Numbers!!!
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Oh Numbers, how doth I love thee, let me
count the ways:
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Naturally
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Integerly
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Rationally
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Irrationally
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Historically
Real Number Sets
Natural Numbers
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Natural Numbers are also known as
“counting numbers.”
The set of natural numbers contains the
numbers 0, 1, 2, 3, 4, …
Mathematically written as:
N = {0, 1, 2, 3, 4, ...}
Real Number Sets
Integers
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The set of integers include all natural
numbers and their negatives.
The set of integers contains the numbers
...-4, -3, -2, -1, 0, 1, 2, 3, 4, …
Mathematically written as:
Z = {…, -2, -1, 0, 1, 2, 3, 4, …}
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Note: Z stands for Zahlen – German for
“numbers.”
Real Number Sets
Rational Numbers
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Rational Numbers are fractions (ratios).
The set of rational
numbers
contains
the
p
numbers written q , where p and q are
integers and q does not equal zero.
Mathematically written as:
Q={
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p
where
q
p and q are integers, q≠0}
Note: Q stands for Quotient
Real Number Sets
Irrational Numbers
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The set of Irrational Numbers include all
decimal numbers that neither terminate
nor repeat.
The set of irrational numbers contains
2
numbers like π, e , √
Irrational Numbers cannot be written as a
ratio between two numbers.
Real Number Sets
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The set of real numbers include all
previous sets of numbers.
Mathematically known as R.
History of Numbers
Today's numbers, also called Hindu-Arabic numbers,
are a combination of just 10 symbols or digits: 1, 2,
3, 4, 5, 6, 7, 8, 9, and 0. These digits were
introduced in Europe within the XII century by
Leonardo Pisano (aka Fibonacci), an Italian
mathematician. L. Pisano was educated in North
Africa, where he learned and later carried to Italy the
now popular Hindu-Arabic numerals.
Hindu numeral system is a pure place-value system,
that is why you need a zero. Only the Hindus, within
the context of Indo-European civilizations, have
consistently used a zero. The Arabs, however,
played an essential part in the dissemination of this
numeral system.
History of Numbers
Numerals as they traveled from India to Europe
Maya Two Systems of
Numbers
Left – 20 digit system
Above-anthropomorphic
system up to ten
Chinese Systems
The Chinese use three
numeral systems: the HinduArabic numerals, along with
two indigenous numeral
systems, one for everyday
writing (simple numerals),
and another one for use in
commercial or financial
contexts (complex
numerals). These last ones
are used on checks and
other transaction forms
because they are much more
difficult to alter. Actually, they
are the equivalent of writing
'one', 'two', 'three', etc.,
rather than 1, 2, 3...
Greenlandic
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Greenlanders use a twenty-base
(vigesimal) system of numeration, divided
into recurring five-based (quinary) periods
of reckoning referring to the fingers and
toes of a person!
When you run out of fingers and toes, you
go on to the next person...and the
next...and the next...
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They call this “man counted out.”
Maori Counting
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Maori are the indigenous people of New
Zealand. They do something very
interesting with numbers. If a number is for
a thing, they prefix it with “ka”, if it is for a
person, they prefix it with “toko.”
One = tahi, about one thing it is katahi, for
a person it would be tokotahi.
Sumerian Numeration
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The Sumerians, who lived around 3200
B.C. on the lower reaches of the river
Euphrates. They had two systems of
numeration: a sexagesimal system (base
60) for astronomical observation, and a
decimal system for everyday use.
Sumerian Numeration
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Building the numbers with two different
size rods.
Then later with finer tools.
Sumerian Numeration
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This is the cuneiform symbolic system.
Other Original Systems of
Numeration
Other original systems of numeration were being used
in the past. The "Notae Elegantissimae" shown
below allow to write numbers from 1 to 9999. They
are useful as a mnemotechnic aid, e.g. the symbol K
in the example may mean 1414 (the first 4 figures of
the square root of 2).
Do other people still think of
numbers differently?