Scientific Notation

The number 123 can be written as 100 + 20 + 3 or as (1 × 10²) + (2 × 10¹) + (3 × 1). The digit 1 stands for 1 times 100; the digit 2 stands for 2 times 10; and the digit 3 stands for 3 times 1.

We say these digits have the following place values:

• 1 is in the hundreds place.
• 2 is in the tens place.
• 3 is in the units (ones) place.

Every digit in a decimal number has a place value. The next places to the left of 123 are the thousands place, the ten-thousands place, the hundred-thousands place, the millions place, and so on.

Digits to the right of the decimal point also have place values. For example 0.56 = 0.5 + 0.06 = (5 × 10^-1) + (6 × 10^-2); the digit 5 is in the tenths place, and the digit 6 is in the hundredths place.

Sometimes, using the concept of place value can let you write a very big or small number in a much shorter form. For example:

• 2,300,000,000,000 = 2.3 × 10¹²
• 0.0000000007 = 7 × 10^-10

Because such numbers often occur in scientific calculations, writing a number as the product of a power of 10 and a number greater than or equal to 1 and less than 10 is called scientific notation.

Example 1

Express the number 39,200,000 in scientific notation.

Place a decimal point after the first digit, and count the places after it.

3.9200000   There are 7 places after the decimal.

Drop the ending zeroes and multiply the number by the power of 10 whose exponent matches the number of places you counted.

3.92 × 10⁷

Example 2

Express the number 0.000000000002818 in scientific notation.

Place a decimal point after the first nonzero digit, and count the places between that decimal and the original one.

0.000000000002.818    There are 12 places between the points.

Drop the old decimal point and all the beginning zeroes, and multiply the number by the negative power of 10 whose exponent's absolute value matches the number of places you counted.

2.818 × 10^-12

Try it yourself

1.)  In 2000, there were approximately 281,000,000 people in the United States.  Express this number in scientific notation.

2.)  In the United States, 15,000,000 households use private wells for their water supply.  Write this number in scientific notation.

3.)  The United States has a total of 1.2916 × 10⁷ acres of land reserved for state parks.  Write this number in standard form.

4.)  Write the number in scientific notation:

    0.00000000342

5.)  Write the numbers in standard form:

a)  6.35 × 10^-6

 

b)  2.83 × 10¹²

 

6.)  About how many times larger is 9.0 × 10⁸ than 4.0 × 10⁶?

Answer key

1.)  2.81 × 10⁸ 

2.)  1.5 × 10⁷ 

3.)  12,916,000

4.)  3.42 × 10^-9 

5.)  a)  0.00000635;  b)  2,830,000,000,000

6.)  225