General (1-9)

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.

 

 

 

 

 

Addition of Positive and 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 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 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 and 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 and 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, 3⁴ means “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.

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, 7³ 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.

 

 

 

 

 

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: 

 

 

 

 

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. 

 

 

 

 

When dividing numbers with powers, the powers can be subtracted as long as the bases are the same. 

 

 

 

 

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 1-9 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 will produce a given number. 

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.(클릭해서 확대해 보세요.)

 

 

 

 

 

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 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