# Scientific Notation, Significant Figures, and GRASS NOTE

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• ### Subtracting and adding sig figs

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SPH4U0 SUMMARY NOTE on Scientific Notation and Significant Figures

Scientific Notation Summary:

1. A number is only in scientific notation if the first nonzero digit is followed by a decimal.

Example 400 X 102 Note: Not in scientific notation. This value in scientific notation is 4.00 X 104.

2. The first digit cannot be zero.

Example 0.789 X 103 Note: Not in scientific notation. This value in scientific notation is 7.89 X 102.

3. The exponent is positive if the value of the number (not in scientific notion) is greater than one. The exponent is negative if the value of the number (not in scientific notion) is less than one.

( Move the decimal behind the first nonzero digit. The number of spaces that the decimal is moved to be placed behind the first digit will be the exponent; unless the number (not in scientific notion is less than one)

then the exponent will be negative the number of spaces that the decimal is moved.

4. When a value is in scientific notation, the number of significant digits it has is the number of digits it has before the “X”.

Example 5.003 X 10-3 has 4 digits in front of X, thus it has 4 significant digits.

Significant figures or significant digits are important for accuracy of numbers as each round number has some amount of uncertainty. Note: Another name for significant digits is significant figures.

Significant Figures Rules:

1. All nonzero digits are significant.

2. Zeros are not significant if they are before all of the nonzero digits and thus are simply spaceholders.

3. Note: zeros after nonzero digits are significant. (Look at Sig Fig methods below)

4. Any zeros between two nonzero digits are significant.

5. Numbers obtained from definitions, constants, and conversion values are considered to be exact. Exact numbers have an infinite number of sig figs.

Examples: 25 students The 25 is an exact number and should be thought of as 25.00000… and zeros going on forever.

1 hour = 60 minutes The 60 is an exact number.

1 km = 1000 m The 1000 is an exact number.

Ek = mv2/2 The 2 is an exact number.

6. Always use the number with the least accuracy to determine the number of significant digits in the final answer of any calculation.

(This means for multiplying or dividing use the least number of significant digits.

(This means for adding or subtracting use the least number of decimal places (the least accurate column).

Sig figs Methods:

Method #1 : All zeros after nonzero digits are significant. This method is used because of numbers such as 100 m. If we assumed that this number had one significant figure then it would be extremely limited in accuracy and not much use for calculations as it would really just be an estimate. Method #1 would give us three significant figures for the 100 m, so that calculation would have some degree of accuracy.

Examples: 10 m ( 2 significant figures)

3700 kg ( 4 significant figures)

30 ( 2 significant figures)

Method #2 : Trailing zeros are not significant. It is assumed that the number is rounded to the last nonzero digit. (Which is true if a measuring device has very limited accuracy such as a beaker )

Examples: 200 mL ( 1 significant figure)

23000 kg ( 2 significant figures)

Note: In this course we will use method #1.

Also for either method the number 100 with only 2 significant figures would have be in scientific notation as 1.0 X 102.

Example 44.1349 with 4 sig figs is 44.13, and not 44.1349( 44.135 (44.14 (BAD Do NOT Double Round

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