# If I have 234 multiplied by 24, how many significant figures do I keep on my answer?

Do I keep three or do I keep two?

That is something I don’t understand, because on my Chemistry lab the professor wanted the numbers rounded to the smallest sf.

I don’t get it

What is the right way to do it?

Relevance

Significant digits.....

234 x 24 = 5616 ............ just plain old multiplication

But if the numbers actually had some significance as measurements, then they would be accompanied by an uncertainty. This is why numbers which are measurements MUST include the units of the measurement.

Suppose that they were the length and width given in cm. NOW you can talk about uncertainty and significant digits. Now we can say that "24 cm" is expressed to two significant digits, and that the answer can only be expressed to two significant digits. Therefore, the area is 5600 cm²

234 cm x 24 cm = 5616 cm² .... must be rounded to 5600 cm²

===========

Assume that the uncertainty is 1 cm either side of the expressed measurement.

234 cm +/- 1 cm.... and .... 24 cm +/- 1 cm

235 x 25 = 5875 ........... highest area

234 x 25 = 5616 ........... middle area

233 x 23 = 5395 ........... lowest area

The "answer" contains all the certain digits, plus one uncertain digit. The uncertain digit is in the 100s column, and that is the last significant digit you can write.

Therefore, the answer is 5600 .... to two significant digits (the trailing zeros are place holders).

================= ===================

Propagation of uncertainty

Z = X * Y

σ(Z) = Z * √((σ(X) / X)² + (σ(Y) / Y)²)

σ(Z) = 5616 * √((1 / 24)² + (1 / 234)²)

σ(Z) = 235

Round σ(Z) to one significant digit (200) to express the uncertainty in the answer. Since 200 is given to the hundred's column, then the uncertainty is in the hundreds, and the product is 5600 cm² +/- 200 cm²

Notice the agreement in the uncertainty using propagation of errors with the high and low areas calculated earlier.

• Login to reply the answers
• ...

it depends on what kind of numbers they are

if they are counts or otherwise known to be exact, then report all generated digits

if they are measurements, then use least number of sf

if there is a mix of types, then the measurement limits sf

example:

assuming measurements*

234*24 = 5616 and rounds to 5600

When you get a good response,

• Login to reply the answers