: −8 − 23 = −8 + (−23) = −31.
• Multiplying or dividing two numbers with the same sign gives a positive result; different signs give a negative one (FAA-H-8083-30B, p. 3-11).

Weight and balance depends on these rules: weight removed from an aircraft gets a minus sign and weight added gets a plus sign (FAA-H-8083-30B, p. 3-11). Calculations follow a fixed order of operations: parentheses first (innermost first, with brackets treated the same way), then exponents and roots, then multiplication and division from left to right in one sweep, then addition and subtraction from left to right (FAA-H-8083-30B, pp. 3-17 to 3-18). The memory aid is PEMDAS. The "one sweep" part matters: 20 − 12 ÷ 4 × 2 is 20 − 6 = 14, because 12 ÷ 4 = 3 comes before 3 × 2 = 6. Doing the multiplication first would give the wrong answer of 18.5.

Powers, roots and number notation

A power, or exponent, shows how many times a base is multiplied by itself: 2³ = 2 × 2 × 2 = 8. A power of 2 is called squared and a power of 3 cubed. Any nonzero number raised to the zero power equals 1, and a negative power equals the reciprocal with the power made positive, so 2⁻³ = 1/8 (FAA-H-8083-30B, pp. 3-11 to 3-12). With the same base, multiplying adds the exponents and dividing subtracts them: 10⁴ ÷ 10² = 10² (FAA-H-8083-30B, p. 3-12). When raising a negative number to a power on a calculator, put it in parentheses first.

A root is the number that, multiplied by itself a given number of times, produces the original number. The square root of 25 is 5 and the cube root of 125 is 5; a root can also be written as a fractional power, so the square root of 31 is 31 to the ½ power (FAA-H-8083-30B, pp. 3-12 to 3-13). Do not confuse the cube of a number with its cube root: 27³ is 19,683, but the cube root of 27 is 3 (FAA-H-8083-30B, p. 3-13).

Scientific notation writes a number as a value of at least 1 and less than 10, times a power of 10. Large numbers have positive powers and small numbers negative ones: 2,350,000 is 2.35 × 10⁶ and 0.000000457 is 4.57 × 10⁻⁷ (FAA-H-8083-30B, pp. 3-13 and 3-16). Binary numbers use only 0 and 1, with place values that are powers of 2 (128, 64, 32, 16, 8, 4, 2 and 1 in an 8-bit number). To convert binary to decimal, add the place values that hold a 1: 10010110 = 128 + 16 + 4 + 2 = 150 (FAA-H-8083-30B, pp. 3-27 to 3-28).

Ratio, proportion and percentage

A ratio compares two numbers and can be written as a fraction, with a colon, or with the word "to" (FAA-H-8083-30B, p. 3-7). Aviation is full of them (FAA-H-8083-30B, pp. 3-7 to 3-8):

• Compression ratio: cylinder volume with the piston at the bottom of its stroke compared with the volume at the top. 140 cubic inches at the bottom and 20 at the top is 7:1 (FAA-H-8083-32B, p. 1-29).
• Aspect ratio: the span of an airfoil compared with its chord.
• Air-fuel ratio: the weight of air compared with the weight of fuel in the mixture.
• Glide ratio: forward distance traveled compared with height lost without power.
• Gear and speed ratios: the gear ratio compares numbers of teeth; the speed ratio is its inverse, because the gear with fewer teeth turns faster.

A proportion states that two ratios are equal, such as 3:4 = 6:8. The first and last terms are the extremes and the middle terms are the means, and the product of the extremes always equals the product of the means. To find an unknown, multiply the two numbers on the diagonal and divide by the third (FAA-H-8083-30B, p. 3-8).

Percentage means parts per hundred. To change a decimal to a percentage, move the decimal point two places right; to go back, move it two places left (FAA-H-8083-30B, p. 3-10). There are three kinds of percentage problem (FAA-H-8083-30B, pp. 3-10 to 3-11): a percentage of a number (20% of 45 is 0.20 × 45 = 9), what percentage one number is of another (9 is 9 ÷ 36 = 25% of 36), and a number when a percentage of it is known (if 18 is 40% of a number, the number is 18 ÷ 0.40 = 45).

Algebra: solving for the unknown

Algebra uses letters or symbols for variables in formulas and equations. To solve for a variable, add, subtract, multiply or divide, always doing the same thing to both sides of the equals sign (FAA-H-8083-30B, p. 3-16). Mechanics rearrange formulas constantly. Pascal's law gives force = pressure × area, so dividing both sides by area gives pressure = force ÷ area (FAA-H-8083-30B, p. 5-28). If a piston with an area of 2.5 square inches must produce 1,250 pounds of force, the pressure needed is 1,250 ÷ 2.5 = 500 psi. Operations inside parentheses are always done first (FAA-H-8083-30B, p. 3-17).

Geometry: terms, areas and volumes

The ACS names polygon, pi, diameter, radius and hypotenuse among the geometric terms to know (FAA-S-ACS-1, AM.I.H.K3). A rectangle has four sides with opposite sides equal and parallel and all angles 90°; a square has four equal sides; a triangle's three angles always add up to 180°; a parallelogram has two pairs of parallel sides; and a trapezoid has one pair (FAA-H-8083-30B, pp. 3-18 to 3-20). The diameter of a circle is twice its radius, the circumference is the diameter times π, and π is about 3.1416 (FAA-H-8083-30B, pp. 3-13 and 3-19).

These formulas are summarized in the handbook (FAA-H-8083-30B, pp. 3-19 and 3-26). A parallelogram's area is length × height and an ellipse's is π × semi-axis A × semi-axis B (FAA-H-8083-30B, p. 3-19). One square foot is 144 square inches and one square yard is 9 square feet (FAA-H-8083-30B, p. 3-20). For a tapered wing, use the mean chord in the same unit as the span (FAA-H-8083-30B, p. 3-20).

Piston displacement is the volume swept by all the pistons in one revolution of the crankshaft: π × (bore ÷ 2)² × stroke × number of cylinders, where the bore is the cylinder's inside diameter and the stroke is how far the piston travels (FAA-H-8083-30B, pp. 3-21 to 3-22). Surface area is a different idea from volume: it is the sheet metal needed to build a tank, while volume is the fuel the tank holds (FAA-H-8083-30B, pp. 3-23 to 3-24).

In a right triangle the side across from the 90° angle is the hypotenuse, and the Pythagorean theorem gives a² + b² = c². The trigonometric ratios compare sides: sine = opposite ÷ hypotenuse, cosine = adjacent ÷ hypotenuse and tangent = opposite ÷ adjacent (FAA-H-8083-30B, pp. 3-24 to 3-25).

Measurement systems and conversions

The United States mainly uses the conventional (U.S. or English) system but continues to use the metric system (SI) as well, so mechanics must convert between them (FAA-H-8083-30B, p. 3-26). Metric units take prefixes: kilo means 1,000, centi one hundredth, milli one thousandth and micro one millionth (FAA-H-8083-30B, p. 3-27). Conversion is often needed because many drawings for new aircraft and components use the metric system with the millimeter as the unit (FAA-H-8083-30B, p. 4-21), and capacitance is measured in microfarads (10⁻⁶ farad) (FAA-H-8083-30B, p. 3-17). Useful factors from the handbook's formula sheet (FAA-H-8083-30B, p. 3-22):

• 1 inch = 25.4 millimeters; 1 foot = 12 inches; 1 mile = 5,280 feet.
• 1 cubic foot = 1,728 cubic inches, or about 7.5 gallons.
• 1 U.S. gallon = 231 cubic inches = 4 quarts = 3.785 liters; 1 liter = 0.264 gallon.
• 1 kilogram = 2.2 pounds; 1 pound = 0.4536 kilogram.

Torque values follow the same logic. A foot is 12 inches, so to get foot-pounds, divide inch-pounds by 12 (FAA-H-8083-30B, p. 7-50); to get inch-pounds, multiply foot-pounds by 12. For example, 480 inch-pounds is 40 foot-pounds. Always take the actual torque value from the manufacturer's data for the job; the conversion only changes the units.

Worked example

A rectangular fuel cell measures 30 inches long, 20 inches wide and 1 foot 2 inches deep inside. Find its capacity in gallons and the weight of the aviation gasoline it holds.

1. Use one unit. 1 foot 2 inches = 12 + 2 = 14 inches.
2. Volume. length × width × height = 30 × 20 × 14 = 8,400 cubic inches (FAA-H-8083-30B, p. 3-21).
3. Gallons. 8,400 ÷ 231 = 36.36 gallons, or 36.4 gallons rounded to the nearest tenth (FAA-H-8083-30B, p. 3-22).
4. Check another way. 8,400 ÷ 1,728 = 4.86 cubic feet, and 4.86 × 7.5 = 36.5 gallons. The small difference comes from 7.5 being a rounded factor, so the result agrees.
5. Weight. The standard weight of avgas for weight and balance is 6 pounds per gallon (FAA-H-8083-30B, p. 6-16), so 36.36 × 6 = 218 pounds. Use the unrounded 36.36 here rather than the rounded 36.4.
6. Metric. 36.36 gallons × 3.785 = 137.6 liters.

Common mistakes

• Squaring the diameter where the formula needs the radius. The area or volume comes out four times too big.
• Mixing feet and inches, or using 144 (square inches per square foot) where a volume needs 1,728.
• Converting the wrong way: millimeters to inches means dividing by 25.4, and inch-pounds to foot-pounds means dividing by 12.
• Doing every multiplication before any division instead of working left to right in one sweep.
• Losing a minus sign, especially for weight removed or for arms forward of a datum.
• Applying a gear ratio the wrong way round: the gear with fewer teeth turns faster.
• For compression ratio, dividing the displaced volume by the clearance volume instead of the total volume at bottom center.
• Confusing the cube of a number with its cube root.