Significant Digits and the GRASP Method

If you have 24 eggs, you are far more inclined to state that you have two dozen eggs. If you say you have two dozen eggs, it is very clear that you actually have 24 eggs, because one dozen consists of 12 eggs.

When is the last time you had to measure an amount of something? In the home, measurements are most often made in the kitchen when we’re baking or cooking. Although specific amounts are important in the kitchen, there are times when the measurement is much more crucial.

• • If someone with diabetes injects himself with too much insulin, there can be serious side effects - in some cases, death.

  • A pharmacist who is preparing medicine for her patients must be very careful to measure the exact amounts of the drugs that have been prescribed.

  • Devices used to perform roadside breathalyzer tests must be properly calibrated so they can accurately measure the concentration of alcohol in someone’s body.

  • Water treatment plants must be able to accurately detect chemicals like mercury and lead that would be toxic to those ingesting it.

Items that are purchased in large quantities are often grouped for convenience and ease of counting. As mentioned, it is common to purchase eggs by the dozen, and ‘dozen’ is a common word that is understood as meaning ‘12’ of something.

Can you think of some other common groupings that are used for convenience? Use your own experience, or print or electronic resources to find out how many items are represented by each of the following:

• • A pair of shoes

  • A ream of paper

  • A gross of pencils

Consider the following question: Why do you think these counting words are used instead of the numbers they represent?

Similar to using dozens to count eggs, chemists have devised a familiar quantity that represents the number of particles (such as atoms or molecules) in a sample of a substance. The reason for this is that atoms are incredibly small, and a small amount of a substance can have a very large number of atoms or molecules. For example, in one litre of pure water, there are about 33460000000000000000000000 water molecules (H₂O molecules) in a 1-litre bottle of water! Instead of writing out this very large number, we can say that a 1-litre bottle of water contains 3.346 × 10²⁵ water molecules.

What is Scientific Notation?

• An ordinary quarter contains about 97,700,000,000,000,000,000,000 atoms.

• The average size of an atom is about 0.00000003 centimeters across.

• Very large or very small numbers like these when written in standard notation makes them awkward to work with.

• Scientific notation is a shorthand way of writing such numbers.

• To express any number in scientific notation, write it as the product of a power of ten and a number greater than or equal to 1 but less than 10.

• Example: In scientific notation the number of atoms in a quarter is 9.77x10²², and the size of each atom is 3.0x10^–8 centimeters across.

Translating Scientific Notation to Standard Notation

Translating Standard Notation to Scientific Notation

Significant Digits

Some key things to remember:

• • non-zero digits (numbers that are not a 0) are always significant

  • zeros are only considered significant if they come after a non-zero digit, otherwise they are just placeholders

  • zeros after a non-zero digit (with no decimal) are not significant

  • zeros after a non-zero digit (with a decimal) are significant

Calculations Using Significant Digits

When multiplying or dividing, the final answer must have the same number of significant digits as the original number with the fewest significant digits (we can’t be any more certain than we were to begin with)

When adding or subtracting, the final answer must have the same number of decimal places as the original number with the fewest number of decimal places

Practice

Use the GRASP method to solve the following question:

A rectangular box has a length of 41.2 m and a width of 25.0 m. Find the perimeter and area of the rectangle.