Question

How do you evaluate `(9.7\cdot 10^ { 6} ) + ( 8.3\cdot 10^ { 5} )`?

Answer 1

As an exact algebraic sum `1.053 xx 10^7`
As a rough approximation: `1.05 xx 10^7`
As a range: `1.05315 xx 10^7 +- 5.65 xx 10^4`

Explanation:

In order to be able to add two values in scientific notation they need to have the same exponent.

To do this we can denormalise the second value to have the same exponent as the first, then renormalise at the end, like this:

`9.7 xx 10^6 + 8.3 xx 10 ^5 = 9.7 xx 10^6 + 0.83 xx 10^6`

`color(white)(9.7 xx 10^6 + 8.3 xx 10 ^5) = 10.53 xx 10^6`

`color(white)(9.7 xx 10^6 + 8.3 xx 10 ^5) = 1.053 xx 10^7`

One consideration remains: Are the given values only guaranteed to be accurate to the given number of significant digits ?

If so, then a rough and ready guide would tell us to only trust digits down to the tenth part of `10^6`, i.e. to a `10^5` resolution. In which case, it would be appropriate to state the sum as `1.05 xx 10^7`.

If we want to be extra precise, note that the maximum possible actual value of the sum is just short of:

`9.75 xx 10^6 + 8.35 xx 10^5 = 9.75 xx 10^6 + 0.835 xx 10^6`

`color(white)(9.75 xx 10^6 + 8.35 xx 10^5) = 10.588 xx 10^6`

`color(white)(9.75 xx 10^6 + 8.35 xx 10^5) = 1.0588 xx 10^7`

while the minimum possible value is:

`9.65 xx 10^6 + 8.25 xx 10^5 = 9.65 xx 10^6 + 0.825 xx 10^6`

`color(white)(9.65 xx 10^6 + 8.25 xx 10^5) = 10.475 xx 10^6`

`color(white)(9.65 xx 10^6 + 8.25 xx 10^5) = 1.0475 xx 10^7`

The average of these two extreme values is:

`1/2(1.0588+1.0475) xx 10^7 = 1.05315 xx 10^7`

which differs from the extreme values by:

`(1.0588-1.05315) xx 10^7 = 0.00565 xx 10^7 = 5.65 xx 10^4`

Hence we could express the value of the sum as:

`1.05315 xx 10^7 +- 5.65 xx 10^4`