Real Numbers on a Line സംഖ്യകളും ജ്യാമിതിയും
You have used natural numbers, fractions, negatives, √2 and π in different chapters. Here they all come together: every one of them is a point on one line, the number line. That simple picture turns "bigger than" into "to the right of", and prepares the way for distances and midpoints.
- Fractions like ½ and 1½ measure parts of a unit; √2 and π are lengths that are not fractions (chapters 2 and 8).
- Negative numbers name positions to the left of a fixed point (chapter 7, ന്യൂനസംഖ്യകൾ).
- a < b means "a is smaller than b".
1. Lengths as numbers അളവുകളും സംഖ്യകളും
Choose any length and call it 1 – the unit of length (ഏകകം). Twice it is 2, half of it ½, one and a half times it 1½.
With a unit fixed, many lengths are natural numbers or fractions of it. But not all of them…
2. Rational and irrational
Natural numbers, fractions, their negatives and 0 are together called rational numbers (ഭിന്നകസംഖ്യകൾ). Numbers that cannot be written as fractions are irrational numbers (അഭിന്നകസംഖ്യകൾ).
Every natural number is also a fraction: 5 = 5/1 = 10/2. With a negative numerator or denominator, negatives are fractions too (−3 = −3/1), and 0 = 0/1. So every rational number can be written as x/y, with x and y natural numbers or their negatives (x may be 0). Irrational numbers – square roots like √2, √3 and numbers like π – have no such common form.
3. Every number is a point
Think of every positive number as a length measured from one fixed point. Put that point at the left end of a line and call it 0; mark the unit to its right.
All numbers – rational and irrational – can be thought of as points on a line. Together they are called real numbers (രേഖീയസംഖ്യകൾ). The line whose points are named by their distance from 0 is the number line (സംഖ്യാരേഖ).
4. Bigger means further right
Which is bigger, −1 or −2? −1 is 1 less than 0; −2 is 2 less than 0, that is, 1 less than −1. So −2 < −1. Going right numbers grow, going left they shrink – from any starting point.
For any two real numbers, the bigger one lies to the right of the smaller one on the number line. The number relation "bigger / smaller" becomes the geometric relation "right / left".
Counting began with 1, 2, 3, …; 0 came later. Measuring length, weight and time needed fractions. That not every length is a fraction of a unit forced irrational numbers. The convenience of algebra brought negatives. The common name for all of them is real numbers. Natural numbers, their negatives and 0 are the integers (പൂർണ്ണസംഖ്യകൾ).
A thermometer is a number line standing up: −5 °C is colder than −2 °C because it is lower (further "left"). A lift panel (−2, −1, 0, 1, 2) and a timeline with years BC and AD are number lines too.
5. Check yourself
- With a unit fixed, lengths become numbers; some (√2, π) are not fractions.
- Rational: natural numbers, fractions, their negatives and 0 – all of the form x/y. Irrational: everything else, like √2, √3, π.
- All of them together are the real numbers; each is a point on the number line.
- Bigger means further right: −2 < −1 < 0 < ½ < √2 < π.