Scientific Notation

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This material was developed with funding from the
National Science Foundation under Grant # DUE 1601612

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What is the average
number of atoms
in a US Quarter (coin)?

Click to reveal the examples

4,840,000,000,000,000,000,000 or 4.84 x 1021

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Introduction to Scientific Notation

Scientific notation (also referred to as scientific form or standard index form) is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. Scientists, mathematicians and engineers use scientific notation.

What is scientific
notation?

0.0000097 lbs. or 9.7 x 10-6 pounds

What is the average
weight of a
grain of sand?

Significant Digits

Integer Digits

Insignificant Digits

Fraction Digits

Any zero in front of the integer digit or after the fraction digits can be dropped or ignored. The Mantissa is the part of a logarithm that follows the decimal point.

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Mantissa

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An integer digit is a whole number
(not a fraction).

When describing numbers, we have common terms called a lexicon to describe the place and purpose of numbers. When describing a value that has a decimal point, the values on the left of the decimal point are called integers digits and the values on the right of the decimal point are called fraction digits. The digits behind the decimal point are also referred to as the mantissa.

Scientific Notation Mathematic Lexicon

The significant figures of a number are digits that carry meaning contributing to its measurement resolution. The leading zeros in an integer or trailing zeros in decimal numbers are not significant.

Match each term to the arrow pointing to the numbers displayed.

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Challenge 1

Challenge 2

Matching Notation Mathematic Lexicon

Challenge 3

BASE

5 is the integer digit,
725 is the fractional digits
and 5.725 is the coefficient;
the coefficient is multiplied by base 10 to the exponent of -4 and is a very small value.

Exponent

Coefficient

Scientific notation has its own format and lexicon. The coefficient consists of a decimal representation of a number with only one integer digit, a decimal point and the fraction integers. The integer is greater than 0 and less than 10. The coefficient is multiplied by base ten to an exponent factor. In very large values, the exponent is a positive number; in very small values, the exponent is a negative value.

Only One Interger Digit

2 is the integer digit,
35 is the fraction digits
and 2.35 is the coefficient;
the coefficient is multiplied by base 10 to the exponent of +7 and is a very large value.

Conventional Notation Mathematic Lexicon

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Drag each term to the appropriate location

Matching Mathematic Lexicon

Integer

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4

Step 4

START

5

Click to
view

Number of hops

6

7

8

9.355 x

Place a decimal point after the last digit on the right.

4

Counter = Exponent
Moving decimal left = Positive

1

Click to
view

Move the decimal point to the left until only one integer is on the left of the decimal point. Count the number of times the decimal point moves.

2

Step 2

To convert very large values to scientific notation:

This count represents the exponent in scientific notation.

10

3

Determining the Scientific Notation
Very Large Value

To complete the conversion to scientific notation write the new number, and multiple it by base ten of the exponent.

Step 1

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Step 3

Determining the Scientific Notation
Very Small Value

Since the decimal point will move to the right until there is one digit with a value greater than zero, the exponent value will be negative.

–

Record the count that the decimal point had to be moved. This count represents the exponent in scientific notation.

Counter = Exponent
Moving decimal right = Negative

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–7

2.35 x

To convert very small numbers to scientific notation:

This count represents the exponent in scientific notation.

Click to add
decimal point

0

<

Click to highlight
decimal point

Decimal Point Moving Left vs Exponent

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8

Decimal
Point

5.71 x 10

-6

9.425 x 10

-5

-1

-3

>

Decimal Point Moving Right vs Exponent

-4

-2

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Click to move decimal point left

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Out of range

Challenge 1:
Converting Very Large Values

blank

2.72 x 10

Click to move decimal to create the correct coefficient and exponent

9

9

SUBMIT

Challenge 2:
Converting Very Large Values

10

5.0725 x 10

11

13

12

13

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18

18

Challenge 3:
Converting Very Large Values

14

15

1.635 x 10

16

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17

6.75 x 10

Challenge 1:
Converting Very Small Values

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-10

-12

Challenge 2:
Converting Very Small Values

-11

-7

4.252 x 10

-8

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-9

5.3333 x 10

-14

Challenge 3:
Converting Very Small Values

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-13

To convert scientific notation to long form:

Hold each digit place with a zero.

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Place the decimal point.

11

2.35 x 10

Scientific Notation to Conventional Long Form
Very Large Value

Move the decimal point to the right by the number shown in the exponent.

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9.72 x 10

Scientific Notation to Conventional Long Form
Very Small Value

-9

Add a zero in front of the decimal.

-11

Move the decimal point to the left by the exponent value.

5.75 x 109 =

2 x 107 =

7.82 x 1012 =

Enter the long form

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2.17 x 1013 =

8.7 x 108 =

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Mastering Converting From Scientific Notation

4.4295 x 1010 =

4.42 x 10-13 =

2.17 x 10-11 =

2.2268 x 10-5 =

2.354 x 10-8 =

7.82 x 10-12 =

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8.7 x 10-10 =

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x 10

6,320,000,000,000 =

132,000,000 =

x 10

x 10

44,200,000,000 =

x 10

22,750,000,000,000 =

Enter the Coefficient

x 10

x 10

95,700,000 =

5,750,000,000 =

Enter the Exponent

0.0000000000625 =

x 10

x 10

x 10

x 10

x 10

0.00000725 =

0.0000000001255 =

0.00000000333 =

0.00000000000225 =

0.0000875 =

x 10

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