Practical Estimation Using Scientific Notation Created for CVCA

Practical Estimation Using Scientific Notation Created for CVCA Physics by Dick Heckathorn 3 June 2 K+1

Table of Contents

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

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

2 2 x 10 = 200 2 x 10 = 2000 2 / 10 = 0. 2 2 / 10 = 0. 02 2 / 10 = 0. 002

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

How a number is formed 1 = 1 10 = 1 x 10 1000 = 1 x 10 x 10 100000 = 1 x 10 x 10 1000000= 1 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 10 102 1, 000 1 x 10 103 10, 000 1 x 10 104 100, 000 1 x 10 x 10 105 1, 000 1 x 10 x 10

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

Power of 10 Number 100 1 10 -2 10 -3 10 -4 10 -5 10 -6 How number is formed 1 0. 1 1 / 10 0. 011 / 10 0. 001 1 / 10 0. 0001 1 / 10 0. 00001 1 / 10 / 10 0. 000001 1 / 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

The Number 63847

Here is an example. 63847 Same number in scientific notation 6. 3847 x 104 Coefficient 6. 3847 Power of Ten 104

Now let us examine the steps to change a number to scientific notation format.

Number to be Changed 76348 4)so It was moved 4 places to the Next The determine number is what 10, 000 you (10 did First How move does the decimal this number point To make the number in scientific We must multiply the number left. with times the decimal smaller. point. that compare onetothe non-zero to the digit original isoriginal to(4). the notation as the by ten thesame fourth power left ofwhat the number? decimal number, must wepoint. do to the powers of ten? 7. 6348 x 4 10

Number to be Changed 0. 0003857 4 Next The It was determine number moved is what 10, 000 4 places you (10 to did First How move does the this decimal number point so To make the number in scientific We thus multiply 7. 6348 by )10 -4 with the times the decimal right. larger. point. that compare one non-zero to the digit original is to the notation the same as the original to the minus four power (10 ). left ofwhat the number? decimal number, must wepoint. do to the powers of ten? 3. 857 x 4 10 -

Change the following number to scientific notation format. 724000 The answer is 7. 24 x 105 Click the mouse to check your answer when finished.

Change the following number to scientific notation format. 0. 0000517 The answer is 5. 17 x 10 -5 Click the mouse to check your answer when finished.

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) Finally finish Next First show separate howthe thecalculation. coefficients powers of ten & powers are combined. of ten. 2 (10 4 10 ) (3. 1 x 2. 0) x x (2 + 4) (3. 1 x 2. 0) x 10 6. 2 x 106

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

The Problem Finally finish Next First show separate howthe thecalculation. coefficients powers of ten & powers are combined. of ten.

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 Next Finally combine write thein coefficients properin and scientific the powers notation. ofof ten. (6. 749 x 103) x (2. 63 x 102 ) (6. 749 x 2. 63 ) x 103 + 2 17. 75 x 5 10 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 Created for CVCA Physics by Dick Heckathorn 3 June 2 K+1

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.

For example: 6. 35 x 5 10 In the desired format is: 6 10

For example: 2. 35 x 5 10 In the desired format is: 5 10

What about 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 1 2 1. 5 10 and 10 is 10 ?

1. 5 10 I hope you agree because is the midpoint between 1 2 10 and 10 ?

Our task then is quite simple. What is the value of 101. 5 ?

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 ‘ 2 nd’ 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.

For Example 0 10 - 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 -5 10 - order of magnitude = -5