Page 51 - ArithBook5thEd ~ BCC

“Left is Less” on the number line:

M < N if and only if M is left of N

M < N M < N

| | | | | | | | | | |
−5 −4 −3 −2 −1 0 1 2 3 4 5

With this rule, we see that −4is less than −2

−4 < −2.
√ √
Recall that 2is less than 3, and that both are greater than 1 but less than 2. (If you’ve forgotten
why, review the discussion in Section 1.4.3.)

√ √
2 3

| | | | |
−2 −1 0 1 2
In symbols,
√ √
1 < 2 < 3 < 2.
Do these relationships remain true if all the signs are changed? Onthe number line, 1 is left of 2, but,
on the negative side, −2is left of −1. Since “left is less,” it follows that −2 < −1. For the same
√ √
reason, − 3 < − 2.
√ √
- 3 - 2

| | | | |
−2 −1 0 1 2
In symbols,
√ √
−2 < − 3 < − 2 < −1.
The mirror analogy helps here: when viewed in a mirror, your right hand‘becomes’ a left hand – the
thumb switches from one side to the other. Putting everything on one number line,

√ √ √ √
- 3 - 2 2 3

| | | | |
−2 −1 0 1 2
we see a chain of inequalities, mirror-symmetric (0 acts as the mirror), except for the presence of negative
signs on the left:
√ √ √ √
−2 < − 3 < − 2 < −1and 1 < 2 < 3 < 2.
If you prefer a rule of thumb to a visual aid, just remember this:

Page 51