Scientific Notation math: *Cool and calculating*

This section details how to calculate using Scientific Notation. A general introduction to Scientific Notation is found in the chapter 2, this section shows how to mathematically manipulate them. As I've already said no one needs to learn this, it is here in case you do want to learn it, if you don't already know it.

Also note that many calculators have a “scientific” mode that let’s you directly work with scientifically notated numbers. Though the Calculator can handle Scientific Notation it can still be handy to know how to manipulate them by hand. Once you have learned to work with these, many answers you need can be found by simple inspection, and conversion from one form to another becomes quite easy.

**Scientific Notation up to a million**

Following is a table of scientific notation up to a million. Once you see the pattern of scientific notation you are able to write any large number (note: __any__ number to the power 0 equals 1):

Math equivalent | Exponential form | |
---|---|---|

1 | 1 | 10^{0} |

10 | 10 | 10^{1} |

100 | 10 × 10 | 10^{2} |

1,000 | 10 × 10 × 10 | 10^{3} |

10,000 | 10 × 10 × 10 × 10 | 10^{4} |

100,000 | 10 × 10 × 10 × 10 × 10 | 10^{5} |

1,000,000 | 10 × 10 × 10 × 10 × 10 × 10 | 10^{6} |

Let’s look at how easy it is to manipulate numbers this way. Scientifically notated numbers have two parts: the first - a simple number (decimal portion) multiplied by the second part: 10 raised to some power (exponent portion).

Scientific notation generally requires that the decimal portion has only one digit to the left of the decimal point. For example, the decimal portion is, generally, written like 1.23456789, not 12.3456789 or 1234.56789, etc.

So to write a number in scientific notation you simply move the decimal point (and count the movements) from the original decimal point until only one digit remains to the left of the decimal point. This decimal portion is then multiplied by 10 raised to a power that is **equal** to the number of decimal point shifts you moved/counted.

For example: To write the number 300 in scientific notation. First note that 300 can also be written as: 300.0

We shift the decimal point *to the left* until only one number, the 3, remains to the left of the decimal point. In this case we shift the decimal point twice (to get 3.00) and so then multiply this decimal portion by 10 raised to the second power, which is 10^{2}.

So 300 = 3.00 × 10^{2} which simplifies to 3 × 10^{2}.

Here’s another way to look at it:

300 = 3 × 10 × 10. We know 10 × 10 = 100 = 10^{2},…

so 300 = 3 × 10 × 10 = 3 × 10^{2}.

To write the number 150,000 in scientific notation is just as simple: we need to move the decimal point to the left 5 times to end up with just one digit to the left of the decimal point. Since we moved the decimal point 5 places we will use 5 as the exponent: 10^{5}:

150,000 = 1.5 × 100,000 = 1.5 × 10^{5}

Reconverting from scientific notation to normal numbers is just as easy. When told there are 6.9 × 10^{9} people in the World, you immediately know you must “add” back the nine missing zeros (since the exponent is 10^{9}, 10^{9} = 1,000,000,000 = 1 billion).

So take the number 6.9 and shift the decimal point 9 places *to the right* to equal 6,900,000,000. Another way to look at it:

6.9 × 10^{9} = 6.9 × 1,000,000,000 = 6,900,000,000.

Multiplication and division are a cinch if you keep in mind the algebraic rule when working with numbers that have exponents. In algebra, when multiplying two powers, the exponents are added.

For instance: X^{2} × X^{3} = X^{2+3} = X^{5}.

In algebra, when dividing powers, the exponent in the denominator is subtracted from the numerator’s exponent. For instance: X^{5} ÷ X^{3} = X^{5-3} = X^{2}.

By keeping in mind the laws of exponents large numbers become tame. Let’s multiply 300 × 150,000 (this is equal to 45 million: 45,000,000).

Following the rules:

300 × 150,000 =

(3 × 10^{2}) × (1.5 × 10^{5}) =

(3 × 1.5) × (10^{2} × 10^{5}) =

4.5 × 10^{2+5} =

4.5 × 10^{7} =

4.5 × 10,000,000 =

45,000,000

To divide 600,000 by 2000 (which equals 300):

600,000 ÷ 2000 =

(6 × 10^{5}) ÷ (2 × 10^{3}) =

(6 ÷ 2) × (10^{5} ÷ 10^{3}) =

3 × 10^{5-3} =

3 × 10^{2} =

3 × 100 =

300

The same rule holds true for small numbers but with a slightly different slant. So far we have worked with numbers that are greater than 1 and all the exponents have been positive numbers. So you might guess that for numbers smaller than 1 the exponents will be *negative* numbers.

A grain of clay is about 0.004 mm in diameter (4 thousandths of a millimeter). To write 0.004 in scientific notation we count the number of decimal points we have to move *to the right* to get the decimal portion to have one digit to the left of the decimal point.

Counting we find we have to move the decimal point 3 places to the right so that the 4 of 0.004 is on the left of the decimal point like so: 0004.

So the exponent for the power of 10 will be "-3":

0.004 = 0004 × 10^{-3} which, removing the useless zeros, simplifies to: 4 × 10^{-3}.

Another way to look at it (see the table below):

0.004 = 4 × 0.001 = 4 × 0.1 × 0.1 × 0.1= 4 × 10^{-3}.

**Scientific Notation down to a millionth**

Following is a table of scientific notation down to a millionth. As before the patter is obvious once you see them arranged like so and you are able to write any small number ([again]note: __any__ number to the power 0 equals 1):

Math equivalent | Exponential form | |
---|---|---|

1 | 1 | 10^{0} |

0.1 | 0.1 | 10^{-1} |

0.01 | 0.1 × 0.1 | 10^{-2} |

0.001 | 0.1 × 0.1 × 0.1 | 10^{-3} |

0.000 1 | 0.1 × 0.1 × 0.1 × 0.1 | 10^{-4} |

0.000 01 | 0.1 × 0.1 × 0.1 × 0.1 × 0.1 | 10^{-5} |

0.000 001 | 0.1 × 0.1 × 0.1 × 0.1 × 0.1 × 0.1 | 10^{-6} |

Let’s imagine that we have a super fine razor and we are able to slice a human hair (0.003 inch) into 900 equally thick pieces. How thick would one of the new pieces be? The piece would be 1/900 of 0.003 inches.

We could either multiply 0.003 by the decimal equivalent of 1/900 (equal to 0.00111), or we could directly divide 0.003 by 900. We’ll try both.

First the multiplication problem:

0.003 × 0.001 11 =

(3 × 10^{-3}) × (1.11 × 10^{-3}) =

(3 × 1.11) × (10^{-3} × 10^{-3}) =

3.33 × 10^{-3+(-3)} =

3.33 × 10^{-6} =

3.33 × 0.000 001 =

0.000 003 33

This is equal to a bit more than 3 millionths of an inch.

Now the division problem:

0.003 ÷ 900 =

(3 × 10^{-3}) ÷ (9 × 10^{2}) =

(3 ÷ 9) × (10^{-3} ÷ 10^{2}) =

0.333 × 10^{-3-(2)} =

0.333 × 10^{-5}

Wait, is there a problem? Just above with the multiplication solution the answer was 3.33 × 10^{-6}, this isn’t the same answer as we have here. Or is it?

Remember that the decimal portion should have one digit to the left of the decimal point, this decimal portion (0.333) doesn’t.

There are two ways to right this. One is by direct multiplication as we have been doing:

0.333 × 10^{-5} =

(3.33 × 10^{-1}) × 10^{-5} =

3.33 × 10^{-1+(-5)} =

3.33 × 10^{-6}

The other way to convert the answer is by simple inspection. You may have noticed that each time we move the decimal point to the right the exponent is reduced by one, each time we move the decimal point to the left the exponent is increased by one. On looking at 0.333 × 10^{-5} we notice we must move the decimal point one place to the right (0.333 to 3.33) and so we must subtract 1 from the exponent (10^{-5} to 10^{-6}). So we change 0.333 × 10^{-5} directly to 3.33 × 10^{-6}.

This means there are many ways to write the same number, though only the way we have talked about (1 digit to the left of the decimal point) is the usually accepted way. There are (approximately) 365 days in a year. This can be written as:

365 (this is also = 365 × 10^{0})

36.5 × 10^{1} (decimal left 1, exponent plus 1, etc.)

3.65 × 10^{2}

0.365 × 10^{3}

3,650 × 10^{-1} (decimal right 1, exponent minus 1, etc.)

36,500 × 10^{-2}

Each of these numbers represents 365.

We have only one more rule for scientific notation. What if we ever have to raise a scientifically notated number to another power? In algebra an exponent raised to a power is simply the product of the two exponents. For instance: (X^{3})^{2} = X^{3×2} = X^{6}.

The square of 900 is 810,000. In scientific notation:

(900)^{2} =

(9 × 10^{2})^{2} =

9^{2} × (10^{2})^{2} =

81 × 10^{2×2} =

81 × 10^{4} =

8.1 × 10^{5} =

8.1 × 100,000 =

810,000

What about a square root, cube root, 5th root, or any other root? Easy, multiply again; but watch out, root exponents are always fractions. You must choose the form your scientific notation is in so that the answer comes out as an even number. The cube root of 27,000,000 is 300 (that is, 300 cubed equals 27,000,000).

First the *unclear way* (this looks rather odd but if you have a calculator that can handle these exponents the answer will be approximately 300, only approximately because the the decimal and exponent are all infinite repeating fractions):

(27,000,000)^{1/3} =

(2.7 × 10^{7})^{1/3} =

2.7^{1/3} × (10^{7})^{1/3} =

1.39248 × 10^{7×(1/3)} =

1.39248 × 10^{7/3} =

1.39248 × 10^{2.333…}

Now the clear way. It is important to choose the form of the notation so that when the exponent is divided by 3 it comes out to an even number, *not a fraction*. To do this here with the "3" of the 1/3 we need the 10’s exponent to be divisible evenly by 3, we will use 6 rather than 7:

(27,000,000)^{1/3} =

(2.7 × 10^{7})^{1/3} =

(27 × 10^{6})^{1/3} =

27^{1/3} × (10^{6})^{1/3} =

3 × 10^{6/3} =

3 × 10^{2} =

3 × 100 =

300

Scientific notation, if it is new to you, may be somewhat confusing for a while. But don’t worry, as with anything new ease comes with use and familiarity. As you read through this book try to make examples of your own and test yourself. You will learn more and more as you proceed.

**One note of caution**: We have worked with multiplication, division, and raising to powers (exponentiation). What about simple addition and subtraction? Let’s look at an example: In adding 150,000 and 9,000 it is obvious that the answer should be 159,000.

How to treat it if the problem is written in scientific notation: (1.5 × 10^{5}) + (9 × 10^{3})?

There is no way as long the notation is in this format.

When adding and subtracting scientifically notated numbers, the powers of ten __ have to be equal__. So to add them we must convert one of the numbers so that the power of ten matches the other.

Let’s convert 1.5 × 10^{5}:

1.5 × 10^{5} = 150 × 10^{3}

Now just add the decimal of the two scientific notations (in adding and subtracting the powers of ten remain the same):

(150 × 10^{3}) + (9 × 10^{3}) =

159 × 10^{3} =

1.59 × 10^{5} =

1.59 × 100,000 =

159,000