Numbers in Science

When scientists measure a quantity, they actually measure two pieces of information-- the value they think they have measured, and the uncertainty. This can be stated as "We measured ten, plus or minus one", and often scientists do use these terms. However, this notation gets cumbersome fast. We need a different strategy. Scientists put only the digits they can reasonably be certain of in their numbers. They might say, for example, that they measured "10." cm. This is actually different from saying that they measured "10" cm. The use of the decimal point indicates that the scientist is sure of both digits to some reasonable degree -- it is 10.something, not 11 or 9, even though rounding both of these numbers to one digit gives 10. The number "10." is said to have two significant digits, or significant figures, the 1 and the 0. The number 1.0 also has two significant digits. So does the number 130, but 10 and 100 only have one "sig fig" as written. Zeros that only hold places are not considered to be significan t. So, how does a scientist indicate that two of the digits in 100 are significant?? We can't put in a decimal point alone to make 100. because that would indicate 3 digits. What should we do? Scientists use scientific notation to handle this problem. Scientific notation makes sure that everything but the first digit of a number is after the decimal place and therefore either certain or not used. Here are some numbers in scientific notation to study:

See the differences? In the first and second example, the zeros are really only place holders. In the third example, the extraneous decimal place is used to mean we are certain of all three digits. In the fourth example the extra zeros (on the right!) are used to indicate that we have extra certainty. In the fifth example, we have finally seen how to represent 100 with exactly two sig figs.

The rules for determining which digits in a measurement are significant are: Every nonzero digit in a recorded measurement is significant m, m and 714 m all have three significant figures. Zeroes appearing between nonzero digits are significant. The measurements 7003 m, m, and m all have four significant figures. Zeroes in front of (before) all nonzero digits are merely placeholders; they are not significant only has two significant figures. Zeroes at the end of the number if a decimal point is present and also zeroes to the right of the decimal are significant. The measurements m, m and all have six significant digits. Zeroes at the end of a measurement and to the left of an omitted decimal point are ambiguous. They are not significant if they are only place holders: 6,000,000 live in New York—the zeroes are just to represent the magnitude of how many people are in N.Y. But the zeroes can be significant if they are the result of precise measurements. A vinculum over the least significant zero is often used. The significant figures in a number in scientific notation is the number of digits in the mantissa. The number 4×10 5 has only one digit in the mantissa, so it has one significant figure ×10 5 has 4 significant figures. Thus the number 1200 which is unclear as to how many significant figures it has is more clearly expressed as 1.200×10 3 as having 4 significant figures or as 1.2×10 3 as having 2.

It is very important to round rather than truncate your results: not You are often instructed to round to so many significant digits or to such and such a level of precision. There are variations, but the standard rule would round anything from $0.50 up to $1.49 all to $1. One variation would round $0.50$ down and $1.50$ up based on the evenness/oddness of destination digit. A common mistake to be avoided is "double rounding," for example, rounding 1.46 first to 1.5 and then to 2.

revise scientific notation = 4 x = 5.1 x = 1 x = 1.25 x 10 6