**Scientific Notation**

Scientific notation is a simple way to write and keep track of large and small numbers without a lot of zeros. It provides a short cut to recording results and doing calculations. The ease of this method is shown below.

**Examples**

100 = (10)(10) = 10 ^{2} = one hundred

1,000 = (10)(10)(10) = 10 ^{3} = one thousand

10,000 = (10)(10)(10)(10) = 10 ^{4} = ten thousand

100,000 = (10)(10)(10)(10)(10) = 10 ^{5} = one hundred thousand

1,000,000 = (10)(10)(10)(10)(10)(10) = 10 ^{6} = one million

1,000,000,000 = (10)(10)(10)(10)(10)(10)(10)(10)(10) = 10 ^{9} = one billion

1/10 = 10 ^{–1} = one tenth

1/100 = 1/(10)(10) = 10 ^{–2} = one hundredth

1/1,000 = 1/(10)(10)(10) = 10 ^{–3} = one thousandth

1/10,000 = 1/(10)(10)(10)(10) = 10 ^{–4} = one ten thousandth

1/1,000,000 = 1/(10)(10)(10)(10)(10)(10) = 10 ^{–6} = one millionth

1/1,000,000,000 = 1/(10)(10)(10)(10)(10)(10)(10)(10)(10) = 10 ^{–9} = one billionth

Since the *English System* was used in the United States for many years with units of inches, feet, yards, miles, cups, quarts, gallons, etc., many people were not comfortable with the metric system until recently. Most students wonder why they ever preferred the older system, when they discover how easy it is to multiply metric units.

**Table 2.3** lists some everyday metric measurements.

**Significant Figures**

Measurements are never exact, but scientists try to record an answer with the least amount of uncertainty. This is why the idea of scientific notation was set up, to standardize measurements with the least uncertainty. The idea of *significant figures* was used in order to write numbers either in whole units or to the highest level of confidence.

**Significant figures** *are the number of digits written after the decimal point to measure a quantity.*

A *counted* significant figure is something that cannot be divided into sub-parts. These are recorded in whole numbers such as 9 chickens, 2 bicycles, or 7 keys. *Defined* significant digits are exact numbers, but not always whole numbers, like 2.54 centimeters equals one inch.

**Examples**

How many significant figures are in the following?

(a) 9.107 (4, zero in the middle is significant)

(b) 401 (3, zero in the middle is significant)

(c) 0.006 (1, leading zeros are never significant)

(d) 800 km (3, zeros are significant in measurements unless otherwise indicated)

(e) 3.002 m (4, zeros in the middle of non-zero digits are significant)

When finding the number of significant figures, the easiest shortcut is to look at the zeros acting as placeholders.

Leading zeros at the beginning of the left-hand side of a number are never significant. You start at the left and count to the right of the decimal point. The measurement 0.096 m has two significant figures. The measurement 13.42 cm has four significant figures. The mass 0.0027 g has two significant figures. (Note: remember to leave off the leading zeros.)

Sandwiched (in the middle) zeros are always significant. The number 26,304 has five significant figures. The measurement 0.000001002 m has four significant figures.

Scientific notation gets rid of guessing and helps to keep track of zeros in very large and very small numbers. If the diameter of the Earth is 10,000,000 m, it is more practical to write 1 × 10 ^{7} m. Or, if the length of a virus is 0.00000004 m, it is easier to write 4 × 10 ^{−8} m.

When multiplying or dividing numbers, the significant digits of the number with the least number of significant digits gives the number of significant digits the answer will have.

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