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

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 ${\displaystyle \mathbb {\mathbb {N} } }$, 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.

Table of small natural numbers

0	1	2	3	4	5	6	7	8	9
10	11	12	13	14	15	16	17	18	19
20	21	22	23	24	25	26	27	28	29
30	31	32	33	34	35	36	37	38	39
40	41	42	43	44	45	46	47	48	49
50	51	52	53	54	55	56	57	58	59
60	61	62	63	64	65	66	67	68	69
70	71	72	73	74	75	76	77	78	79
80	81	82	83	84	85	86	87	88	89
90	91	92	93	94	95	96	97	98	99
100	101	102	103	104	105	106	107	108	109
110	111	112	113	114	115	116	117	118	119
120	121	122	123	124	125	126	127	128	129
130	131	132	133	134	135	136	137	138	139
140	141	142	143	144	145	146	147	148	149
150	151	152	153	154	155	156	157	158	159
160	161	162	163	164	165	166	167	168	169
170	171	172	173	174	175	176	177	178	179
180	181	182	183	184	185	186	187	188	189
190	191	192	193	194	195	196	197	198	199
200	201	202	203	204	205	206	207	208	209
210	211	212	213	214	215	216	217	218	219
220	221	222	223	224	225	226	227	228	229
230	231	232	233	234	235	236	237	238	239
240	241	242	243	244	245	246	247	248	249
250	251	252	253	254	255	256	257	258	259
260	261	262	263	264	265	266	267	268	269
270	271	272	273	274	275	276	277	278	279
280	281	282	283	284	285	286	287	288	289
290	291	292	293	294	295	296	297	298	299
300	301	302	303	304	305	306	307	308	309
310	311	312	313	314	315	316		318
				400	500	600	700	800	900
	1000	2000	3000	4000	5000	6000	7000	8000	9000
	10,000	20,000	30,000	40,000	50,000	60,000	70,000	80,000	90,000
105	106	107	108	109	1010	1011
larger numbers, including 10100 and 1010100

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.^[1] 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".^[2]
• 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 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.

The first 100 prime numbers

2	3	5	7	11	13	17	19	23	29
31	37	41	43	47	53	59	61	67	71
73	79	83	89	97	101	103	107	109	113
127	131	137	139	149	151	157	163	167	173
179	181	191	193	197	199	211	223	227	229
233	239	241	251	257	263	269	271	277	281
283	293	307	311	313	317	331	337	347	349
353	359	367	373	379	383	389	397	401	409
419	421	431	433	439	443	449	457	461	463
467	479	487	491	499	503	509	521	523	541

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

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

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.^[3]

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

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 ${\displaystyle p}$. The denominator(which cannot be zero) is written as ${\displaystyle q}$.^[4] 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 (${\displaystyle {\frac {3}{25}}}$), nine seventy-fifths (${\displaystyle {\frac {9}{75}}}$), etc.

Table of notable rational numbers

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

Irrational numbers

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

Main article: Algebraic number

Name	Expression	Decimal expansion	Reason
Square root of two	${\displaystyle {\sqrt {2}}}$	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	${\displaystyle {\frac {{\sqrt {17}}-1}{2}}}$	1.561552812808830274910704927987
Phi, Golden ratio	${\displaystyle \phi }$	1.618033988749894848204586834366	The golden ratio is a famous number used in both math and science.

Transcendental numbers

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...[5]	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[broken anchor], 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

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

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