(COC) Significant Digits Lesson

Significant Digits

When we express measured values, we can only list as many digits as we initially measured with our measuring tool plus one estimated digit.  For example, if you use a ruler to measure the length of a stick, you may measure it to be 36.7 cm. You could not express this value as 36.71234 cm because your measuring tool was not precise enough to measure or estimate this many places after the decimal.  

Please watch the videos below to learn more about how to determine the number of significant digits.

Assignment Check:

Dimensional Analysis (Factor-Label Method) for Converting

When converting metric units to other metric units, you can choose the method you would like to use to find the final answer.  However, when using the English system of measurements in which moving the decimal point is not the solution; using dimensional analysis is essential.  

Chemists also use dimensional analysis frequently.  It is a method by which the scientist can keep track of the units of numbers during the mathematical operation.   Units are converted by organizing one or more conversion factors into a logical series that cancels or eliminates all except for the units wanted in the answer. This can be compared to playing dominoes in which you link corresponding units together just as you would string together dominoes.

 

Look at the table below for conversion factors within the English System.

Conversion with the English System

12 in = 1 ft	1 kg = 2.2 lbs	32 oz = 1 qt	1 day = 24 hours
3 ft = 1 yd	16 oz = 1 lb	4 qts = 1 gal	1 hour = 60 mins
1760 yds = 1 mile	2000 lbs = 1 ton	1 L = 1.06 qts	1 min = 60 seconds
1 mile = 1.61 km	1 oz = 28.35 g	1 qt = 2 pints	365 days = 1 yr

Please watch the video below explaining step-by-step how to use the factor-label method to convert between units.  

In summary, it is important to follow the steps below:

1. Place the given quantity as a fraction.
2. Insert the necessary conversion factors as fractions.
3. Cross out units that cancel out.
4. Perform the math calculation

Example:   How many hours are in 18 weeks?

  18 weeks = _______ hours

$\frac{18\: weeks}{1}\times\frac{7\: days}{1 \:week}\times\frac{24\: hours}{1\: day}=3,024\: hours$

Scientific Notation

Can you read this number?

5,973,600,000,000,000,000,000,000

(Earth's Mass in Kilograms)

What about this number?

0.000000000000000000000000000000910938

(Mass of an electron in Kilograms)

Scientists often have to work with very large or very small numbers. It can be too cumbersome to write out these large or small numbers. Scientific notation is a shorthand representation of these very large or very small numbers.  When written in scientific notation, a number is expressed as a decimal number between 1 and 10 that is multiplied by a power of 10.  

To write a large number in scientific notation, we first have to move the decimal point to a number between 1 and 10. Since moving the decimal point changes the value, we have to apply multiplication by the power of 10 which will yield an equivalent value to the original. To figure out the exponent, we just count the number of places we moved the decimal sideways. That number is the exponent for the power of 10.

Let's look at an example. To rewrite 180,000 in scientific notation, we first move the decimal point to the left until we have a number greater than or equal to 1 and less than 10. The decimal point is not written in 180,000 but if it were it would be after the last zero.  If we start moving the decimal sideways one place at a time, we'll get to 1.8 after 5 shifts:

180000.

18000.0

1800.00

180.000

18.0000

1.80000

So now we know both the number (1.8) and the exponent for the power of 10 multiplier that preserves the original value (5).  In scientific notation 180,000 is written as 1.8 x 10⁵.

The process for moving between decimal and scientific notation is the same for small numbers (between 0 and 1), but in this case, the decimal moves to the right, and the exponent will be negative. Consider the small number 0.0004:

0.0004

00.004

000.04

0000.4

00004.

We moved the decimal point sideways until we got the number 4, which is between 1 and 10 as required. It took 4 moves, but they were moves that made the number bigger than the original. So we'll have to multiply by a negative power of 10 to bring the new number back down to the equivalent of the original value. In scientific notation, 0.0004 is written as 4.0 x 10^-4.

Watch this video explaining how to convert large numbers to scientific notation.

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