# Order of Magnitude

## Practice

### practice problem 1

- in the life of an average human
- in all of recorded history
- since modern humans first appeared
- since the dinosaurs went extinct
- since the Earth was formed
- since the beginning of time

#### solution

Since all of these times are normally stated in years, we should start by determining the number of seconds in a year.

1 year = | (60 s/min) |

1 year = | 3.156 × 10^{7} s |

Life expectancy varies from country to country. According to the United Nations Statistics Division, Japan had the longest life expectancy at 82 years and Zambia the shortest at 32 years (2000–2005). Women typically live longer than men. In the United States, the difference is roughly six years. To answer this part of the question, we'll use the global mean of 66 years for both genders (2003).

*t*=(66 years) (3.156 × 10 ^{7}s/year)*t*=2.1 × 10 ^{9}sWriting was independently invented by humans in at least three and possibly five different places…

The invention of writing location years before present mesopotamia 5300 china 3500 mesoamerica 2500 egypt? 5100 indus river valley? 4600 Let's go with the oldest of these, Mesopotamia, where cuneiform was invented more than 50 centuries ago.

*t*=(5300 years) (3.156 × 10 ^{7}s/year)*t*=1.7 × 10 ^{11}sThe oldest fossils of anatomically modern humans date back about 195,000 years, which agrees with molecular evidence for the age of

*Homo sapiens*. Genetic diversity and the rate at which mutations occur point to a common ancestor who lived in Africa some 200,000 years ago. Let's use the nice round number.*t*=(200,000 years) (3.156 × 10 ^{7}s/year)*t*=6.3 × 10 ^{12}sThis number is known by many nonscientists, since dinosaurs have worked their way into popular culture. All of the nonavian dinosaurs (the ones that didn't evolve into birds) and about 50% of the other species inhabiting the Earth disappeared 65 million years ago at a point in geological history known as the Cretaceous-Paleogene extinction. This event was most certainly influenced by some cataclysmic natural catastrophe — probably a collision between the Earth and an asteroid.

*t*=(65 × 10 ^{6}years)(3.156 × 10 ^{7}s/year)*t*=2.1 × 10 ^{15}sThe oldest known terrestrial rocks are 3~4 billion years old as determined by radioisotopic dating. Mineral grains found in sandstone have been dated to 4.4 billion years. The oldest known meteorites go back 4.5~4.6 billion years. Since all the bodies of the solar system are assumed to have formed more or less at the same time, this last number hints at the actual age of the Earth — 4.54 billion years. The earth is a geologically dynamic body. Its surface is constantly being ground down by erosion, consumed at subduction zones, and regurgitated from volcanoes. As a result, none of the surface rocks are in a form that would enable them to be used for dating the Earth as a whole. Meteorites are small bits of material left over from the birth of the solar system that did not manage to coalesce into a body of any significant size. They do not share the same complicated history as the Earth's crust and are in near pristine condition (in terms of their radioisotopes, at least). The age of the Earth is therefore the age of the oldest known meteorites.

*t*=(4.54 × 10 ^{9}years)(3.156 × 10 ^{7}s/year)*t*=1.4 × 10 ^{17}sThe universe as we know it began when a microscopic bubble of space and time appeared in the background foam of quantum uncertainty and then suddenly and mysteriously found itself rapidly increasing in size. It is estimated that the diameter of the universe increased 10

^{50}times in a mere 10^{−34}s. This is known as the period of inflation — that is, the period when space was being pushed outward by some as yet unknown form of dark energy — but the momentum of the ordinary matter and energy was enough to keep space expanding for a long time afterwards. In fact, it's still expanding now at the relatively slow rate of one part in 10^{18}every second or about the diameter of an atom every ten years. Stepping back in time again, the very early universe was too dense and hot for light to propagate any significant distance and space was basically opaque. Approximately 380,000 years after the initial period of inflation, the universe had expanded and cooled enough so that light was able to live an existence independent of matter. The universe is about 1000 times bigger now than it was then and the wavelength of this radiation has shifted from dangerous gamma rays to innocuous microwaves. Studies of the variations in the temperature of this cosmic microwave background radiation have yielded the most accurate measurement of the age of the universe to date. The universe began in a big bang explosion some 13.8 ± 0.3 billion years ago.*t*=(13.8 × 10 ^{9}years)(3.156 × 10 ^{7}s/year)*t*=4.3 × 10 ^{17}sNote that even though the mantissa (the decimal portion of the number that comes before the exponential part) is less than five this number rounds up to the next higher power of ten. That's because the rule for rounding to the nearest exponent is different from the rule for rounding to the nearest decimal. We use five for decimals because it divides a decimal interval in half. For example, 1.5 is midway between 1 and 2, while 25,000 is midway between 20,000 and 30,000. If we apply similar reasoning to exponents we find that it's 3 that divides an exponential interval — or, more precisely…

10

^{0.5}= √10 = 3.16228… ≈ 3For example, 30 is midway between 10 and 100, while 0.003 is midway between 0.01 and 0.001.

Since…

4.3 x10

^{17}s > 3.16228 x10^{17}s ~ 10^{17.5}s…the age of the universe rounds up to 10

^{18}s.

To summarize…

order | event | comparison |
---|---|---|

10^{07} s |
one year | ten million seconds |

10^{09} s |
one human lifetime | one hundred years |

10^{11} s |
all of written history | one hundred lifetimes |

10^{13} s |
entirety of human existence | one hundred times older than written history |

10^{15} s |
extinction of the dinosaurs | one hundred times older than hummanity |

10^{17} s |
age of the earth | one hundred times older than the last dinosaur |

10^{18} s |
age of the universe | an order of magnitude older than the Earth |

### practice problem 2

#### solution

Answer it.

### practice problem 3

#### solution

Answer it.

### practice problem 4

#### solution

Answer it.