Number Systems:
Naturals, Integers, Rationals, Irrationals, Reals, and Beyond
Complete number System and Algebra note
The Natural Numbers
The natural (or counting) numbers are 1,2,3,4,5,1,2,3,4,5, etc. There are infinitely many natural numbers. The set of natural numbers, {1,2,3,4,5,...}{1,2,3,4,5,...}, is sometimes written NN for short.
The whole numbers are the natural numbers together with 00.
(Note: a few textbooks disagree and say the natural numbers include 00.)
The sum of any two natural numbers is also a natural number (for example, 4+2000=20044+2000=2004), and the product of any two natural numbers is a natural number (4×2000=80004×2000=8000). This is not true for subtraction and division, though.
The Integers
The integers are the set of real numbers consisting of the natural numbers, their additive inverses and zero.
{...,−5,−4,−3,−2,−1,0,1,2,3,4,5,...}{...,−5,−4,−3,−2,−1,0,1,2,3,4,5,...}
The set of integers is sometimes written JJ or ZZ for short.
The sum, product, and difference of any two integers is also an integer. But this is not true for division... just try 1÷21÷2.
The Rational Numbers
The rational numbers are those numbers which can be expressed as a ratio between two integers. For example, the fractions 1313 and −11118−11118 are both rational numbers. All the integers are included in the rational numbers, since any integer zz can be written as the ratio z1z1.
All decimals which terminate are rational numbers (since 8.278.27 can be written as 827100827100.) Decimals which have a repeating pattern after some point are also rationals: for example,
0.0833333....=1120.0833333....=112.
The set of rational numbers is closed under all four basic operations, that is, given any two rational numbers, their sum, difference, product, and quotient is also a rational number (as long as we don't divide by 00).
The Irrational Numbers
An irrational number is a number that cannot be written as a ratio (or fraction). In decimal form, it never ends or repeats. The ancient Greeks discovered that not all numbers are rational; there are equations that cannot be solved using ratios of integers.
The first such equation to be studied was 2=x22=x2. What number times itself equals 22?
2√2 is about 1.4141.414, because
Other famous irrational numbers are the golden ratio, a number with great importance to biology:
1+5√2=1.61803398874989...1+52=1.61803398874989...
ππ (pi), the ratio of the circumference of a circle to its diameter:
π=3.14159265358979...π=3.14159265358979...
and ee, the most important number in calculus:
e=2.71828182845904...e=2.71828182845904...
Irrational numbers can be further subdivided into algebraic numbers, which are the solutions of some polynomial equation (like 2√2 and the golden ratio), and transcendental numbers, which are not the solutions of any polynomial equation. ππ and ee are both transcendental.
The Real Numbers
The real numbers is the set of numbers containing all of the rational numbers and all of the irrational numbers. The real numbers are “all the numbers” on the number line. There are infinitely many real numbers just as there are infinitely many numbers in each of the other sets of numbers. But, it can be proved that the infinity of the real numbers is a bigger infinity.
The "smaller", or countable infinity of the integers and rationals is sometimes called ℵ0ℵ0(alef-naught), and the uncountable infinity of the reals is called ℵ1ℵ1(alef-one).
There are even "bigger" infinities, but you should take a set theory class for that!
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