$ \left[ a , b \right] + \left[ c , d \right] = \left[ a + c , b + d \right] $
$ \left[ a , b \right] - \left[ c , d \right] = \left[ a - d , b - c \right] $
$ \left[ a , b \right] \times \left[ c , d \right] = \left[ \min \left( a \times c , a \times d , b \times c , b \times d \right) , \max \left( a \times c , a \times d , b \times c , b \times d \right) \right] $
If $ 0 \notin \left[ c , d \right] $, then $ \left[ a , b \right] / \left[ c , d \right] = \left[ \min \left( a / c , a / d , b / c , b / d \right) , \max \left( a / c , a / d , b / c , b / d \right) \right] $
If $ a \geq 0 $, then $ \sqrt { \left[ a , b \right] } = \left[ \sqrt a , \sqrt b \right] $
$ a \approx a ^ { \prime } $ to $ n $ decimal significant places after the period if and only if $ a \in \left[ a ^ { \prime } - 5 \times 10 ^ { - \left( n + 1 \right) }, a ^ { \prime } + 5 \times 10 ^ { - \left( n + 1 \right) } \right) $, assuming I'm rounding half up.
If $ a \approx a ^ { \prime } $, $ b \approx b ^ { \prime } $ to $ n + 1 $ decimal significant places after the period, then $ a + b \approx a ^ { \prime } + b ^ { \prime } $, $ a - b \approx a ^ { \prime } - b ^ { \prime } $ to $ n $ decimal significant places after the period because $ a + b $ $ \in \left[ a ^ { \prime } - 5 \times 10 ^ { - \left( \left( n + 1 \right) + 1 \right) } + b ^ { \prime } - 5 \times 10 ^ { - \left( \left( n + 1 \right) + 1 \right) } , a ^ { \prime } + 5 \times 10 ^ { - \left( \left( n + 1 \right) + 1 \right) } + b ^ { \prime } + 5 \times 10 ^ { - \left( \left( n + 1 \right) + 1 \right) } \right) $ $ = \left[ a ^ { \prime } + b ^ { \prime } - 10 ^ { - \left( n + 1 \right) } , a ^ { \prime } + b ^ { \prime } + 10 ^ { - \left( n + 1 \right) } \right) $ $ \subset \left[ a ^ { \prime } + b ^ { \prime } - 5 \times 10 ^ { - \left( n + 1 \right) }, a ^ { \prime } + b ^ { \prime } + 5 \times 10 ^ { - \left( n + 1 \right) } \right) $, $ a - b $ $ \in \left[ a ^ { \prime } - 5 \times 10 ^ { - \left( \left( n + 1 \right) + 1 \right) } - \left( b ^ { \prime } + 5 \times 10 ^ { - \left( \left( n + 1 \right) + 1 \right) } \right) , a ^ { \prime } + 5 \times 10 ^ { - \left( \left( n + 1 \right) + 1 \right) } - \left( b ^ { \prime } - 5 \times 10 ^ { - \left( \left( n + 1 \right) + 1 \right) } \right) \right) $ $ = \left[ a ^ { \prime } - b ^ { \prime } - 10 ^ { - \left( n + 1 \right) } , a ^ { \prime } - b ^ { \prime } + 10 ^ { - \left( n + 1 \right) } \right) $ $ \subset \left[ a ^ { \prime } - b ^ { \prime } - 5 \times 10 ^ { - \left( n + 1 \right) }, a ^ { \prime } - b ^ { \prime } + 5 \times 10 ^ { - \left( n + 1 \right) } \right) $. So, to calculate $ a + b $ to $ n $ decimal significant places after the period, I need to first calculate $ a $, $ b $ to $ n + 1 $ decimal significant places after the period.
I'm still working on multiplication, division, root. My brain is frying...