Scientific measurements often produce long numbers. Consider the speed of light, which is 3,000,000,000 m/sec, or the mass of a dust particle, which is 0.000000000753 kg. **Scientific notation** is a method of writing long, standard numbers in a condensed format based on powers of 10. Using scientific notation, the speed of light (3,000,000,000 m/sec) is written as:

3.00 x 10^{9} m/sec

The first number (3.00) is called the **coefficient**. The coefficient must be between 1.00 and 9.99. The second number (10^{9}) is called the **base**. The base is always written in exponent form with positive exponents for numbers greater than or equal to 1.00, or negative exponents for numbers less than 1.00. Using scientific notation, the mass of a dust particle weighing 0.000000000753 kg is written as:

7.53 x 10^{-10} kg

Notice that the exponent is a negative number and the units remain the same.

Converting standard numbers to scientific notation requires four steps.

- Write the number.

360000

- Place the decimal after the first non-zero digit.

3.60000

- Write the base. Determine the exponent by counting the number of places the decimal moved. If the decimal moved left, the exponent is positive; if the decimal moved right, the exponent is negative.

3.60000 x 10^{5}

- Create the coefficient by dropping the zeros, and then record the units.

3.6 x 10^{5} kg

Here is an example of converting a standard number less than 1.00 to scientific notation:

- Write the number.

0.00023

- Place the decimal after the first non-zero digit.

00002.3

- Write the base. Determine the exponent by counting the number of places the decimal moved. If the decimal moved left, the exponent is positive; if the decimal moved right, the exponent is negative.

00002.3 x 10^{-4}

- Create the coefficient by dropping the zeros, and then record the units.

2.3 x 10^{-4} kg

Similarly, scientific notation can be converted to standard notation in four steps.

- Write the number.

5.70 x 10^{-3} kg

- Determine the direction the decimal will move. Positive exponents indicate that the decimal will move to the right, and negative exponents indicate that the decimal will move to the left.

10^{-3} indicates that the decimal will move 3 places to the left

- Move the decimal the number of places indicated by the exponent and drop the base.

5.70 becomes 0.00570

- Indicate the units of the standard number.

0.00570 kg

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