Decimal form and standard form
Definition
Standard form
A way of writing very large or very small numbers as a value between 1 and 10 multiplied by a power of ten, for example 2400 written as 2.4 x 10 to the power 3.
- Most of the numbers you meet in psychology are in decimal form, meaning ordinary digits with a decimal point, such as a reaction time of 0.42 seconds.
- Standard form rewrites an awkwardly large or small number as a value between 1 and 10 multiplied by a power of ten.
- The power of ten is simply a record of how many places the decimal point has moved.
Key Idea
- A number in standard form always looks like A x 10 to the power n, where A is at least 1 and less than 10.
- A positive power tells you the number is large, and a negative power tells you it is small.
Converting between the two forms
- To put a large number into standard form, move the decimal point to the left until exactly one non-zero digit sits in front of it.
- Count the moves, and that count is your positive power of ten.
- To put a small number into standard form, move the decimal point to the right until exactly one non-zero digit sits in front of it.
- Count the moves, and that count becomes your negative power of ten.
- To go back to decimal form, move the point the same number of places in the opposite direction.
Example
- A stopwatch records 2400 milliseconds, so move the point three places left to reach 2.4, giving 2.4 x 10 to the power 3.
- A second reading is 0.0045 seconds, so move the point three places right to reach 4.5, giving 4.5 x 10 to the power minus 3.
- Reading 4.5 x 10 to the power minus 3 back again, move the point three places left and you recover 0.0045.
Estimating a result before you trust it
- An estimate is a quick, rough answer you work out in your head as a check on the exact one.
- Round every number in the calculation to one significant figure, which usually leaves one non-zero digit followed by zeros.
- Add the rounded numbers up in your head, then compare that estimate with the figure your calculator has given you.
Example
Six pupils revise for 47, 52, 38, 61, 44 and 56 minutes, and you want the mean.
- Estimate: round each value to 50, 50, 40, 60, 40 and 60, which total 300, and 300 divided by 6 is 50 minutes.
- Exact: 47 + 52 + 38 + 61 + 44 + 56 = 298, and 298 divided by 6 = 49.67 minutes to two decimal places.
- The exact answer sits right next to the estimate, so you can trust it.
- Had the calculator shown 496.7, the gap from 50 would have told you a decimal point had slipped.
Spotting a slip with your estimate
- An estimate can tell you an answer is in the wrong region, which is where most marks are actually lost.
- Dividing by the wrong number is the classic slip, and an estimate catches it at once.
Common Mistake
- Do not round part way through a calculation, because those small errors build on each other.
- Do not hand in the estimate as your answer, because the question wants the exact value unless it says otherwise.
Choosing an appropriate number of significant figures
Definition
Significant figures
The digits in a number that carry meaning, counted from the first non-zero digit. Reporting a value to a set number of them shows how precise the result is.
- You count significant figures from the first non-zero digit on the left.
- In 0.00427 the first significant figure is the 4, so the three significant figures are 4, 2 and 7 and the leading zeros do not count.
- To round to a set number of significant figures, look at the digit immediately after the last one you are keeping.
- If that digit is 5 or more, round the last kept figure up.
- If it is less than 5, leave the last kept figure exactly as it is.
Example
Your calculator shows the mean revision time as 49.666666 minutes.
- To three significant figures you keep 4, 9 and 6, and the next digit is 6, which is 5 or more, so the answer is 49.7 minutes.
- To two significant figures you keep 4 and 9, and the next digit is 6, so you round up to 50 minutes.
- To one significant figure you keep only the 4, and the next digit is 9, so you round up to 50 minutes again.
Matching the precision to the data you collected
- Your answer cannot be more precise than the measurements it was built from.
- If you timed revision to the nearest minute, reporting 49.666666 minutes claims a precision you never had, so give 50 minutes.
- If you counted whole people, round to a whole person, because there is no such thing as 7.4 pupils.
- Two or three significant figures is usually the right level of detail for the data you will handle here.
Turning this into marks
- Show the working line as well as the answer, and write the unit next to it.
- When a number of significant figures or decimal places is named, do the rounding, because it carries a mark of its own.
- The mark most often lost comes from copying the full calculator display and stopping there.
Self review
- Write 0.00062 in standard form.
- Write 3.7 x 10 to the power 4 in decimal form.
- Estimate the mean of 19, 31, 42 and 28 by rounding each value to one significant figure first.
- Round 8.3472 to three significant figures.
