Measurements and Significant Figures

__Measured Numbers__: Numbers/Amounts that are determined using a measuring tool

(Examples includes: height or weight)

__Significant Figures:__ The “figures”, or digits that are important in measurements and calculations

(Simply speaking, it’s any digit that you measure plus one digit that you estimate.)

Example: The following image shows that it’s around 1.2 cm. It’s the last accurate digit in this picture. However, because they’re not perfect, we need to estimate a bit more past the last accurate digit. The following could range anywhere from 1.20-1.30… as long as it’s less than 1.30

What determines the number of significant figures in a Measurement?

The size of divisions on your measuring device

The size of the object

The difficulty in measuring the object

Which numbers are significant?

To begin, there are two types of numbers as far as significant figures are concerned: zeroes, and nonzero digits (1-9)

ALL nonzero digits are SIGNIFICANT !

Example: 1,234 has four significant figures

Some zeroes are significant, while others aren’t:

Leading zeroes

These are the zeroes in front of a number… they start it off. These zeroes NEVER count.

Example: 0.00001 has 1 significant figure (the 1)

Captive zeroes

These are the zeroes in between nonzero numbers. They ALWAYS count.

Example: 1.0091 has 5 significant figures (they all count)

Trailing zeroes

These are the zeroes at the end of a number. When a decimal point is in place, it counts… otherwise, they will not count.

Example: 1,000 has 1 significant figure (the 1), while 1,000.0 has 5 significant figures (all of them)

Exact numbers are numbers that are…

Obtained when you count objects

Obtained from a defined relationship

**NOT**obtained with measuring tools

Example: **12** eggs in a dozen

__Exact numbers are known with absolute certainty, so they’re viewed as numbers with an infinite number of significant figures__

(2 could be 2.0000000000000000…)

Adding and Subtracting:

The answer has the same number of decimal places as the measurement with the fewest decimal places

Example:

Multiplying and Dividing:

Round the answer to the same number of significant figures as the measurement with the fewest significant figures

Example:

Scientific notations are used for very large numbers, or really small numbers

consists of a number between 1-10 followed by the power of 10

If the number you start with is greater than 1, the exponent will be

__positive__Example: 39923 = 3.9923 x 10^4

If the number you start with is less than 1, the exponent will be

__negative__Example: 0.0052 = 5.2 x 10^-3

Dimensional Analysis is a method that uses the idea that any number can be multiplied by one without changing its value

(Basically, it’s used to go from one unit to another)

Using conversion factors, you could set an equation up and multiply them to determine the new value in a new unit

You would have your starting value, and the conversion rate in a fraction next to it. The key to this method is to have the units cancel out, so that the value is left in the unit you want it in.

Example:

Oftentimes, this means that the unit you want to convert to is on the top, while the unit you’re trying to cancel out is on the bottom.

Exact numbers are obtained by

Counting

Definition

Measured numbers are obtained by

Using a measuring tool

Significant figures are counted when

Nonzero numbers

Captive zeroes

Trailing zeroes WHEN there is a decimal point

Adding/Subtracting with Significant Figures: Round to number with fewest decimal places

Multiplying/Dividing with Significant Figures: Round to number with fewest significant numbers

When using scientific notation, make sure the first number is a number from 1-10 and the exponent is a power of 10

__Measured Numbers__: Numbers/Amounts that are determined using a measuring tool

(Examples includes: height or weight)

__Significant Figures:__ The “figures”, or digits that are important in measurements and calculations

(Simply speaking, it’s any digit that you measure plus one digit that you estimate.)

Example: The following image shows that it’s around 1.2 cm. It’s the last accurate digit in this picture. However, because they’re not perfect, we need to estimate a bit more past the last accurate digit. The following could range anywhere from 1.20-1.30… as long as it’s less than 1.30

What determines the number of significant figures in a Measurement?

The size of divisions on your measuring device

The size of the object

The difficulty in measuring the object

Which numbers are significant?

To begin, there are two types of numbers as far as significant figures are concerned: zeroes, and nonzero digits (1-9)

ALL nonzero digits are SIGNIFICANT !

Example: 1,234 has four significant figures

Some zeroes are significant, while others aren’t:

Leading zeroes

These are the zeroes in front of a number… they start it off. These zeroes NEVER count.

Example: 0.00001 has 1 significant figure (the 1)

Captive zeroes

These are the zeroes in between nonzero numbers. They ALWAYS count.

Example: 1.0091 has 5 significant figures (they all count)

Trailing zeroes

These are the zeroes at the end of a number. When a decimal point is in place, it counts… otherwise, they will not count.

Example: 1,000 has 1 significant figure (the 1), while 1,000.0 has 5 significant figures (all of them)

Exact numbers are numbers that are…

Obtained when you count objects

Obtained from a defined relationship

**NOT**obtained with measuring tools

Example: **12** eggs in a dozen

__Exact numbers are known with absolute certainty, so they’re viewed as numbers with an infinite number of significant figures__

(2 could be 2.0000000000000000…)

Adding and Subtracting:

The answer has the same number of decimal places as the measurement with the fewest decimal places

Example:

Multiplying and Dividing:

Round the answer to the same number of significant figures as the measurement with the fewest significant figures

Example:

Scientific notations are used for very large numbers, or really small numbers

consists of a number between 1-10 followed by the power of 10

If the number you start with is greater than 1, the exponent will be

__positive__Example: 39923 = 3.9923 x 10^4

If the number you start with is less than 1, the exponent will be

__negative__Example: 0.0052 = 5.2 x 10^-3

Dimensional Analysis is a method that uses the idea that any number can be multiplied by one without changing its value

(Basically, it’s used to go from one unit to another)

Using conversion factors, you could set an equation up and multiply them to determine the new value in a new unit

You would have your starting value, and the conversion rate in a fraction next to it. The key to this method is to have the units cancel out, so that the value is left in the unit you want it in.

Example:

Oftentimes, this means that the unit you want to convert to is on the top, while the unit you’re trying to cancel out is on the bottom.

Exact numbers are obtained by

Counting

Definition

Measured numbers are obtained by

Using a measuring tool

Significant figures are counted when

Nonzero numbers

Captive zeroes

Trailing zeroes WHEN there is a decimal point

Adding/Subtracting with Significant Figures: Round to number with fewest decimal places

Multiplying/Dividing with Significant Figures: Round to number with fewest significant numbers

When using scientific notation, make sure the first number is a number from 1-10 and the exponent is a power of 10