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0.10Significant Figures

An engineer is designing a car engine, and has been told that the

diameter of the pistons (which are being designed by someone else) is 5 cm.

He knows that 0.02 cm of clearance is required for a piston of this size, so

he designs the cylinder to have an inside diameter of 5.04 cm. Luckily, his

supervisor catches his mistake before the car goes into production. She

explains his error to him, and mentally puts him in the “do not promote”

category.

What was his mistake. The person who told him the pistons were 5 cm

in diameter was wise to the ways of significant figures, as was his boss, who

explained to him that he needed to go back and get a more accurate num-

ber for the diameter of the pistons. That person said “5 cm” rather than

“5.00 cm” specifically to avoid creating the impression that the number was

extremely accurate. In reality, the pistons’ diameter was 5.13 cm. They

would never have fit in the 5.04-cm cylinders.

The number of digits of accuracy in a number is referred to as the

number of significant figures, or “sig figs” for short. As in the example

above, sig figs provide a way of showing the accuracy of a number. In most

cases, the result of a calculation involving several pieces of data can be no

more accurate than the least accurate piece of data. In other words, “garbage

in, garbage out.” Since the 5 cm diameter of the pistons was not very

accurate, the result of the engineer’s calculation, 5.04 cm, was really not as

accurate as he thought. In general, your result should not have more than

the number of sig figs in the least accurate piece of data you started with.

The calculation above should have been done as follows:

5 cm(1 sig fig)

+ 0.04 cm(1 sig fig)

= 5 cm(rounded off to 1 sig fig)

The fact that the final result only has one significant figure then alerts you

to the fact that the result is not very accurate, and would not be appropriate

for use in designing the engine.

Note that the leading zeroes in the number 0.04 do not count as

significant figures, because they are only placeholders. On the other hand, a

number such as 50 cm is ambiguous — the zero could be intended as a

significant figure, or it might just be there as a placeholder. The ambiguity

involving trailing zeroes can be avoided by using scientific notation, in

which 5 x 10

1

cm would imply one sig fig of accuracy, while 5.0 x 10

1

cm

would imply two sig figs.

Self-Check

(a) The following quote is taken from an editorial by Norimitsu Onishi in the

New York Times, August 18, 2002.

Consider Nigeria. Everyone agrees it is Africa’s most populous nation.

But what is its population. The United Nations says 114 million; the

State Department, 120 million. The World Bank says 126.9 million, while

the Central Intelligence Agency puts it at 126,635,626.

What should bother you about this.

Chapter 0Introduction and Review

The various estimates differ by 5 to 10 million. The CIA’s estimate includes a ridiculous number of gratuitous

significant figures. Does the CIA understand that every day, people in are born in, die in, immigrate to, and

emigrate from Nigeria.