Operations with rational numbers
The idea
Fractions, decimals, whole numbers, and all of their negatives belong to one family: the rational numbers, every number you can write as one integer over another. You already know how to add, subtract, and multiply fractions and how decimal place value works — the new challenge is letting signs ride along during those same operations. A temperature drop, a debt, or a depth below sea level is just a familiar number with a direction attached.
A practical strategy is to pick one outfit and dress every number in it before combining: all decimals or all fractions, whichever divides cleanly. Then read each operation as movement — adding a positive moves right on the number line, adding a negative moves left, and subtracting a number is the same as adding its opposite. The classic trap is panicking at something like 6.2 − (−3): subtracting a negative removes a loss, which is a gain, so it lands you further right, at 9.2.
Worked example
A research submarine starts at −15.5 m relative to the surface, dives another 8 3/4 m to film a reef, then rises 6.2 m to follow a turtle. What is its final position?
- Get every number into the same form first: 8 3/4 = 8.75, so all three values are decimals and can be combined directly.
- Diving deeper means subtracting from the current position: −15.5 − 8.75 = −24.25 m after the dive.
- Rising means adding: −24.25 + 6.2 = −18.05 m, since moving up 6.2 brings the position closer to zero.
- Check with the net change: overall the sub moved −8.75 + 6.2 = −2.55 m from its start, and −15.5 − 2.55 = −18.05 m, which matches.
Answer. The submarine ends at −18.05 m, which is 18.05 m below the surface.
Check your understanding
- Why is subtracting a negative number the same as adding a positive one — can you justify it with money or temperature?
- When would you rather convert everything to fractions instead of decimals, and what kind of numbers force that choice?
- How does the number line help you predict the sign of an answer before you compute anything?
- What changes about the rules you already knew for fractions when negative signs are attached to them?
Build the foundations first
Operations with rational numbers builds on these concepts. If any feel shaky, start there.