Numbers in the Real World

3

Unit 3B

Putting Numbersin Perspective

Scientific Notation

Scientific notation is a format in which a number is expressed as a number between 1 and 10 multiplied by a power of 10. Examples: 6,700,000,000 in scientific notation is 6.7 109 0.000000000000002 is 2.0 1015

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Perspective through estimation An order of magnitude estimate specifies a broad range of values, usually within one or two powers of ten, such as “in the ten thousands” or “in the millions.”.

Giving Meaning to Numbers

Is the total annual ice cream spending in the United States measured in thousands of dollars, millions of dollars, or billions of dollars?“Reasonably guess that an average person has something like 50 servings of ice cream per year.”

Example

Perspective through comparisons

Giving Meaning to Numbers

Example

Compare the U.S. population to the world population and U.S. energy consumption to world energy consumption. What does this tell you about energy usage by Americans?Solution World population = 7 billion (7 × 109)U.S. population = 300 million (3 × 108).

Example

The U.S. population is only about 4% of the world’s population but, as shown in Table 3.1, the United States uses about 1/5.3, or about 19%, of the world’s energy. That is, Americans use more than four times as much energy per person as the world average.

Giving Meaning to Numbers

Perspective through scaling Verbally: “1 cm = 1 km”

Graphically:

As a ratio: 1 cm = 1 km means a scale ratio of 1 to 100,000

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Example

The distance from the Earth to the Sun is about 150 million kilometers. The diameter of the Sun is about 1.4 million kilometers, and the equatorial diameter of the Earth is about 12,760 kilometers. Put these numbers in perspective by using a scale model of the solar system with a 1 to 10 billion scale.Solution

Example

The distance from the Earth to the Sun is about 150 million kilometers. The diameter of the Sun is about 1.4 million kilometers, and the equatorial diameter of the Earth is about 12,760 kilometers. Put these numbers in perspective by using a scale model of the solar system with a 1 to 10 billion scale.Solution

Example (cont)

Note that, in the last step, we converted the distance from kilometers to meters because it is easier to understand “15 meters” than “0.015 kilometer.” We find the scaled Sun and Earth diameters similarly, this time converting the units to centimeters and millimeters, respectively:

Example (cont)

Example (cont)

The model Sun, at 14 centimeters in diameter, is roughly the size of a grapefruit. The model Earth, at about 1.3 millimeters in diameter, is about the size of the ball point in a pen, and the distance between them is 15 meters.

Case Study: What is a Billion Dollars?

One way to put $1 billion in perspective is to ask a question like “How many people can you employ with $1 billion per year?” Let’s suppose that employees receive a fairly high average salary of $100,000 and that it costs a business an additional $100,000 per year in overhead for each employee (costs for office space, computer services, health insurance, and other benefits). The total cost of an employee is therefore $200,000, so $1 billion would allow a business to hire

Case Study: What is a Billion Dollars?

One billion dollars per year could support a work force of some 5000 employees.

Example (cont)

Another way to put $1 billion in perspective also points out how different numbers can be, even when they sound similar (like million, billion, and trillion). Suppose you become a sports star and earn a salary of $1 million per year. How long would it take you to earn a billion dollars? We simply divide $1 billion by your salary of $1 million/year:Even at a salary of $1 million per year, earning a billion dollars would take a thousand years.