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You will learn the basics of scientific notation in this activity. Scientific notation is a way of writing very large and very small numbers with a minimum of fuss and without having to count a lot of zeros.

In science we are often forced to use numbers that are inconveniently large or small. For example, the number of stars in the Andromeda Galaxy is

200,000,000,000 or two hundred billion stars

Another example is the mass of a single He (helium) atom:

0.000 000 000 000 000 000 000 000 006 649 kg

Because these numbers are such a pain to write out, and more importantly, so difficult to do calculations with we use scientific notation. Scientific notation is like an abbreviation for numbers.

The number 200,000,000,000 can be re-written as the product of 2
and 100,000,000,000 (2 × 100,000,000,000). Conveniently, the
number 100,000,000,000 can be re-written as a power of ten:
10^{11}. (Count the zeros: there are 11 of them.) To write
200,000,000,000 in scientific notation you write the two factors
together like this:

2 × 10^{11} stars

Numbers in scientific notation have two parts: a
**coefficient** and a **base with an
exponent**.

In this example 2 is the coefficient and 10 is the base. The exponent is 11.

Coefficients in scientific notation are always between 1 and 10. The base is always ten.

In this example 2 is the coefficient and 10 is the base. The exponent is 11.

Coefficients in scientific notation are always between 1 and 10. The base is always ten.

When using a calculator to do calculations with this number you
will need one that can handle scientific notation. You can tell
whether your calculator can do it by looking for a button that
says `
e`, `
ee`, or `
exp`. When you use these buttons you are telling the
calculator “times ten to the power of”. Never use any
key combination that results in 10^X since this can cause problems
when the calculator follows the rules of order of operations. Many
calculators (and documents that cannot produce
superscripts^{like this}) use an alternate way to write
scientific notation. They would display the number 2 ×
10^{11} as

2e+11

This is read the same way as 2 × 10^{11}: two times
ten to the eleventh power.

The mass of the He atom is written this way using scientific notation:

6.649 × 10^{-27} kg or 6.649e-27 kg

Here is how to go about changing a number from normal notation to scientific notation.

Write 6,750,000 using scientific notation- Find the
**coefficient**: the number between 1 and 10 that is the first part of a number expressed in scientification notation: 6,750,000. The coefficient does not include any zeros after the last non-zero digit and in this example it is 6.75. - Place a new decimal point after the first non-zero digit: 6.750000
- Count the number of digits to the right of the decimal in its new position: there are six digits.
- The number you just counted is the
**exponent**of the**base**so the base (10) has the exponent 6 which is written 10^{6}. - Re-write the number using scientific notation: 6.75 ×
10
^{6}

Similarly a number like 1.1 millionths of a second (0.000 001 1 s)
can be written as 1.1 × 10^{-6} s. The difference is
that you count digits to the left of the new decimal position.
When you count to the left the exponent of the base will be
negative. Negative exponents indicate that a number is a fraction:
that is, that the number is less than one. Be careful not to count
the zero in front of the old decimal point!

Examples2 × 10^{3}= 2,000 4.1 × 10^{–5}= 0.000041 2.9 × 10^{14}= 290,000,000,000,000

Write the following numbers in decimal form.

- 2.3 × 10
^{4}= __________ - 4.71 × 10
^{9}= __________ - 3.14 × 10
^{–3}= __________ - 1.2 × 10
^{–1}= __________

- 1.805 × 10
^{0}= __________ - 6.02 × 10
^{23}= __________ - 1.602 × 10
^{–19}= __________ - 8.21 × 10
^{–2}= __________

Examples602,200,000,000,000,000,000,000 = 6.022 × 10^{23}0.00132 = 1.32 × 10^{–3}0.0193 = 1.93 × 10^{–2}

Write the following numbers in scientific notation.

- 0.068 0 = __________
- 0.005 9 = __________
- 502,000 = __________
- 137,000,000,000 = __________

- 14,258 = __________
- 1,580 = __________
- 1,002,010 = __________
- 0.000 002 5 = __________

Write the following numbers so that they have the exponent of 10 shown. For example, 1.2 x 10^{11} =
0.012 x 10^{13} = 120. x 10^{9}

- 2.7 x 10
^{1}=

__________ x 10^{3}

__________ x 10^{–1}

- 3.9 x 10
^{4}=

__________ x 10^{6}

__________ x 10^{2} - 4.2 x 10
^{13}=

__________ x 10^{14}

__________ x 10^{12}

- 12.7 x 10
^{6}=

__________ x 10^{7}

__________ x 10^{4} - 1.45 x 10
^{–11}=

__________ x 10^{–13}

__________ x 10^{–9} - 1.37 x 10
^{–5}=

__________ x 10^{–7}

__________ x 10^{–4}

Now you should know how to write numbers using scientific notation. If you do not yet understand any of the exercises on the previous page, please ask for help before going on.

Numbers written in scientific notation are easy to multiply and divide We’ll start with some problems using just powers of ten.

When you multiply numbers with the same base you just add the
exponents:

x^{a} × x^{b} = x^{a + b}.

When you divide numbers with the same base you just subtract the exponents:

x^{a} ÷ x^{b} = x^{a – b}.

x

When you divide numbers with the same base you just subtract the exponents:

x

Multiplication Examples10^{2}× 10^{4}= 10^{2 + 4}= 10^{6}10^{14}× 10^{–6}= 10^{14 + (-6)}= 10^{8}Division Examples10^{2}÷ 10^{4}= 10^{2 – 4}= 10^{–2}10^{14}÷ 10^{–6}= 10^{14 – (-6)}= 10^{20}10^{6}—— = 10^{6 – 4}= 10^{2}10^{4}

- 10
^{3}× 10^{–9}= __________ - 10
^{5}÷ 10^{4}= __________ - 10
^{–7}× 10^{9}= __________ - 10
^{7}÷ 10^{–4}= __________ - 10
^{3}÷ 10^{1}= __________

- 10
^{3}× 10^{8}= __________ - 10
^{–4}÷ 10^{3}= __________ - 10
^{6}× 10^{3}= __________ - 10
^{23}× 10^{14}= __________ - 10
^{–1}÷ 10^{–16}= __________

When multiplying, multiply the numbers and add the exponents. When dividing, divide the numbers and subtract the exponents. Finally, once you have an answer in scientific notation change the answer so that the number multiplied by a power of 10 is between 1 and 10:

Multiplication and Division Examples(2 x 10^{2}) × (6 x 10^{3}) = (6 x 10^{3}) ÷ (2 x 10^{2}) = 12 x 10^{5}= 1.2 x 10^{6}3 x 10^{1}(2.7 x 10^{9}) × (4.2 x 10^{3}) = (2.16 x 10^{2}) ÷ (1.2 x 10^{6}) = 11.34 x 10^{12}= 1.134 x 10^{13}1.8 x 10^{–4}(5.6 x 10^{–9}) × (2.3 x 10^{3}) = (4.5 x 10^{2}) ÷ (9.0 x 10^{–5}) = 12.88 x 10^{–6}= 1.288 x 10^{–5}0.50 x 10^{7}= 5.0 x 10^{6}

Multiply or divide as indicated. Simplify all answers so that the number multiplied by a power of 10 is between 1 and 10.

- (2.5 x 10
^{–1}) × (5.0 x 10^{2}) = __________ - (1.80 x 10
^{2}) ÷ (9.0 x 10^{–1}) = __________ - (3.2 x 10
^{13}) × (4.0 x 10^{8}) = __________ - (4.2 x 10
^{23}) ÷ (7.0 x 10^{31}) = __________ - (9.0 x 10
^{3}) × (2.1 x 10^{–9}) = __________ - (3.6 x 10
^{55}) ÷ (6.0 x 10^{42}) = __________

- (5.0 x 10
^{-15}) × (11 x 10^{4}) = __________ - (2.4 x 10
^{-4}) ÷ (4.0 x 10^{3}) = __________ - (8.0 x 10
^{67}) × (4.0 x 10^{34}) = __________ - (3.6 x 10
^{18}) ÷ (1.2 x 10^{6}) = __________ - (4.8 x 10
^{13}) × (8.0 x 10^{15}) = __________ - (9.0 x 10
^{42}) ÷ (3.0 x 10^{24}) = __________

**Do this word problem, too!**A box contains 1.81 x 10^{24}atoms. One third of them are carbon, the other two thirds are oxygen. How many carbon atoms are in the box? How many oxygen atoms?

Bonus: what chemical compound might this be?

509 + 71 ------ 580

**In order to add or subtract two numbers in scientific notation both numbers must be expressed using the same power of ten**. Otherwise, the digits with the same place value will not be lined up and you will end up with a wrong answer. For example, it is clear that when adding 509 and 71 you should line up the 7 under the 0 and the 1 under the nine:

(5.09 × 10^{2}) + (7.1 × 10^{1}) (5.09 × 10^{2}) = (7.1 × 10^{1}) --------------- and this is wrong! ==> 12.19 × 10^{3}

However, when the same two numbers are written in scientific notation our usual instinct to line numbers up on the decimal point is betrayed (see at right).

This result is clearly incorrect because the answer we know is correct (580) is written as 5.80 × 10^{2} in scientific notation. The problem is that we did not line up the numbers based on their place values. Instead we lined them up on the decimal points. When using scientific notation the decimal point alone is not enough of a guide to the value of the digits in the number. The exponent communicates the correct place value of each digit based on the following rule: **the place value of the digit to the left of the decimal point is given by the exponent.**

Thousands | Hundreds | Tens | Ones | Tenths | Hundredths |

10^{3} | 10^{2} | 10^{1} | 10^{0} | 10^{-1} | 10^{-2} |

We normally think about place value based on names: ones, tens, hundreds or tenths, hundredths, thousandths. Place value can be usefully abbreviated by using the powers of ten that are used when writing a number using scientific notation. See the table at right. The '5' in '509' is in the hundreds place or, as we may now call it, the 10^{2} place. So when we write it in scientific notation with the decimal to the right of the 5 it becomes 5.09 × 10^{2}. We could write it in other ways, too. For example, if the decimal were placed after the '0', which is in the tens place, then it becomes 50.9 × 10^{1}. The number has the same value but we have just re-written it with a new power of ten. The power of ten we use just depends on the place value of the digit just to the left of the decimal.

This discussion suggests a way in which we might set up the arithmetic for adding and subtracting numbers expressed in scientific notation. Take a look at the following example.

5.4 × 10Write the digits in a chart with each digit in the column that correctly gives its place value (see the table at left).^{12}+ 2.3 × 10^{11}

10^{12} | 10^{11} | 10^{10} |

5 | 4 | 0 |

+ | 2 | 3 |

5 | 6 | 3 |

Now that we have lined up the digits correctly and added the two numbers we have only to re-write the answer to give each digit its proper place value. By putting the decimal to the right of the '5' we can look at the top of the chart to see that the correct power of ten to give its place value is 12. So the answer is 5.63 × 10^{12}. Of course, another way to do the problem is to adjust one number to have the same power of ten as the other number. Then it is simply a matter of lining up the decimals to add or subtract. Worked out in this way the problem looks like this:

(5.4 x 10^{12}) + (2.3 x 10^{11}) = (5.4 x 10^{12}) + (0.23 x 10^{12}) = 5.63 x 10^{12}

Addition and Subtraction Example(3 x 10^{4}) + (2 x 10^{3}) = (3 x 10^{4}) + (0.2 x 10^{4}) = 3.2 x 10^{4}= 32,000

Add or subtract. Express all answers in scientific notation.

- 1.2 x 10
^{3}+ 2.4 x 10^{5}= __________ - 2.7 x 10
^{5}– 1.3 x 10^{4}= __________ - 3.9 x 10
^{–4}+ 4.7 x 10^{–5}= __________ - 4.2 x 10
^{45}+ 6.82 x 10^{46}= __________

- 1.27 x 10
^{6}– 3.11 x 10^{5}= __________ - 1.45 x 10
^{9}– 9.12 x 10^{7}= __________ - 1.33 x 10
^{–5}+ 1.49 x 10^{–3}= __________ - 4.97 x 10
^{–6}– 3.2 x 10^{–7}= __________

Do as many of the following problems as you need to do to get good at the arithmetic. They have all been written to be done with no help at all from a calculator. As an additional exercise you might try checking your mental arithmetic using the calculator’s scientific notation function. But be careful! Some of the problems will not be possible on the calculator because the numbers are so large the calculator can’t handle them!

- ( 2 × 10
^{ 7 }) X ( 4 × 10^{ 5 }) _______ - ( 5 × 10
^{ 8 }) ÷ ( 7 × 10^{ 4 }) _______ - ( 6 × 10
^{ 4 }) X ( 5 × 10^{ 9 }) _______ - ( 3 × 10
^{ 9 }) ÷ ( 2 × 10^{ 12 }) _______ - ( 4 × 10
^{ 12 }) X ( 9 × 10^{ 7 }) _______ - ( 8 × 10
^{ 16 }) ÷ ( 8 × 10^{ 17 }) _______ - ( 9 × 10
^{ 25 }) X ( 2 × 10^{ 23 }) _______ - ( 2 × 10
^{ 42 }) ÷ ( 1 × 10^{ 21 }) _______ - ( 1 × 10
^{ 85 }) X ( 4 × 10^{ 15 }) _______ - ( 4 × 10
^{ 99 }) ÷ ( 5 × 10^{ 26 }) _______ - ( 5 × 10
^{ 104 }) X ( 2 × 10^{ 42 }) _______ - ( 3 × 10
^{ 112 }) ÷ ( 7 × 10^{ 57 }) _______ - ( 6 × 10
^{ 420 }) X ( 8 × 10^{ 125 }) _______ - ( 7 × 10
^{ 987 }) ÷ ( 7 × 10^{ 729 }) _______ - ( 45 × 10
^{ 9 }) X ( 2 × 10^{ 5 }) _______ - ( 52 × 10
^{ 5 }) ÷ ( 3 × 10^{ 7 }) _______ - ( 125 × 10
^{ 7 }) X ( 4 × 10^{ 9 }) _______ - ( 3 × 10
^{ -5 }) ÷ ( 7 × 10^{ -3 }) _______ - ( 8 × 10
^{ -6 }) X ( 8 × 10^{ -9 }) _______ - ( 4 × 10
^{ -17 }) ÷ ( 7 × 10^{ -25 }) _______ - ( 5 × 10
^{ -12 }) X ( 1 × 10^{ -5 }) _______ - ( 2 × 10
^{ -42 }) ÷ ( 4 × 10^{ -16 }) _______ - ( 7 × 10
^{ -12 }) X ( 5 × 10^{ -89 }) _______ - ( 8 × 10
^{ -45 }) ÷ ( 8 × 10^{ -15 }) _______ - ( 5 × 10
^{ -29 }) X ( 6 × 10^{ 60 }) _______ - ( 4 × 10
^{ -15 }) ÷ ( 5 × 10^{ 42 }) _______ - ( 6 × 10
^{ -32 }) X ( 9 × 10^{ 24 }) _______ - ( 9 × 10
^{ -9 }) ÷ ( 7 × 10^{ 20 }) _______

- ( 5 × 10
^{ 5 }) + ( 4 × 10^{ 6 }) _______ - ( 8 × 10
^{ 6 }) – ( 9 × 10^{ 5 }) _______ - ( 2 × 10
^{ 7 }) + ( 5 × 10^{ 8 }) _______ - ( 7 × 10
^{ 5 }) – ( 8 × 10^{ 4 }) _______ - ( 3 × 10
^{ 4 }) + ( 7 × 10^{ 6 }) _______ - ( 4 × 10
^{ 8 }) – ( 6 × 10^{ 7 }) _______ - ( 6 × 10
^{ 10 }) + ( 9 × 10^{ 7 }) _______ - ( 9 × 10
^{ 12 }) – ( 1 × 10^{ 11 }) _______ - ( 8 × 10
^{ -32 }) + ( 9 × 10^{ -33 }) _______ - ( 3 × 10
^{ -25 }) – ( 4 × 10^{ -27 }) _______ - ( 7 × 10
^{ -65 }) + ( 8 × 10^{ -66 }) _______ - ( 2 × 10
^{ -99 }) – ( 3 × 10^{ -101 }) _______ - ( 1 × 10
^{ -542 }) + ( 2 × 10^{ -543 }) _______ - ( 42 × 10
^{ -15 }) – ( 52 × 10^{ -16 }) _______ - ( 67 × 10
^{ -24 }) + ( 77 × 10^{ -26 }) _______ - ( 34 × 10
^{ -18 }) – ( 44 × 10^{ -20 }) _______ - ( 1 × 10
^{ 15 }) + ( 3 × 10^{ 17 }) _______ - ( 4 × 10
^{ 28 }) – ( 7 × 10^{ 26 }) _______ - ( 28 × 10
^{ 42 }) + ( 42 × 10^{ 59 }) _______ - ( 42 × 10
^{ 52 }) – ( 52 × 10^{ 51 }) _______ - ( 19 × 10
^{ 72 }) + ( 24 × 10^{ 70 }) _______ - ( 63 × 10
^{ 102 }) – ( 37 × 10^{ 100 }) _______ - ( 9 × 10
^{ -14 }) + ( 8 × 10^{ -15 }) _______ - ( 6 × 10
^{ -12 }) – ( 7 × 10^{ -13 }) _______ - ( 5 × 10
^{ -21 }) + ( 6 × 10^{ -22 }) _______ - ( 4 × 10
^{ -36 }) – ( 5 × 10^{ -38 }) _______ - ( 19 × 10
^{ -32 }) + ( 29 × 10^{ -34 }) _______ - ( 84 × 10
^{ -987 }) – ( 94 × 10^{ -988 }) _______

Do these problems as homework.

- Express the following numbers in
scientific notation:

(a) 0.000000027 (b) 356 (c) 47,764 (d) 0.096 - Express the following numbers as
decimals:

(a) 1.52 × 10^{-2}(b) 7.78 × 10^{-8}(c) 6.02 × 10^{23} - Express the answers to the following
calculations in scientific notation:

(a) 145.75 + (2.3 × 10^{-1})

(b) 79,500 ÷ (2.5 × 10^{2})

(c) (7.0 × 10^{-3}) – (8.0 × 10^{-4})

(d) (1.0 × 10^{4}) × (9.9 × 10^{6}) - Express the answers to the following
calculations in scientific notation:

(a) 0.0095 + (8.53 × 10^{-3})

(b) 653 ÷ (5.75 × 10^{-8})

(c) 850,000 – (9.0 × 10^{5})

(d) (3.6 × 10^{-4}) × (3.6 × 10^{6}) - The surface area and average depth of the Pacific Ocean are
1.8 × 10
^{14}m^{2}and 3.9 × 10^{ 3}m, respectively. Calculate the volume of water in the ocean in cubic meters. (*Hint*: the volume of an object with straight sides is L × W × H. The L × W is really the area of the base. Simplify the problem by pretending the ocean has a simple 3-D shape.)

- The US national debt is now around $8.819
× 10
^{12}. The population of the US is about 3.02 × 10^{8}. If everyone in the country were to pay an equal share of the debt how much would each person owe? - The Mole is a unit used by chemists to
count atoms and molecules. Think of it as the Chemist’s
Dozen. It is defined as the number of carbon-12 atoms which have a
mass of exactly 12 g. The Mole is therefore a very large number:
6.02 × 10
^{23}. Atoms are unbelievably tiny. If there are 6.02 × 10^{23}objects per mole then multiplying a number of moles by this number will give you the total number of objects. Try it with dozens first.- How many kittens are there in 3 dozen?
- How many water molecules are in 7 dozen?
- How many oxygen molecules are there in 5 moles?
- How many water molecules are there in 3.4 moles?
- How many sodium atoms are there in
1.0 × 10
^{-3}moles? - How many nitrogen atoms are there in
1.0 × 10
^{-6}moles of nitrogen molecules, each of which contains two atoms (N_{2})?