Writing Numbers in Scientific Notation
There are two steps to writing a large positive number in
scientific notation. Consider the number 1,649,000,000. The first
step is to move the decimal point to the right of the leftmost
digit. Remove all of the zeros on the far right.
The second step is to multiply this number by a power of 10.
To find the power of 10, count the number of places you moved the
decimal point.
So, in scientific notation, 1,649,000,000 is written 1.649 Ã—10^{
9}.
Example 1
Write 3,587,000,000,000 in scientific notation.
Solution
3,587,000,000,000 = 3.587 Ã—10^{ 12} Move the decimal
point 12 places to the left. Multiply by 10^{ 12}.
To write numbers less than 1 in scientific notation, there is
a similar method. Consider the number 0.0000387. The first step
is to move the decimal point to the right of the first nonzero
digit. Remove all of the zeros on the far left.
The second step is to multiply this number by a power of 10.
To find the power of 10, count the number of places you moved the
decimal point. Then use the negative of that number as the
exponent of 10.
In this case, the decimal point was moved 5 places. So, 5
will be used as the exponent of ten. In scientific notation,
0.0000387 is written 3.87 Ã—10^{ 5}.
Example 2
Write 0.000000634 in scientific notation.
Solution
0.000000634 = 6.34 Ã—10^{ 7} Move the decimal point 7
places to the right. Multiply by 10^{ 7}.
Comparing and Ordering Numbers in Scientific Notation
It is easy to compare two numbers that are given in scientific
notation.
Key Idea
To compare two numbers given in scientific notation, first
compare the exponents. The one with the greater exponent will be
greater. If the exponent is the same, compare the two numbers
that are being multiplied by comparing their decimals.
Example 3
Compare 6.23 Ã— 10^{ 14 }and 8.912 Ã— 10^{ 12}
.
Solution
Since the exponent in the first number is greater than the
exponent in the second number, 6.23 Ã— 10^{ 14} is
greater than 8.912 Ã— 10^{ 12}.
Example 4
Which is greater, 5.15 Ã— 10^{ 4} or 6.35 Ã— 10^{
5} ?
Solution
Since 4 is greater than 5, 5.15 Ã— 10^{ 4} is
greater than 6.35 Ã— 10^{ 5} .
Example 5
Compare 3.28 Ã— 10^{ 17} and 4.25 Ã— 10^{ 17}
.
Solution
The exponents are both 17. So, we need to compare the numbers
that are being multiplied. Since 4.25 is greater than 3.28, 4.25
Ã— 10^{ 17} is greater than 3.28 Ã— 10^{ 17} .
