Change of Position: Subtraction സ്ഥാനമാറ്റം
In the last section we knew where a point started and how it moved, and found where it finished. Now turn the question round: we know where it starts and where it must reach – how far, and which way, must it move? Answering that for every pair of positions forces us to give a meaning to sums like 4 − (−7).
- Right of O is positive, left is negative; a move right is a positive change, a move left a negative change.
- x + (−y) = x − y (page 125): 5 + (−3) = 2.
- For positive x < y: x − y = −(y − x); and −x − y = −(x + y).
1. How far, and which way?
From 4 to 7 the point must move 3 – but "3" alone is not enough: from 4, a move of 3 to the left reaches 1, not 7. So we must say 3 to the right. From 7 to 4 it is also 3, but to the left. And from 4 to −7?
2. The textbook table (page 128)
| Starting position x | Target y | Change (words) | Change (number) | y − x |
|---|---|---|---|---|
| 4 | 7 | 3 right | 3 | 7 − 4 = 3 |
| 7 | 4 | 3 left | −3 | 4 − 7 = −3 |
| 4 | −7 | 11 left | −11 | −7 − 4 = −11 |
| −7 | 4 | 11 right | 11 | 4 − (−7) = 11 |
| −4 | 7 | 11 right | 11 | 7 − (−4) = 11 |
| 7 | −4 | 11 left | −11 | −4 − 7 = −11 |
| −4 | −7 | 3 left | −3 | (−7) − (−4) = −3 |
| −7 | −4 | 3 right | 3 | (−4) − (−7) = 3 |
The first three rows and the sixth work with what we know: 7 − 4 = 3, 4 − 7 = −3, −7 − 4 = −(7 + 4) = −11. But rows 4, 5, 7 and 8 contain something new: subtracting a negative number. What should 4 − (−7) mean?
3. Subtraction as "how much must be added?"
10 − 6 can be read as a question: how much must be added to 6 to make 10? Since 6 + 4 = 10, the answer is 4. Read 4 − (−7) the same way: how much must be added to −7 to make 4?
7 − (−4): from −4 to 7 is 4 + 7 = 11, so 7 − (−4) = 7 + 4 = 11.
(−7) − (−4): following the same idea, take it as −7 + 4 = 4 − 7 = −3. From −4 to −7 is indeed 3 to the left.
(−4) − (−7): −4 + 7 = 7 − 4 = 3. From −7 to −4 is 3 to the right. Now every row fits: change = target − start.
x ഏതു സംഖ്യയും, y ഏത് അധിസംഖ്യയും ആയാലും
x − (−y) = x + yFor any number x and any positive number y, subtracting a negative number means adding.
4. The negative of a negative
x ഏതു അധിസംഖ്യയായാലും
−(−x) = 0 − (−x) = 0 + x = xThe negative of the negative of a number is the number itself: −(−3) = 3.
5. One rule for all numbers: x − y = −(y − x)
The first rule on page 119 said: for positive x < y, x − y = −(y − x). Does it still hold when x > y? Try x = 10, y = 3.
x, y ഏതു സംഖ്യകളായാലും
x − y = −(y − x)For any numbers x and y. If x < y, y − x is positive and x − y is its negative; if x > y, y − x is negative and x − y is its negative.
6. Try it: change = target − start
Whatever the two positions, the change is always target − start. Rows with a negative start use x − (−y) = x + y.
Negative numbers first appeared for quantities less than zero. Then we used them for opposite directions. To do that we gave new meanings to adding and subtracting. Following on, "the negative of a negative" became a new operation too: taking a negative of a negative brings you back to where you started.
The temperature rises from −7 °C to 4 °C: the change is 4 − (−7) = 11 °C. A diver at −12 m rises to −5 m: the change is −5 − (−12) = 7 m upward. A lift from basement −2 to floor 6 travels 6 − (−2) = 8 floors.
7. Check yourself
- Change of position = target − start, with right positive and left negative.
- x − (−y) = x + y: 4 − (−7) = 11, (−7) − (−4) = −3, (−4) − (−7) = 3.
- −(−x) = x: the negative of a negative is the number itself.
- x − y = −(y − x) for all numbers x, y.