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Single Idea 10102
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
]
Full Idea
In defining the integers in set theory, our definition will be motivated by thinking of the integers as answers to subtraction problems involving natural numbers.
Gist of Idea
The integers are answers to subtraction problems involving natural numbers
Source
A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3)
Book Ref
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.63
A Reaction
Typical of how all of the families of numbers came into existence; they are 'invented' so that we can have answers to problems, even if we can't interpret the answers. It it is money, we may say the minus-number is a 'debt', but is it? Cf Idea 10106.
Related Idea
Idea 10106
Rational numbers give answers to division problems with integers [George/Velleman]
The
25 ideas
with the same theme
[the various families of numbers]:
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9603
|
An assumption that there is a largest prime leads to a contradiction
[Euclid, by Brown,JR]
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15905
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Cantor proved the points on a plane are in one-to-one correspondence to the points on a line
[Cantor, by Lavine]
|
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9886
|
Cardinals say how many, and reals give measurements compared to a unit quantity
[Frege]
|
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14438
|
New numbers solve problems: negatives for subtraction, fractions for division, complex for equations
[Russell]
|
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18254
|
Russell's approach had to treat real 5/8 as different from rational 5/8
[Russell, by Dummett]
|
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14144
|
Ordinals result from likeness among relations, as cardinals from similarity among classes
[Russell]
|
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9896
|
A prime number is one which is measured by a unit alone
[Dummett]
|
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15273
|
Points can be 'dense' by unending division, but must meet a tougher criterion to be 'continuous'
[Harré/Madden]
|
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12333
|
Each type of number has its own characteristic procedure of introduction
[Badiou]
|
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12322
|
Must we accept numbers as existing when they no longer consist of units?
[Badiou]
|
|
18100
|
ω + 1 is a new ordinal, but its cardinality is unchanged
[Bostock]
|
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18101
|
Each addition changes the ordinality but not the cardinality, prior to aleph-1
[Bostock]
|
|
3330
|
Negatives, rationals, irrationals and imaginaries are all postulated to solve baffling equations
[Benardete,JA]
|
|
3337
|
Natural numbers are seen in terms of either their ordinality (Peano), or cardinality (set theory)
[Benardete,JA]
|
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10573
|
Dedekind cuts lead to the bizarre idea that there are many different number 1's
[Fine,K]
|
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13641
|
Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals
[Shapiro]
|
|
8763
|
The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex
[Shapiro]
|
|
10106
|
Rational numbers give answers to division problems with integers
[George/Velleman]
|
|
10102
|
The integers are answers to subtraction problems involving natural numbers
[George/Velleman]
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8454
|
The whole numbers are 'natural'; 'rational' numbers include fractions; the 'reals' include root-2 etc.
[Orenstein]
|
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15712
|
1 and 0, then add for naturals, subtract for negatives, divide for rationals, take roots for irrationals
[Kaplan/Kaplan]
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15942
|
Every rational number, unlike every natural number, is divisible by some other number
[Lavine]
|
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8667
|
The 'integers' are the positive and negative natural numbers, plus zero
[Friend]
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8668
|
The 'rational' numbers are those representable as fractions
[Friend]
|
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8670
|
A number is 'irrational' if it cannot be represented as a fraction
[Friend]
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