0.999... (also denoted [0.9 with a bar on top of 9] or
[0.9 with a dot on top of 9] is a recurring decimal which is exactly
to 1. In other words, the symbols 0.999… and 1 represent the same real
number. Mathematicians have formulated a number of proofs of this
identity, which vary with their level of rigor, preferred development
of the real numbers, background assumptions, historical context and
target audience. The equality 0.999… = 1 has long been taught in
textbooks, and in the last few decades, researchers of mathematics
education have studied the reception of this equation among students,
who often vocally reject the equality. Their reasoning is often based
on an expectation that infinitesimal quantities should exist, that
arithmetic may be broken, or simply that 0.999… should have a last 9.
These ideas are false in the real numbers, as can be proven by
explicitly constructing the reals from the rational numbers, and such
constructions can also prove that 0.999… = 1 directly.
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Today's selected anniversaries:
Reconquista: Forces under King Afonso I of Portugal captured Lisbon
from the Moors after a four-month siege during the Second Crusade.
Hundred Years' War: Henry V of England and his lightly armoured
infantry and archers defeated the heavily armoured French cavalry in
the Battle of Agincourt on Saint Crispin's Day.
The Dutch sailing ship Eendracht reached Shark Bay on the western
coastline of Australia, as documented on the Hartog Plate.
The Third Dáil adopted the Constitution of the Irish Free State.
The People's Republic of China replaced the Republic of China as
China's representative at the United Nations.
Wikiquote of the day:
A man doesn't begin to attain wisdom until he recognizes that he is no
longer indispensable. -- Richard E. Byrd