Practical Estimation Using Scientific Notation

Created by Dick Heckathorn

Can you quickly approximate an answer to the following problem? How many seconds are there in one year?

Can you quickly determine which is larger? 3684 x 27.36 x 416.3 ? or 63.72 x 273 x 4175 ?

Can you quickly approximate an answer to the following problem?

A review of place holders

57,246.9318 5 ten-thousands 9 tenths 7 thousands 3 hundredths 2 hundreds 1 thousandths 4 tens 8 ten-thousandths 6 ones

A review of what happens to a number when it is multiplied by ten (10).

2 2 x 10 = 20 2 x 10 x10 = 200 2 x 10 x 10 x 10 = 2000 2 / 10 = 0.2 2 / 10 / 10 = 0.02 2 / 10 / 10 / 10 = 0.002

A review of numbers that are different by a power of 10.

Number How number is formed 1 = 1 10 = 1 x 10 100 = 1 x 10 x 10 1000 = 1 x 10 x 10 x 10 10000 = 1 x 10 x 10 x 10 x 10 100000 = 1 x 10 x 10 x 10 x 10 x 10 1000000 = 1 x 10 x 10 x 10 x 10 x 10 x 10

A review of how numbers are changed to power of ten format.

Power Number How number is formed of 10 1 1 x no tens 100 10 1 x 10 101 100 1 x 10 x 10 102 1,000 1 x 10 x 10 x 10 103 10,000 1 x 10 x 10 x 10 x 10 104 100,000 1 x 10 x 10 x 10 x 10 x 10 105 1,000,000 1 x 10 x 10 x 10 x 10 x 10 x 10 106

A review of how numbers are changed to power of ten format.

Power of 10 Number How number is formed 100 1 1 10-1 0.1 1 / 10 10-2 0.01 1 / 10 / 10 10-3 0.001 1 / 10 / 10 / 10 10-4 0.0001 1 / 10 / 10 / 10 / 10 10-5 0.00001 1 / 10 / 10 / 10 / 10 / 10 10-6 0.000001 1 / 10 / 10 / 10 / 10 / 10 x 10

Scientific Notation Is a system in which the numbers are expressed as the product of the COEFFICIENT which is a number that is equal to or greater than one but less than ten and the appropriate POWER OF TEN

Here is an example.

The Number 63847

Same number in scientific notation 63847 Same number in scientific notation 6.3847 x 104 Coefficient Power of Ten

Let us now examine the steps to change a number to scientific notation format.

Number to be Changed 76348

76348 First move the decimal point so that one non-zero digit is to the left of the decimal point. 7.6348

Next determine what you did with the decimal point.

It was moved 4 places to the left. 76348 7.6348 It was moved 4 places to the left.

How does this number compare to the original number?

The number is 10,000 (104) times smaller. 76348 7.6348 The number is 10,000 (104) times smaller.

76348 7.6348 To make the number in scientific notation the same as the original number, what must we do to the powers of ten?

We must multiply the number by ten to the fourth power (4). 76348 7.6348 We must multiply the number by ten to the fourth power (4). 7.6348 x 104

Number to be Changed 0.000385

0.0003857 First move the decimal point so that one non-zero digit is to the left of the decimal point. 3.857

Next determine what you did with the decimal point.

It was moved 4 places to the right. 0.0003857 3.857 It was moved 4 places to the right.

How does this number compare to the original number?

The number is 10,000 (104) times larger. 0.0003857 3.857 The number is 10,000 (104) times larger.

0.0003857 3.857 To make the number in scientific notation the same as the original number, what must we do to the powers of ten?

0.0003857 3.857 We thus multiply 7.6348 by 10 to the minus four power (10-4). 3.857 x 10-4

Check your answer on the next slide when finished. Change the following number to scientific notation format. 724000 Check your answer on the next slide when finished.

724000 The answer is 7.24 x 104

Check your answer on the next slide when finished. Change the following number to scientific notation format. 0.0000517 Check your answer on the next slide when finished.

0.0000517 The answer is 5.17 x 10-5

Let us now solve a problem where the numbers are written in scientific notation.

The Problem (3.1 x 102) x (2.0 x 104)

First separate the coefficients & powers of ten. (3.1 x 102) x (2.0 x 104) First separate the coefficients & powers of ten. (3.1 x 2.0) x (102 x 104)

(3.1 x 102) x (2.0 x 104) (3.1 x 2.0) x (102 x 104) Next show how the powers of ten are combined. (3.1 x 2.0) x 10(2 + 4)

(3.1 x 102) x (2.0 x 104) (3.1 x 2.0) x (102 x 104) (3.1 x 2.0) x 10(2 + 4) Finally finish the calculation. 6.2 x 106

Let us solve another problem where the numbers are written in scientific notation.

The Problem

First separate the coefficients & powers of ten.

Next show how the powers of ten are combined.

Finally finish the calculation.

Let us now solve a problem by first changing the number to scientific notation format and then determine the answer.

The Problem 6749 x 263

First write the number in 6749 x 263 First write the number in scientific notation. (6.749 x 103) x (2.63 x 102)

Next combine the coefficients and powers of ten. (6.749 x 103) x (2.63 x 102) Next combine the coefficients and powers of ten. (6.749 x 2.63 ) x 103 + 2

Next combine the coefficients and the powers of ten. (6.749 x 103) x (2.63 x 102) (6.749 x 2.63 ) x 103 + 2 Next combine the coefficients and the powers of ten. 17.75 x 105

Finally write in proper scientific notation. 6749 x 263 (6.749 x 2.63 ) x 103 + 2 17.75 x 105 Finally write in proper scientific notation. 1.775 x 106

Set up the following problem on a piece of paper showing each step. 0.00463 x 6734 x 27 When finished check your answer on the next slide.

The answer is: 8.42 x 102

Fermi Questions By Dick Heckahtorn

To solve Fermi questions such as how may blades of grass are there inside the fence that surrounds the soccer field, one needs to learn how to round a number to its nearest order of magnitude.

In the desired format is: For example: 6.35 x 105 In the desired format is: 106

In the desired format is: For example: 2.35 x 105 In the desired format is: 105

But what about 4.35 x 105?

4.35 x 105? The answer is: 106 How many of you said 105 ? How can this be?

Since we are dealing only with orders of magnitude, we need to determine the half-way point between one order of magnitude and the next. For example between 101 and 102.

Would you agree that the half way point between 101 and 102 is 101.5 ?

I hope you agree because 101.5 is the midpoint between 101 and 102 ?

Our task then is quite simple. What is the value of 101.5 ?

If you did, you have responded with an answer that most students give. Did you say 50? If you did, you have responded with an answer that most students give. But 50 is an incorrect answer.

You see, we are not dealing with an ordinary number scale. We are dealing with an exponential scale. This is true because each order of magnitude is 10 times larger than the previous.

Without going into a lot of mathematical explanation, perform the following steps on your calculator.

1. Press the ‘2nd’ key 2. Press the ‘LOG’ key 3. Enter 1.5 4. Press the ‘ENTER’ key The answer is ……..

Did you get 31.6227766? You should if you have the calculator set on float. Otherwise you will get a rounded value.

An Example

Examine the following: 100 - order of magnitude = 0 101 - order of magnitude = 1 105 - order of magnitude = 5 10-1 - order of magnitude = -1 10-3 - order of magnitude = -3 10-5 - order of magnitude = -5