- Standard Form to Scientific Form
- Scientific Form to Standard Form
- Working with Numbers in Scientific Notation

Inside Collection (Textbook): Basic Mathematics Review

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The basic operations with real numbers are presented in this chapter. The concept of absolute value is discussed both geometrically and symbolically. The geometric presentation offers a visual understanding of the meaning of |x|. The symbolic presentation includes a literal explanation of how to use the definition. Negative exponents are developed, using reciprocals and the rules of exponents the student has already learned. Scientific notation is also included, using unique and real-life examples. Objectives of this module: be able to convert a number from standard form to scientific form and from scientific form to standard form, be able to work with numbers in scientific notation.

- Standard Form to Scientific Form
- Scientific Form to Standard Form
- Working with Numbers in Scientific Notation

Very large numbers such as 43,000,000,000,000,000,000 (the number of different possible configurations of Rubik’s cube) and very small numbers such as

To see how this is done, let us start with a somewhat smaller number such as 2480. Notice that

The last form is called the scientific form of the number. There is *one* nonzero
digit to the left of the decimal point and the absolute value of the exponent on 10
records the number of places the original decimal point was moved to the *left*.

There is *one* nonzero digit to the left of the decimal point and the absolute value of the exponent of 10 records the number of places the original decimal point was moved to the *right*.

Numbers written in scientific form are also said to be written using scientific notation. In scientific notation, a number is written as the product of a number between and including 1 and 10

To write a number in scientific notation:

- Move the decimal point so that there is one nonzero digit to its left.
- Multiply the result by a power of 10 using an exponent whose absolute value is the number of places the decimal point was moved. Make the exponent positive if the decimal point was moved to the left and negative if the decimal point was moved to the right.

Write the numbers in scientific notation.

981

The number 981 is actually

The decimal point is now two places to the left of its original position, and the power of 10 is 2.

The decimal point is one place to the left of its original position, and the power of 10 is 1.

The decimal point is twelve places to the right of its
original position, and the power of 10 is

The decimal point is two places to the right of its original position, and
the power of 10 is

Write the following numbers in scientific notation.

346

87,000,000

179,000,000,000,000,000,000

100,000

1,000,000

A number written in scientific notation can be converted to standard form by reversing the process shown in Sample Set A.

To convert a number written in scientific notation to a number in standard form, move the decimal point the number of places prescribed by the exponent on the 10.

Move the decimal point to the right when you have a positive exponent, and move the decimal point to the left when you have a negative exponent.

The exponent of 10 is 4 so we must move the decimal point to the right 4 places (adding

The exponent of 10 is 7 so we must move the decimal point to the right 7 places (adding

The exponent of 10 is 27 so we must move the decimal point to the right 27 places (adding

The exponent of 10 is

The exponent of 10 is

Convert the following numbers to standard form.

925

401000

There are many occasions (particularly in the sciences) when it is necessary to find the product of two numbers written in scientific notation. This is accomplished by using two of the basic rules of algebra.

Suppose we wish to find

Then, by the rules of exponents,

The product of

We need to move the decimal point one place to the *left* to put this number in scientific notation.

Thus, we must also change the exponent of 10.

Thus,

Perform each multiplication.

Convert the numbers used in the following problems to scientific notation.

Mount Kilimanjaro is the highest mountain in Africa. It is 5890 meters high.

The planet Mars is about 222,900,000,000 meters from the sun.

There is an irregularly shaped galaxy, named NGC 4449, that is about 250,000,000,000,000,000,000,000 meters from earth.

The farthest object astronomers have been able to see (as of 1981) is a quasar named 3C427. There seems to be a haze beyond this quasar that appears to mark the visual boundary of the universe. Quasar 3C427 is at a distance of 110,000,000,000,000,000,000,000,000 meters from the earth.

The smallest known insects are about the size of a typical grain of sand. They are about

Atoms such as hydrogen, carbon, nitrogen, and oxygen are about

The island of Manhattan, in New York, is about 57,000 square meters in area.

The second largest moon of Saturn is Rhea. Rhea has a surface area of about 735,000 square meters, roughly the same surface area as Australia.

A star, named Epsilon Aurigae B, has a diameter (distance across) of 2,800,000,000,000 meters. This diameter produces a surface area of about 24,630,000,000,000,000,000,000,000 square meters. This star is what astronomers call a red giant and it is the largest red giant known. If Epsilon Aurigae were placed at the sun’s position, its surface would extend out to the planet Uranus.

The volume of the planet Venus is 927,590,000,000,000,000,000 cubic meters.

The average mass of a newborn American female is about 3360 grams.

The largest brain ever measured was that of a sperm whale. It had a mass of 9200 grams.

The mass of the Eiffel tower in Paris, France, is 8,000,000 grams.

In 1981, a Japanese company built the largest oil tanker to date. The ship has a mass of about 510,000,000,000 grams. This oil tanker is more than 6 times as massive as the U.S. aircraft carrier, U.S.S. *Nimitz*.

In the constellation of Virgo, there is a cluster of about 2500 galaxies. The combined mass of these galaxies is 150,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 grams.

The mass of an amoeba is about

Cells in the human liver have masses of about

The human sperm cell has a mass of about

The principal protein of muscle is myosin. Myosin has a mass of

Amino acids are molecules that combine to make up protein molecules. The amino acid tryptophan has a mass of

An atom of the chemical element bromine has 35 electrons. The mass of a bromine atom is

Physicists are performing experiments that they hope will determine the mass of a small particle called a neutrino. It is suspected that neutrinos have masses of about

The approximate time it takes for a human being to die of asphyxiation is 316 seconds.

On the average, the male housefly lives 1,468,800 seconds (17 days).

Aluminum-26 has a half-life of 740,000 years.

Manganese-53 has a half-life of 59,918,000,000,000 seconds (1,900,000 years).

In its orbit around the sun, the earth moves a distance one and one half feet in about

A pi-meson is a subatomic particle that has a half-life of about

A subatomic particle called a neutral pion has a half-life of about

Near the surface of the earth, the speed of sound is 1195 feet per second.

For the following problems, convert the numbers from scientific notation to standard decimal form.

The sun is about

The mass of the earth is about

Light travels about

One year is about

Rubik’s cube has about

A photon is a particle of light. A 100-watt light bulb emits

There are about

A car traveling at an average speed will travel a distance about equal to the length of the smallest fingernail in

A ribosome of *E. coli* has a mass of about

A mitochondrion is the energy-producing element of a cell. A mitochondrion is about

There is a species of frogs in Cuba that attain a length of at most

Perform the following operations.

If Mount Kilimanjaro was 1,000,000 times as high as it really is, how high would it be? (See problem 1.)

If the planet Mars was 300,000 times as far from the sun as it really is, how far from the sun would it be? (See problem 2.)

If 800,000,000 of the smallest insects known were lined up head to tail, how far would they stretch? (See problem 5.)

If Rhea, the moon of Saturn, had a surface area

If the star Epsilon Aurigae B had a surface area

If the mass of all the galaxies in the constellation Virgo was only

What is the mass of 15,000,000,000,000 bromine atoms? (See problem 21.)

*((Reference))* What integers can replace

*((Reference))* Simplify

*((Reference))* Determine the value of

*((Reference))* Write

*((Reference))* Write

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Comments:"Reviewer's Comments: 'I recommend this book for courses in elementary algebra. The chapters are fairly clear and comprehensible, making them quite readable. The authors do a particularly nice job […]"