Adding Irrationals & Decimals അഭിന്നകസംഖ്യകളുടെ സങ്കലനം
Before multiplying irrational numbers, let us be fluent in writing them (4√2, ½√3), adding them (3√2 + 5√2 = 8√2), and turning them into decimals of any accuracy we need. These habits are used on every page of the chapter.
1. What 4√2 means
Four diagonals of unit squares laid end to end make a length √2 + √2 + √2 + √2. We write this as 4√2 ("four root two"), dropping the × sign, just as 4x means 4 × x in algebra.
k√2 = k × 1.41421… Each extra diagonal adds another 1.414 (and the decimal of the total grows by the same amount).
In a√b the number a in front (the coefficient) counts how many √b's we have. So 4√2 = 4 × √2, ½√2 = half of √2 ≈ 0.7071, and 3√5 = 3 × 2.236… ≈ 6.708. The number under the root is called the radicand.
2. Like terms: when can we add?
3√2 + 5√2 = 8√2 for the same reason that 3 apples + 5 apples = 8 apples. But √2 + √3 is like 1 apple + 1 orange – two different kinds of length, and no shorter way of writing the total exists.
With only √2 strips the answer is a single term a√2. Add √3 strips and the total has two kinds of pieces – you can still find its decimal, but you cannot write it as one square root term.
2√3 + 5√3 − √3 = (2 + 5 − 1)√3 = 6√3. 7√5 − 7√5 = 0. √2 + √2 = 2√2 (not √4!). 3√2 + 2√3 stays as it is.
Later in the chapter you will see that some roots that look different are really like terms: √8 = 2√2, so √8 + √2 = 3√2.
3. Decimal values and the rounding rule ദശാംശക്കണക്ക്
√2 = 1.41421356… To one decimal place it is 1.4, to two places 1.41, to three 1.414, to four 1.4142. Rule: look at the first digit you are throwing away – if it is 5 or more, add 1 to the last digit you keep.
4. Exact or approximate? Keep the √ until the end
4√2 is an exact answer – one short symbol that is perfectly correct. 5.66 is an approximation. In this chapter you will give exact answers like 4√2 or 6√3 and convert to decimals only when a measurement in metres or millimetres is asked for.
A millimetre is 0.001 m. Multiplying a rounded value of √2 by k also multiplies the error by k – so bigger lengths need more digits. Three decimal places of √2 is the textbook's usual choice.
5. Preview: adding is not multiplying
Two diagonals end to end give a length √2 + √2 = 2√2 ≈ 2.83. A square whose side is √2 has an area √2 × √2 = (√2)² = 2 exactly. Adding stacks lengths; multiplying makes areas – and that is where Section 1 of the chapter begins.
A tiler lays four square tiles corner to corner along a diagonal line: the line is 4√2 tile-widths long, about 5.66 widths – so the fifth tile will not fit. A fence around a square plot of 200 m² has side √200 = 10√2 ≈ 14.14 m, and the perimeter is 40√2 ≈ 56.6 m. Keep the root while calculating, round when you buy the fencing.
6. Check yourself
- a√b = a × √b: 4√2 = √2 + √2 + √2 + √2 ≈ 5.657; ½√2 ≈ 0.7071.
- Only like terms add: 3√2 + 5√2 = 8√2, but √2 + √3 cannot be combined.
- Rounding: look at the next digit; 5 or more → add 1. 1.41421 → 1.414 (3 places), 2.4393 → 2.44.
- Keep the √ for exact answers; convert to decimals only for measurements.