Non-rational number approximation

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A rational number is a number that can be exactly represented as the division of two integers, say, a and b.

$Rational number$
The real numbers include numbers that are not rational. Integer ratios can be used to approximate real numbers that are not rational. Pythagoras had proven that the square root of 2 has no rational exact representation, but integer rational approximation were useful, nonetheless.

In antiquity, there were many attempts to approximate non-rational numbers as a ratio of two integers (i.e., rational number).

Such methods are still used today and called floating point approximations (albeit using very large integers). Consider the approximation of 1 over the square root of 3. This is done to get a ratio between 0.0 and 1.0.

$Square root of three$

A manual approach is to start trying values for numerator and denominator and see which is closest. There may be clever ways to prune the search. Here is a grid to show the best approximations for denominators ranging from 1 to 10 and numerators ranging from 0 to 10. The best approximation below and above the desired result is in light green while the others are in gray.

With the use of computer programming code, is easy to find rational approximations to non-rational numbers. Note: The program is intended to be somewhat clear, but not necessarily the most efficient.

Here is the Lua code [#1]

local n1 = 3
local x1 = 1.0 / math.sqrt(n1)
local n1 = 300
local maxLen1 = 20
local list1 = {}
for bot1=1,n1 do
	for top1=0,bot1 do
		local ratio1 = top1 / bot1
		local dif1 = math.abs(x1 - ratio1)
		if #list1 < maxLen1 then
			table.insert(list1,{dif1,top1,bot1,ratio1})
			table.sort(list1, function(a, b) return a[1] < b[1] end)
		else
			local lastItem1 = list1[#list1]
			local cutoff1 = lastItem1[1]
			if dif1 < cutoff1 then
				table.remove(list1)
				table.insert(list1,{dif1,top1,bot1,ratio1})
				table.sort(list1, function(a, b) return a[1] < b[1] end)
				end
			end
		end
	end
for pos1,item1 in ipairs(list1) do
	local top1 = item1[2]
	local bot1 = item1[3]
	local ratio1 = item1[4]
	print(string.format("%2d. %9.8f ≈ %d / %d",pos1,ratio1,top1,bot1))
	end

The output shows the best 20 integer ratio approximations where the denominator ranges from 1 to 300. The ratio of 153/265, found by Archimedes, is the best given these constraints.

0.57735026918963

Here is the output of the Lua code.

 1. 0.57735849 ≈ 153 / 265
 2. 0.57731959 ≈ 112 / 194
 3. 0.57731959 ≈ 168 / 291
 4. 0.57731959 ≈ 56 / 97
 5. 0.57738095 ≈ 97 / 168
 6. 0.57740586 ≈ 138 / 239
 7. 0.57727273 ≈ 127 / 220
 8. 0.57723577 ≈ 142 / 246
 9. 0.57723577 ≈ 71 / 123
10. 0.57746479 ≈ 82 / 142
11. 0.57746479 ≈ 41 / 71
12. 0.57746479 ≈ 123 / 213
13. 0.57746479 ≈ 164 / 284
14. 0.57720588 ≈ 157 / 272
15. 0.57718121 ≈ 86 / 149
16. 0.57718121 ≈ 172 / 298
17. 0.57751938 ≈ 149 / 258
18. 0.57754011 ≈ 108 / 187
19. 0.57714286 ≈ 101 / 175
20. 0.57711443 ≈ 116 / 201

That number 153 was taken to be a special number.

... more to be added ...

Here is the above approach repeated for the inverse of the square root of 2.

Consider the approximation of 1 over the square root of 2. This is done to get a ratio between 0.0 and 1.0.

$Square root of three$

Here is the Lua code [#2]

local n1 = 2
local x1 = 1.0 / math.sqrt(n1)
local n1 = 300
local maxLen1 = 20
local list1 = {}
for bot1=1,n1 do
	for top1=0,bot1 do
		local ratio1 = top1 / bot1
		local dif1 = math.abs(x1 - ratio1)
		if #list1 < maxLen1 then
			table.insert(list1,{dif1,top1,bot1,ratio1})
			table.sort(list1, function(a, b) return a[1] < b[1] end)
		else
			local lastItem1 = list1[#list1]
			local cutoff1 = lastItem1[1]
			if dif1 < cutoff1 then
				table.remove(list1)
				table.insert(list1,{dif1,top1,bot1,ratio1})
				table.sort(list1, function(a, b) return a[1] < b[1] end)
				end
			end
		end
	end
for pos1,item1 in ipairs(list1) do
	local top1 = item1[2]
	local bot1 = item1[3]
	local ratio1 = item1[4]
	print(string.format("%2d. %9.8f ≈ %d / %d",pos1,ratio1,top1,bot1))
	end

The output shows the best 20 integer ratio approximations where the denominator ranges from 1 to 300. The ratio of 153/265, found by Archimedes, is the best given these constraints.

0.70710678118655

Here is the output of the Lua code.

 1. 0.70711297 ≈ 169 / 239
 2. 0.70707071 ≈ 70 / 99
 3. 0.70707071 ≈ 210 / 297
 4. 0.70707071 ≈ 140 / 198
 5. 0.70714286 ≈ 198 / 280
 6. 0.70714286 ≈ 99 / 140
 7. 0.70703125 ≈ 181 / 256
 8. 0.70718232 ≈ 128 / 181
 9. 0.70700637 ≈ 111 / 157
10. 0.70720721 ≈ 157 / 222
11. 0.70722433 ≈ 186 / 263
12. 0.70697674 ≈ 152 / 215
13. 0.70695971 ≈ 193 / 273
14. 0.70689655 ≈ 205 / 290
15. 0.70689655 ≈ 123 / 174
16. 0.70689655 ≈ 82 / 116
17. 0.70689655 ≈ 41 / 58
18. 0.70689655 ≈ 164 / 232
19. 0.70731707 ≈ 174 / 246
20. 0.70731707 ≈ 145 / 205