- #1
PcumP_Ravenclaw
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- 4
Q4) Let a and b be real numbers with a < b. 1) Show that there are infinitely
many rational numbers x with a < x < b, and 2) infinitely many irrational
numbers y with a < y < b. Deduce that there is no smallest positive
irrational number, and no smallest positive rational number.
1)
a < x < b
a + (b - a)/n or b - (b - a) /n
n = Natural numbers, N
n = 0 to infinity
This shows that there are infinitely many rational numbers between a and b which can be written in the form of integers
Deduce that there is no smallest positive rational number.
This is because say a = 0 and b = 1. Note 0 is not a positive or negative number it has no sign.
so 0 < x < 1
as there are infinitely many n. there is no smaller number e.g.
0 + (1 - 0) / n
it starts from 1 when n = 1 and decreases to very very small... but never stops... as positive numbers keep decreasing
Please check if my proof above is correct and suggest how I can prove that there are infinitely many irrational numbers between two real numbers?
Danke!
many rational numbers x with a < x < b, and 2) infinitely many irrational
numbers y with a < y < b. Deduce that there is no smallest positive
irrational number, and no smallest positive rational number.
1)
a < x < b
a + (b - a)/n or b - (b - a) /n
n = Natural numbers, N
n = 0 to infinity
This shows that there are infinitely many rational numbers between a and b which can be written in the form of integers
Deduce that there is no smallest positive rational number.
This is because say a = 0 and b = 1. Note 0 is not a positive or negative number it has no sign.
so 0 < x < 1
as there are infinitely many n. there is no smaller number e.g.
0 + (1 - 0) / n
it starts from 1 when n = 1 and decreases to very very small... but never stops... as positive numbers keep decreasing
Please check if my proof above is correct and suggest how I can prove that there are infinitely many irrational numbers between two real numbers?
Danke!