Geometric Measurements

Until now, we have been writing numbers in "decimal notation." Sometimes, especially with large numbers, we need to convert numbers into scientific notation.

To write a number in scientific notation, we write it as a product of a single digit and a power of 10. Here are the steps to writing a number in scientific notation:

1. Write the first non-zero digit of the number times a power of ten--see Exponents and Negative Exponents.
2. Place a decimal point after the single digit number, and put the remaining digits in the same order after the decimal point. If the number is a whole number that ends with zeros, drop the zeros.

Thus, to write 527 in scientific notation:

1. Write the first digit times a power of ten: 500 = 5×10²
   
2. Put the remaining digits in order after a decimal point: 5.27×10²

527 = 5.27×10²

To write 1,108.4 in scientific notation:

1. 1, 000 = 1×10³
2. 1.1084×10³ (Note that there is no decimal point between 8 and 4)

1, 108.4 = 1.1084×10³

To write 0.0963 in scientific notation:

1. 0.09 = 9×10^-2
   
2. 9.63x10^
   

0.0963 = 9.63×10^-2

To write 78,000 in scientific notation:

1. 70, 000 = 7×10⁴
   
2. 7.8×10⁴ (Note that 78,000 is a whole number, so we dropped the zeros)

78, 000 = 7.8×10⁴

To write 15.200 in scientific notation:

1. 10 = 1×10¹
   
2. 1.5200×10¹ (Note that 15.200 is a decimal, so we did not drop the zeros)

15.200 = 1.5200×10¹

Observe: the exponent on "10" corresponds to the number of places that the decimal point has moved--it is positive if the decimal point has moved to the left and negative if it has moved to the right.

One of the trickiest things about scientific notation is remembering the rules for zeros: if a number ends in one or more zeros, do not include the zeros if the number is a whole number, but do include the zeros if the number is a decimal. For example, 820 = 8.2×10² in scientific notation, and 0.820 = 8.20×10^-1 in scientific notation. Zeros in the middle of a number are treated as normal digits.

Scientific notation makes it easy to compare very large (or very small numbers). The number with a larger exponent on "10" is always greater. For example, 6.7103×10¹³ is greater than 9.2×10⁷ and 8.3×10^-5 is greater than 2.3×10^-11.