1.7 E) Ordering Numbers
We are now going to have a look at ordering number in standard form in either ascending or descending order.
If we are ordering numbers in descending order, we start by with the numbers that have the highest power of 10 because the numbers that have the highest power of 10 will be larger than numbers that have a lower power of 10/ a negative power of 10.
If there were two or more numbers that had the same power of 10, we use the values for A to determine which number is larger; we determine which number is larger in the standard way.
We do the opposite if we are placing a set of numbers that are in standard form in ascending order; we start with the lowest power of 10 and finish with the highest power of 10.
Example 1
Place the following numbers in descending order.
6.3 x 104, 3.4 x 108, 2.1 x 10-3, 9.9 x 1018, 5.4 x 108, 6.55 x 10-6 and 7.23 x 10-3
The first step is to look at the powers of 10 and order them according to the highest power of 10 to the lowest power of 10. If any of the numbers are raised to the same power of 10, we would order them according to their values of A.
The highest value of 10 is 18. Therefore, the largest number out of all of these numbers is 9.9 x 1018.
9.9 x 1018
There are two numbers that are both to the power of 8; 3.4 x 108 and 5.4 x 108. This means that we order them by looking at the values of A. 5.4 is larger than 3.4, which means that 5.4 x 108 is the second largest number and 3.4 x 108 is the third largest number.
9.9 x 1018, 5.4 x 108, 3.4 x 108
The next highest power of 10 is 4, which means that the list becomes:
9.9 x 1018, 5.4 x 108, 3.4 x 108, 6.3 x 104
The next highest power is -3 and there are two numbers that are to the power of -3; 2.1 x 10-3 and 7.23 x 10-3. This means that we need to compare the values for A in each of these numbers. When we do this, we see that 7.23 is larger than 2.1, which means that 7.23 x 10-3 is the fifth largest number and 2.1 x 10-3 is the sixth.
9.9 x 1018, 5.4 x 108, 3.4 x 108, 6.3 x 104, 7.23 x 10-3, 2.1 x 10-3
The last number has the smallest power of 10; it’s power of 10 is -6. Therefore, the numbers in descending order are as follows:
9.9 x 1018, 5.4 x 108, 3.4 x 108, 6.3 x 104, 7.23 x 10-3, 2.1 x 10-3 and 6.55 x 10-6