Comparing expressions without calculating

We compare numbers with =, < and >. We can compare expressions the same way, by their values. 10 + 2 > 7 + 1 (12 is more than 8). 13 − 2 < 4 × 3 (11 is less than 12).

Often you can compare without working anything out. Which is more: 1023 + 125 or 1022 + 128? Think of a story.

Raja had 1023 marbles and got 125 more. Joy had 1022 and got 128 more. Raja started 1 ahead, but Joy got 3 more today. So Joy ends 2 ahead: 1023 + 125 < 1022 + 128. No adding needed!

Another way to see it: write 1023 as 1022 + 1. Then it is 1022 + 126 against 1022 + 128. Same start, so the bigger addition wins.

Now subtraction: 113 − 25 or 112 − 24? Raja had 113 and lost 25; Joy had 112 and lost 24. Raja had 1 more, but also lost 1 more. The two cancel, so they have the same: 113 − 25 = 112 − 24.

Try it yourself: 245 + 289 against 246 + 285, and 273 − 145 against 272 − 144.