Integers Subtraction

imagine line segment of length b left end labeled , right end labeled c. starting a, takes b steps right reach c. movement right modeled mathematically addition:

a + b = c.

from c, takes b steps left a. movement left modeled subtraction:

c − b = a.

now, line segment labeled numbers 1, 2, , 3. position 3, takes no steps left stay @ 3, 3 − 0 = 3. takes 2 steps left position 1, 3 − 2 = 1. picture inadequate describe happen after going 3 steps left of position 3. represent such operation, line must extended.

to subtract arbitrary natural numbers, 1 begins line containing every natural number (0, 1, 2, 3, 4, 5, 6, ...). 3, takes 3 steps left 0, 3 − 3 = 0. 3 − 4 still invalid since again leaves line. natural numbers not useful context subtraction.

the solution consider integer number line (..., −3, −2, −1, 0, 1, 2, 3, ...). 3, takes 4 steps left −1:

3 − 4 = −1.