Approximations & Error

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Rounding Numbers

Discussion

Martha is at the doctor's office for her annual checkup. The diagram below depicts her standing next to a ruler so the doctor can measure her height. Try moving the bottom leveler to measure the very top of Martha's head.

Can you find Martha's exact height?
The doctor's office only records height to the nearest centimeter. What height should the doctor write?

Part (a)

Martha's exact height appears to be about 170.6cm.

Part (b)

Looking at the scale, the centimeter marking which is nearest to the top of Martha's head is 171.

There are many occasions where an approximation is preferred to an exact value.

Approximating the number 1873.8 as 1874, for example, is called rounding to the nearest integer. The digit in the "ones place" -- here, 3 -- is the digit "being rounded off." When rounding, we pay attention to the digit after the one we are rounding to decide what number we will round "to."

We can round to the nearest value of any digit "place," and delete all decimal places after the one we have rounded. We use the symbol ≈ to show that an answer has been approximated.

Rounding rules

There are standardized rules for how to approximate numerical values:

If the digit after the one being rounded off is LESS than 5 (0,1,2,3, or 4), we round down.
If the digit after the one being rounded off is 5 OR MORE (5,6,7,8, or 9), we round up.
1.34∣5→1.35
1.34∣4→1.34
(image)

Checkpoint

Round off 932.14 to:

one decimal place
the nearest integer
the nearest ten

Select the correct option

Discussion

After her appointment, Martha returns to school for chemistry class. She and her lab partner, Luis, are doing a science experiment where they need to measure out 50 milliliters of water.

Martha and Luis both have graduated cylinders with a capacity of 30 mL, so they decide to measure out 25 mL each. Luis's cylinder has markings every 0.25 mL, but since Martha got to class late after her appointment, she had last pick of the equipment: her cylinder only has markings for every 1 mL.

They both measure out their water. Martha reports 25 mL, and Luis reports 25.25 mL.

When Martha and Luis add their water together, should they report the volume as 50 mL or 50.25 mL? Why?

Because Martha’s measurement is only given to the units place (no decimal places), any digits beyond the units place in the final sum aren’t supported by her reading. In addition (and subtraction), the result is reported to the same decimal place as the least precise measurement.

Aligning the decimals gives

25.0+25.25=50.25

but Martha’s 25.0 has zero decimal places, so we keep only the units place in the answer. Rounding 50.25 to the nearest whole milliliter gives

50 mL

The significant figures (sometimes called "significant digits" or just "sig figs") of a given number are its "reliably certain" digits. They ensure that the result of a calculation or measurement isn't presented with more accuracy than is actually known. Not all digits are significant figures, though: some, like a string of zeros at the end of a really big number or the beginning of a really small one, are typically just placeholders.

Significant figures

A significant figure is any digit that is not a leading or trailing zero. To round off to a number of significant figures, count off the specified number of significant figures, then round based off of rounding rules (down if the next digit is less than 5, up if the digit is 5 or more).

NOTE: Unless a question states otherwise, on IB exams, you are expected to give answers to 3 significant figures.

Here are some examples of rounding off to three significant figures:

2.579 → 2.58
43.283 →43.3
0.0239 →0.0239
3,985,782.63 →3,980,000
0.0041990 →0.00420
0.41→0.410

Exercise

Let z=27.00999995.

Round z to four decimal places.
Round z to four significant figures.

Select the correct option

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