Absolute Value കേവലമൂല്യം
Every point of the number line is named by its distance from 0 – with a minus sign if it is on the left. Throw away the sign and you are left with just the distance. That distance has a name and a symbol: the absolute value |x|.
- Every real number is a point on the number line; positives to the right of 0, negatives to the left.
- The negative of a negative is the number itself: −(−2) = 2 (chapter 7, ന്യൂനസംഖ്യകൾ).
- Squares are never negative: 5² = 25 and (−5)² = 25.
1. Distance from 0
The point 2 is 2 units from 0. The point −2 is on the other side, the same distance away.
- The distance between 0 and a positive number is the number itself.
- The distance between 0 and a negative number is the positive number obtained by removing its minus sign.
2. "Removing the minus sign" in algebra
When we write numbers as letters we cannot see whether x has a minus sign. But removing the minus sign is the same as taking the negative: −(−2) = 2. So if x is negative, the positive number we want is −x. For example, if x = −3 then −x = −(−3) = 3.
3. The definition of |x|
x > 0 ആണെങ്കിൽ |x| = x · x < 0 ആണെങ്കിൽ |x| = −x · x = 0 ആണെങ്കിൽ |x| = 0
|x| is read "the absolute value of x" (x ന്റെ കേവലമൂല്യം). Examples: |5| = 5, |−5| = 5, |⅔| = ⅔, |−⅔| = ⅔, |π| = π, |−π| = π, |0| = 0.
Drag x to either side of 0. The bracket length never goes negative – and the highlighted line shows which of the three cases is being used.
സംഖ്യാരേഖയിൽ 0 സൂചിപ്പിക്കുന്ന ബിന്ദുവും, x എന്ന സംഖ്യ സൂചിപ്പിക്കുന്ന ബിന്ദുവും തമ്മിലുള്ള അകലം |x|.
On the number line, the distance between 0 and x is |x|.
4. A number and its negative: |−x| = |x|
ഏതു സംഖ്യയുടെയും അതിന്റെ ന്യൂനത്തിന്റെയും കേവലമൂല്യം ഒരേ (അധി)സംഖ്യയാണ്.
A number and its negative have the same (positive) absolute value: for any x, |−x| = |x|. For example |3| = 3 = |−3|, and |−3| = |3| = |−(−3)|.
5. Squares: |x|² = x²
x ഏതു സംഖ്യ ആയാലും |x|² = x².
For any number x, |x|² = x²: 5² = 25, (−5)² = 25, and |5|² = |−5|² = 5² = 25.
6. The graph of |x| beyond the textbook
Plot y = |x| for every x: the right half is the line y = x, the left half is y = −x. The graph is a "V" with its tip at 0 – never below the x-axis, because a distance is never negative.
A lift on floor −3 and a lift on floor 3 are both 3 floors from the ground. A temperature of −8 °C is "8 degrees from freezing" just as 8 °C is. Whenever only "how far", not "which way", matters, we use absolute value.
7. Check yourself
- |x| is the distance of x from 0 on the number line.
- |x| = x if x > 0, |x| = −x if x < 0, |0| = 0. So |x| is never negative.
- |−x| = |x| for every x, and |x|² = x².