Summary & Quick Quiz രേഖീയസംഖ്യകൾ
Every rule of the chapter on one page, the mistakes to avoid, flashcards, a scored quiz and your study checklist.
1. The chapter in one picture
|−4| = 4 is the distance of −4 from 0. The distance between −4 and 3 is 3 − (−4) = 7. Their midpoint is ½(−4 + 3) = −½.
2. The key rules
Natural numbers, fractions, their negatives and 0 are rational (ഭിന്നകസംഖ്യകൾ), all of the form x/y. The rest, like √2, √3, π, are irrational (അഭിന്നകസംഖ്യകൾ). Together they are the real numbers (രേഖീയസംഖ്യകൾ): the points of the number line. Bigger means further right.
സംഖ്യാരേഖയിൽ പൂജ്യം സൂചിപ്പിക്കുന്ന ബിന്ദുവും, മറ്റൊരു സംഖ്യ സൂചിപ്പിക്കുന്ന ബിന്ദുവും തമ്മിലുള്ള അകലം, ഈ സംഖ്യയുടെ കേവലമൂല്യമാണ്.
|x| is the distance of x from 0.
ഏതു സംഖ്യയുടെയും അതിന്റെ ന്യൂനത്തിന്റെയും കേവലമൂല്യം ഒരേ (അധി)സംഖ്യയാണ്. Also, from practice set 1:
|x + y| = |x| + |y| exactly when x and y have the same sign (or one is 0).
സംഖ്യാരേഖയിൽ രണ്ടു സംഖ്യകൾ തമ്മിലുള്ള അകലം, വലിയ സംഖ്യയിൽനിന്നു ചെറിയ സംഖ്യ കുറച്ചു കിട്ടുന്ന സംഖ്യയാണ്.
It is the bigger minus the smaller: 4½ and −6¼ are 4½ + 6¼ = 10¾ apart.
സംഖ്യാരേഖയിൽ ഏതു രണ്ടു ബിന്ദുക്കളുടെയും മധ്യബിന്ദു, അവയെ സൂചിപ്പിക്കുന്ന സംഖ്യകളുടെ തുകയുടെ പകുതിയെ സൂചിപ്പിക്കുന്ന ബിന്ദുവാണ്.
Between any two real numbers, however close, there are infinitely many others: take the midpoint, then midpoints of the halves, and so on (സംഖ്യാസാന്ദ്രത).
3. Recipe: equations and inequalities with |x − a|
| Statement | Meaning on the line | Answer | Example |
|---|---|---|---|
| |x − a| = r | x is r away from a | x = a − r or a + r | |x − 1| = 3 ⇒ x = −2 or 4 |
| |x − a| < r | x is less than r away from a | a − r < x < a + r | |x − 1| < 3 ⇒ −2 < x < 4 |
| |x − a| > r | x is more than r away from a | x < a − r or x > a + r | |x − 3| > 5 ⇒ x < −2 or x > 8 |
| |x + a| = r | write x + a = x − (−a): r away from −a | x = −a ± r | |x + 1| = 3 ⇒ x = 2 or −4 |
| |x − a| = |x − b| | x is equally far from a and b | x = ½(a + b) | |x − 1| = |x − 4| ⇒ x = 2½ |
- |−3| is not −3. An absolute value is a distance, never negative: |−3| = 3.
- |x + y| is not always |x| + |y|. |−3 + 5| = 2, but |−3| + |5| = 8. They are equal only when x and y have the same sign.
- Forgetting the second solution. |x − 4| = 6 has two answers, 10 and −2 – one on each side of 4.
- |x − 3| > 5 is two rays, not an interval. The answer is x < −2 or x > 8, not −2 < x < 8.
- The midpoint is not half the difference. The midpoint of −2 and 4 is ½(−2 + 4) = 1; half the distance, 3, is how far the midpoint is from each end.
- −2 is smaller than −1, even though 2 is bigger than 1: it is further left.
4. Flashcards
Click a card to flip it.
5. Scored quiz
Answer all ten, then press Check my answers. Each wrong answer shows the page to revise.
6. Your study checklist
Next: the model test paper ▶ – attempt it on paper first, then check with the solved paper.