Question

How do you add `2.113 xx 10^4 + 9.2 xx 10^4` in scientific format?

Answer 1

`(2.113 xx 10^4) + (9.2 xx 10^4) = 1.1313 xx 10^5 ~~ 1.13 xx 10^5`

Explanation:

To add two expressions in scientific format:

• First ensure that they both have the same exponent, by rescaling one of them if necessary.

In our example both of the numbers are expressed with exponent `4`, so we do not need to do anything.

• Add the mantissas.

`2.113 + 9.2 = 11.313`

So:

`(2.113 xx 10^4) + (9.2 xx 10^4) = 11.313 xx 10^4`

• Rescale into scientific format by adjusting the mantissa and exponent to bring the mantissa into the range `[1, 10)` if required.

So in our example:

`11.313 xx 10^4 = 1.1313 xx 10^5`

• Round to an appropriate number of significant digits as appropriate.

If the original numbers were approximations - only expressed to a limited precision - then it is appropriate to give the result to a similar precision.

In our example, the least precise figure is `9.2 xx 10^4`, which is quoted with a precision of `0.1 xx 10^4 = 0.01 xx 10^5`. So we should only quote the answer with two decimal places in the mantissa:

`1.13 xx 10^5`