of an expression too complicated to be easily calculated.
| 1 + x | lies between | 1 | and | x | , or 1 and | 1 | ; |
|---|---|---|---|---|---|---|---|
| 1 + xx | 1 | xx | x | ||||
| ax + by | lies between | ax | and | by | , | ||
| axx + bbyy | axx | bbyy |
that is, between 1/x and 1/by. And it has been shewn that (a + b)/2 lies between a and b, the denominator being considered as 1 + 1.
- It may also be proved that a fraction such as
| a + b + c + d |
|---|
| p + q + r + s |
always lies among
| a | , | b | , | c | , and | d | , |
|---|---|---|---|---|---|---|---|
| p | q | r | s |
that is, is less than the greatest of them, and greater than the least. Let these fractions be arranged in order of magnitude; that is, let a/p be greater than b/q, b/q be greater than c/r, and c/r greater than d/s. Then by (200)
| is less than | and greater than | |||||||
|---|---|---|---|---|---|---|---|---|
| a + b | a | b | and | c | ||||
| p + q | p | q | r | |||||
| a + b + c | a + b | and | a | c | and | d | ||
| p + q + r | p + q | p | r | s | ||||
| a + b + c + d | a + b + c | and | a | d | ||||
| p + q + r + s | p + q + r | p | s |
whence the proposition is evident.
- It is usual to signify “a is greater than b” by a > b and “a is less than b” by a < b; the opening of V being turned towards the greater quantity. The pupil is recommended to make himself familiar with these signs.