From standard form to scientific notation:

Example: Write the number 300,000,000 in scientific notation. (300,000,000 m/s is the speed of light.)

Solution:

(a) The corresponding number between 1 and 10 is 3.0.

(b) For 300,000,000=300,000,000.0 to "become" 3.0 we need to move the decimal point 8 places to the left.

(b) For 300,000,000=300,000,000.0 to "become" 3.0 we need to move the decimal point 8 places to the left.

Answer: 300,000,000=3.9 \times 10^8

Example: Write the number -23780 in scientific notation.

Solution:

(a) The corresponding number between 1 and 10 is 2.378, or rather -2.378 because we are working with a negative number.

(b) For -23780= -23780.0 to "become" -2.378 we need to move the decimal point 4 places to the left.

(b) For -23780= -23780.0 to "become" -2.378 we need to move the decimal point 4 places to the left.

Answer: -23780=-2.378 \times 10^4

Example: Write the number 0.000000000753 in scientific notation. (0.000000000753 kg is the mass of a dust particle.)

Solution:

(a) The corresponding number between 1 and 10 is 7.53.

(b) For 0.000000000753=0.000\,000\,000\,753 to "become" 7.53 we need to move the decimal point 10 places to the right.

(b) For 0.000000000753=0.000\,000\,000\,753 to "become" 7.53 we need to move the decimal point 10 places to the right.

Answer: -0.000000000753=7.53 \times 10^{-10}

From scientific notation to standard form:

Example: Write the number 3.4 \times 10^{-1} in standard form.

Solution:

(a) A negative exponent means we are talking about a "small" number, that is a number between 0 and 1.

(b) We move the decimal point 1 place to the left.

(b) We move the decimal point 1 place to the left.

Answer: 3.4 \times 10^{-1}=0.34

Example: Write the number 1.03 \times 10^6 in standard form.

Solution:

(a) A positive exponent means we are talking about a "big" number.

(b) We move the decimal point 6 places to the right.

(b) We move the decimal point 6 places to the right.

Answer: 1.03 \times 10^6 =1,030,000

Example: Write the number -2.0 \times 10^{-4} in standard form.

Solution:

(a) A negative exponent means we are talking about a "small" number, that is a number between 0 and 1; the minus sign means we are talking about a negative number. So the result must be a number between -1 and 0.

(b) We move the decimal point 4 places to the left.

(b) We move the decimal point 4 places to the left.

Answer: -2.0 \times 10^{-4} = -0.0002