You will see a few more examples of scientific notation in this handout. Scientific notation is a way of writing very large and very small numbers with a minimum of fuss and without having to count a lot of zeros.

2 × 10^{11}

Numbers in scientific notation have two parts: a
**coefficient** and a **base with an
exponent**.

In this example 2 is the coefficient and 10 is the base. The exponent is 11.

Coefficients in scientific notation are always between 1 and 10. The base is always ten.

In this example 2 is the coefficient and 10 is the base. The exponent is 11.

Coefficients in scientific notation are always between 1 and 10. The base is always ten.

When using a calculator to do calculations with this number you will need
one that can handle scientific notation. You can tell whether your
calculator can do it by looking for a button that says `
e`, `
ee`, or `
exp`. When you use these buttons you are telling the calculator
“times ten to the power of”. Never use any key combination that
results in 10^X since this can cause problems when the calculator follows
the rules of order of operations. Many calculators (and documents that
cannot produce superscripts^{like this}) use an alternate way to
write scientific notation. They would display the number 2 ×
10^{11} as

2E+11

This is read the same way as 2 × 10^{11}: two times ten to the
eleventh power.

Procedure: Put a decimal point after the first non-zero digit. Look at where the decimal point used to be and count how many spaces you jumped over to move it to its new position. If you moved the decimal point to the *left* then the power of ten is positive and equal to the number you counted. If you moved the decimal point to the *right* then the power of ten is negative and equal to the number you counted. Do this in reverse to write numbers expressed using scientific notation as ordinary decimal numbers.

Special Note: numbers with exponents of 1, 0, or -1 are usually written without scientific notation. That is 10 is **not** written 1 × 10^{1} and 4.1 is **not** written 4.1 × 10^{0} and 0.34 is **not** written 3.4 × 10^{-1}.

- 4,000 = 4 × 10
^{3} - 8,548,000 = 8.548 × 10
^{6} - 151,369 = 1.51369 × 10
^{5} - 602,000,000,000,000,000,000,000 = 6.02 × 10
^{23} - 512 = 5.12 × 10
^{2} - 0.051 2 = 5.12 × 10
^{-2} - 0.000 000 000 000 000 000 000 060 2 = 6.02 × 10
^{-23} - 0.000 015 136 9 = 1.51369 × 10
^{-5} - 0.000 008 548 = 8.548 × 10
^{-6} - 0.004 = 4 × 10
^{-3}

Procedure: Multiply or divide the coefficients. Add the exponents if multiplying. Subtract the exponents if dividing. If the coefficient is greater than 10 or less than 1 then change the position of the decimal and adjust the exponent to fix the problem.

- (2.5 × 10
^{2}) x (6.2 × 10^{3}) = 15.5 × 10^{5}--> 1.55 × 10^{6} - (8.90 × 10
^{4}) x (7.77 × 10^{3}) = 69.2 × 10^{7}--> 6.92 × 10^{8} - (3.6 × 10
^{-2}) x (4.8 × 10^{3}) = 17.3 × 10^{1}--> 1.73 × 10^{2} - (2.10 × 10
^{-7}) x (9.01 × 10^{-2}) = 18.9 × 10^{-9}--> 1.89 × 10^{-8} - (1.25 × 10
^{9}) x (5.00 × 10^{12}) = 6.25 × 10^{21} - (2.5 × 10
^{2}) ÷ (6.2 × 10^{3}) = 0.40 × 10^{-1}--> 4.0 × 10^{-2} - (8.90 × 10
^{4}) ÷ (7.77 × 10^{3}) = 1.15 × 10^{1}--> 11.5 - (4.8 × 10
^{-2}) ÷ (3.6 × 10^{3}) = 1.33 × 10^{-5} - (2.10 × 10
^{-2}) ÷ (9.01 × 10^{-7}) = 0.233 × 10^{5}--> 2.33 × 10^{4} - (5.00 × 10
^{9}) ÷ (1.25 × 10^{12}) = 4.00 × 10^{-3}

Procedure: Change the position of the decimal and adjust the exponent of the number that is smaller to match the number that is larger. Then add or subtract: the operation will not affect the exponent at all. Occasionally the coefficient will not be between 1 and 10: make the necessary adjustment.

Special Note: adding or subtracting numbers that differ by 2 or more in the exponent will result in little change to the larger number...unless you are subtracting a large number from a small one.

- (4.9 × 10
^{2}) + (5.8 × 10^{3}) = (0.49 × 10^{3}) + (5.8 × 10^{3}) = 6.29 × 10^{3} - (6.5 × 10
^{-9}) + (2.7 × 10^{-10}) = (6.5 × 10^{-9}) + (0.27 × 10^{-9}) = 6.77 × 10^{-9} - (2.31 × 10
^{7}) + (6.57 × 10^{8}) = (0.231 × 10^{8}) + (6.57 × 10^{8}) = 6.801 × 10^{8} - (5.02 × 10
^{2}) + (5.02 × 10^{8}) = ~5.02 × 10^{8}(5.020 005 02 × 10^{8}) - (4.23 × 10
^{-8}) + (2.78 × 10^{-6}) = (0.0423 × 10^{-6}) + (2.78 × 10^{-6}) = ~2.82 × 10^{-6}(2.8223 × 10^{-6}) - (6.5 × 10
^{5}) – (6.2 × 10^{4}) = (6.5 × 10^{5}) – (0.62 × 10^{5}) = 5.88 × 10^{5} - (2.3 × 10
^{6}) – (8.5 × 10^{5}) = (2.3 × 10^{6}) – (0.85 × 10^{5}) = 1.45 × 10^{6} - (1.54 × 10
^{-3}) – (8.90 × 10^{-4}) = (1.54 × 10^{-3}) – (0.890 × 10^{-3}) = 0.65 × 10^{-3}--> 6.5 × 10^{-4} - (9.5 × 10
^{19}) – (2.5 × 10^{18}) = (9.5 × 10^{195}) – (0.25 × 10^{19}) = 9.25 × 10^{19} - (7.2 × 10
^{1}) – (9.0 × 10^{3}) = (0.072 × 10^{3}) – (9.0 × 10^{3}) = -8.928 × 10^{3}