List of numbers

This is a list of numbers. This list will always be not finished. This happens because there are an infinite amount of numbers. Only notable numbers will be added. Numbers can be added as long as they are popular in math, history or culture.

This means that numbers can only be notable if they are a big part of history. A number isn't notable if it is only related to another number. For example, the number (3,4) is a notable number when it is a complex number (3+4i). When it is only (3,4), however, it's not notable. If a number is not important, that can also make the number important. This is called the interesting number paradox.

Natural numbers

See the main article: Natural number

Natural numbers are a type of integer. They can be used for counting. Natural numbers can also be used to find out about other number systems. The symbol N (or blackboard bold $\mathbb {\mathbb {N} }$, Unicode U+2115 ℕ DOUBLE-STRUCK CAPITAL N) is used to symbolize all of the natural numbers.. A negative number is not a natural number.

0 is argued on whether or not it is a natural number. In computer science and set theory, 0 is a natural number. In number theory, it is not. To fix this, people use the terms "non-negative integers". Non-negative integers includes the number 0. "Positive integers" does not.

Natural numbers can be used as cardinal numbers or ordinal numbers.

Importance in math

Natural numbers can be important in mathematics. This might because it is part of a set. A number can also be important because it has a unique property.

List of important natural numbers in math

• 1. If you multiply any number by one, you will get the same number. This is called the multiplicative identity. It is also the only natural number (other than 0) that is not prime or composite.
• 2. It is the base of the binary number system. This system is used in almost all computers.
• 3. It is 2²-1. It is also the first Mersenne prime. It is the first odd prime number.
• 4. It is the first composite number.
• 6. It is the first perfect number.
• 9. It is the first odd number that is composite.
• 11. It is the fifth prime.
• 12. It is the first sublime number.
• 17. It is the sum of the 1st 4 prime numbers. It is the only prime number that is the sum of 4 prime numbers in a row.
• 24. All Dirichlet characters mod n are real if n is a divisor of 24.
• 25. It is the first centered square number (other than 1) that is also a square number.
• 27, the cube of 3. This means that it is 3³.
• 28. It is the second perfect number.
• 30. It is the smallest sphenic number.
• 32. It is the smallest fifth power (other than 1 or 0).
• 36. This is the smallest number that is not a perfect power, but not a prime power.
• 70. It is the smallest weird number.
• 72. It is the smallest Achilles number.

Importance in culture

Many numbers are also important in culture. They can also be important for measurement.

List of important natural numbers in culture

• 3. It is important in Christianity as the Trinity. It is also important in Hinduism (Trimurti, Tridevi). It is also important in other older mythologies.
• 4. It is an "unlucky" number in modern China, Japan, and Korea. In their languages, 4 sounds similar to the word "death".
• 7. It is the number of days in a week. It is also called a "lucky" number in Western cultures.
• 8. 8 is called a "lucky" number in Chinese culture. This is because the Chinese word for "Eight" sounds like prosperity.
• 12. 12 things is called a dozen. There are 12 months in a year. There are 12 constellations of the Zodiac and astrological signs. In Christianity, there were 12 Apostles of Jesus.
• 13, is called an "unlucky" number in Western superstition. It is also called a "Baker's dozen".
• 17. It is called unlucky in Greek and Latin countries.
• 18. In Jewish numerology, 18 is a "lucky" number. This is because it is also the Hebrew word for life
• 40. 40 is important in Tengrism and Turkish folklore. There are many traditions that use 40.
• 42. According to the The Hitchhiker's Guide to the Galaxy, 42 the "answer to the ultimate question of life, the universe, and everything."
• 69. Is a word that can mean oral sex.

List of natural numbers important in measurements and scales.

• 10. 10 is the number of digits in the decimal number system.
• 12. 12 is the number base to measuring time in many places.
• 14. Is the number of days in a fortnight.
• 16. 16 is the number of digits in the hexadecimal number system.
• 24. 12 is also the number of hours in a day.
• 31. 31 is the most number of days a month can have.
• 60. 60 is the number base for some ancient counting systems. For example, the Babylonians used this number base. Also, 60 is used in some modern measuring systems.
• 360. 360 is the number of degrees in a full circle.
• 365. 365 is the number of days in the normal year. In leap years, there are 366 days.

List of natural numbers in computers

• 4. 4 is the number of bits in a nibble.
• 8. 8 is the number of bits in an octet and a byte.
• 256. 256 is the number of possible numbers in 8 bits.
• 1024. 1024 is the number of bytes in a kibibyte. It is the number of bits in a kibibit.
• 65535. 2¹⁶ − 1. It is the largest number in a 16-bit number.
• 65536. 2¹⁶. It is the number of possible numbers in 16-bits.

Classes of natural numbers

Prime numbers

Main articles: Prime number and List of prime numbers

A prime number is a type of natural number. It only has two divisors: 1 and itself.

Highly composite numbers

A highly composite number is a type of natural number. It has more divisors than any smaller natural number. They are used a lot in geometry, grouping, and time measurement.

The first 20 highly composite numbers are:

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560

Perfect numbers

See the main article: Perfect number

A perfect number is a type of integer. It has the sum of its positive divisors (all divisors except itself).

The first 10 perfect numbers:

1.   6
2.   28
3.   496
4.   8128
5.   33 550 336
6.   8 589 869 056
7.   137 438 691 328
8.   2 305 843 008 139 952 128
9.   2 658 455 991 569 831 744 654 692 615 953 842 176
10.   191 561 942 608 236 107 294 793 378 084 303 638 130 997 321 548 169 216

Integers

See the main article: Integer

Integers are a set of numbers. They usually are in arithmetic and number theory. There are many subsets of integers. These can cover natural numbers, prime numbers, perfect numbers, etc.

Popular integers are −1 and 0.

Orders of magnitude

Integers can be written in orders of magnitude. This can be written as 10^k, where k is an integer. If k = 0, 1, 2, 3, then the powers of ten for them are 1, 10, 100 and 1000. This is used in scientific notation.

Each number has its own prefix. Each prefix has its own symbol. For example, kilo- may be added to the beginning of gram. This changes the meaning of gram to mean that the gram is 1000 times more than a gram: one kilogram is the same as 1000 grams.

Number	1000m	Name	Symbol
0.0000000001	10-24	Yokto	y
0.000000001	10-21	Zepto	z
0.00000001	10-18	Atto	a
0.0000001	10-15	Femto	f
0.000001	10-12	Pico	p
0.00001	10-9	Nano	n
0.0001	10-6	Micro	μ
0.001	10-3	Mili	m
0.01	10-2	Centi	c
0.1	10-1	Deci	d
10	101	Deca	da
100	102	Hecto	h
1000	103	Kilo	k
1000000	106	Mega	M
1000000000	109	Giga	G
1000000000000	1012	Tera	T
1000000000000000	1015	Peta	P
1000000000000000000	1018	Exa	E
1000000000000000000000	1021	Zetta	Z
1000000000000000000000000	1024	Yotta	Y

Rational numbers

See the main article: Rational number

A rational number is a number that can be written as a fraction with two integers. The numerator is written as $p$. The denominator(which cannot be zero) is written as $q$. Every integer is a rational number. This is because, in integers, 1 is always the denominator of a fraction.

Rational numbers can be written in infinitely many ways. For example, 0.12 can be written as three twenty-fifths (${\frac {3}{25}}$), nine seventy-fifths (${\frac {9}{75}}$), etc.

Table of notable rational numbers

Decimal expansion	Fraction	Reason
1.0	${\bar {1}}$	${\bar {1}}$ is equal to 1, a notable real number.
1
0.5	${\bar {2}}$	${\bar {2}}$ is a popular number in math. For example, you can use ${\bar {2}}$ to find the area of a Triangle.
3.142 857...	${\frac {22}{7}}$	${\frac {22}{7}}$ is a number slightly above $\pi$ and is an approximation of $\pi$.
0.166 666...	${\bar {6}}$	One sixth is seen in a lot of equations. For example, the solution to the Basel problem uses 1/6.

Irrational numbers

See the main article: Irrational number

Irrational numbers are numbers that cannot be written as a fraction. These are written as algebraic numbers or transcendental numbers.

Algebraic numbers

See the main article: Algebraic number

Name	Expression	Decimal expansion	Reason
Square root of two	${\sqrt {2}}$	1.414213562373095048801688724210	The Square root of 2(also called Pythagoras' constant) is a number used in math a lot. It can be used to find the ratio of diagonal to side length in a square.
Triangular root of 2	${\frac {{\sqrt {17}}-1}{2}}$	1.561552812808830274910704927987
Phi, Golden ratio	$\phi$	1.618033988749894848204586834366	The golden ratio is a famous number used in both math and science.

Transcendental numbers

See the main article: Transcendental number

Name	Symbol or Formula	Decimal expansion	Reason
e, Euler's number	e	2.718281828459045235360287471352662497757247...	e is the base of a natural logarithm.
Pi	π	3.141592653589793238462643383279502884197169...	Pi is an irrational number that is the result of dividing the circumference of a circle by its diameter.

Real numbers

The real numbers are a superset(or category) of numbers. They cover algebraic and transcendental numbers.

Real but not known if irrational or transcendental

Name and symbol	Decimal expansion	Notes
Euler–Mascheroni constant, γ	0.577215664901532860606512090082...	The Euler–Mascheroni constant is used in limits and logarithms. It is thought to be transcendental but not proven to be so.
Twin prime constant, C2	0.660161815846869573927812110014...

Hypercomplex numbers

A hypercomplex number is a word for an element of a unital algebra over the field of real numbers.

Algebraic complex numbers

• i, Imaginary unit: ${\textstyle i={\sqrt {-1}}}$

Transfinite numbers

See the main article: Transfinite number

Transfinite numbers are numbers that are "infinite". They are larger than any finite number. They are, however, not absolutely infinite.

Physical Constants

See the main article: Physical constant

Physical constants are constants that can be used in the universe to figure out information.

• Avogadro constant
• Gravitational constant
• Planck constant
• Rydberg constant
• Speed of light in vacuum

Named numbers

• Googol, 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
• Googolplex, 10^{(10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000)}
• Graham's number, G
• Skewes's number, S

Bibliography

• Finch, Steven R. (2003), "Anmol Kumar Singh", Mathematical Constants (Encyclopedia of Mathematics and its Applications, Series Number 94), Cambridge University Press, pp. 130–133, ISBN 0521818052
• Apéry, Roger (1979), "Irrationalité de $\zeta (2)$ et $\zeta (3)$", Astérisque, 61: 11–13.

Further reading

• Kingdom of Infinite Number: A Field Guide by Bryan Bunch, W.H. Freeman & Company, 2001. ISBN 0-7167-4447-3

Other websites

• The Database of Number Correlations: 1 to 2000+ Archived 2017-07-24 at the Wayback Machine
• What's Special About This Number? A Zoology of Numbers: from 0 to 500
• Name of a Number
• See how to write big numbers
• About big numbers at the Wayback Machine (archived 27 November 2010)
• Robert P. Munafo's Large Numbers page
• Different notations for big numbers – by Susan Stepney
• Names for Large Numbers, in How Many? A Dictionary of Units of Measurement by Russ Rowlett
• What's Special About This Number? Archived 2018-02-23 at the Wayback Machine (from 0 to 9999)