Chapter 1

Scientific Notation

1.1 Scientific Notation gives us a convenient way to express very large or very small numbers. Apply the following rules when expressing numbers in Scientific Notation:

1. If a decimal is placed at the end of a number then we will need to “float” that decimal to left until only one nonzero number is written before that decimal point. For Example: We would express a large number such as 214 000 000.0 as 2.14 x 108 in Scientific Notation. Note: I rounded this number off to two places after the decimal point. Students are encouraged to follow their instructors requests for rounding these numbers.

2. If a decimal point is placed before a number then we will need to “float” that decimal to the right until only one nonzero number is written before that decimal point. For Example: We would express a small number such as write 0.000 000 419 as 4.19 x10-7 in Note: Again See your instructor about rounding numbers after the decimal point.  

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William Gary2

LECTURE

Let’s examine the following examples below.

2000 0.1200 0 0x10

1

. 1 1

220 0 0 0x10.

32 0 0 0 0x10

1 1 1

.

Mathematically this number is expressed as: 2000.0 x 100 which is 2000.0 x 1 or 2000.0. Thus, the original number remains unchanged when we raise 10 to the power of zero.

When we move the decimal point one place to the left, the exponent now increases by a factor of 10+1. The number is now written as: 20.000 x 10+2.

When we move the decimal point two places to the left, the exponent now increases by a factor of 10+2. The number is now written as: 20.000 x 10+2.

When we move the decimal point three places to the left, the exponent now increases by a factor of 10+3. The number is now written as: 2.0000 x 10+3.

Illustration 1.1 – Positive Exponents

02000 0x10.

1200 0 0x10

1

.

1 1

220 0 0 0x10.

32 0 0 0 0x10

1 1 1

.

32 0x10.

32 0x10. In Scientific Notation only one nonzero number is written before the decimal point.

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William Gary3

4

0 00456.

00 00456 x10.

10 0 0456 x10

1

.

20 0 0 456 x10

1 1

.

30 0 0 4 456 x10

1 1 1

.

Mathematically this number can be expressed as: 0.00456 x 100 which is 0.00456 x 1 or 0.00456. Thus, the original number remains unchanged when we raise 10 to the power of zero.

When we move the decimal point one place to the right, the exponent now increases by a factor of 10-1. The number is now written as: 00.0456 x 10+2.

When we move the decimal point two places to the right, the exponent now increases by a factor of 10-2. The number is now written as: 000.456 x 10-2.

When we move the decimal point three places to the right, the exponent now increases by a factor of 10-3. The number is now written as: 0004.45 x 10+3.

Illustration 1.2 – Negative Exponents

34 56 x10.

34 56 x10. In Scientific Notation only one nonzero number is written before the decimal point.

10 0 0456 x10

1

.

20 0 0 456 x10

1 1

.

30 0 0 4 456 x10

1 1 1

.

All written materials and visual presentations are the intellectual Property of

William Gary

Good Practices You should always write numbers so that there is no mistake or confusion about where a decimal point is placed. Let the reader know where the decimal point is placed. Although you may be thinking “what’s the point” anyone can read what I wrote. This is not always the case. You should always place a decimal point at the end of a number greater than zero. For Example: 2,456 should be written as 2,456.0 Likewise you should always place a decimal point before numbers less than zero. For Example: .0045 should be written as 0.0045.

Note: Students who are enrolled in AP Physics B/C exams will not receive credit for work or answers they write in Scientific Notation as displayed by your calculator. The correct way to write a number in Scientific Notation is: 6.023 x 1023. The incorrect way to write a number in Scientific Notation is: 6.023E23.

All written materials and visual presentations are the intellectual Property of

William Gary5

All written materials and visual presentations are the intellectual Property of

William Gary6

Practice Problems

1 a. 3500.00 b. 174 000 000.00c. 0.000 000 343 d. 0.541

2 a. 130 000.00 b. 000 000 000 17c. 140 000 000 000.00 d. 17 423.00

3 a. 85 000.00 b. 258 000 000.0c. 985 000 000 000.00 d. 000 000 000 000 000 327

4. a. 589 457 654.00 b. 0.235 667 290c. 0.000 324 451 d.