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175 (number)
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| ||||
|---|---|---|---|---|
| Cardinal | one hundred seventy-five | |||
| Ordinal | 175th (one hundred seventy-fifth) | |||
| Factorization | 52 × 7 | |||
| Divisors | 1, 5, 7, 25, 35, 175 | |||
| Greek numeral | ΡΟΕ´ | |||
| Roman numeral | CLXXV, clxxv | |||
| Binary | 101011112 | |||
| Ternary | 201113 | |||
| Senary | 4516 | |||
| Octal | 2578 | |||
| Duodecimal | 12712 | |||
| Hexadecimal | AF16 | |||
175 (one hundred [and] seventy-five) is the natural number following 174 and preceding 176.
In mathematics
[edit]Raising the decimal digits of 175 to the powers of successive integers produces 175 back again: 175 = 11 + 72 + 53.[1]
175 is a figurate number for a rhombic dodecahedron, the difference of two consecutive fourth powers: 175 = 44 − 34.[2] It is also a decagonal number and a decagonal pyramid number, the smallest number after 1 that has both properties.[3]
See also
[edit]References
[edit]- ^ Sloane, N. J. A. (ed.). "Sequence A032799 (Numbers n such that n equals the sum of its digits raised to the consecutive powers (1,2,3,...))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A005917 (Rhombic dodecahedral numbers: a(n) = n^4 - (n - 1)^4)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A344280 (Numbers that are both 10-gonal numbers (A001107) and 10-gonal pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
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175 (number)
View on Grokipediafrom Grokipedia
Mathematical properties
Prime factorization and divisors
The prime factorization of 175 is .[8] This decomposition arises from dividing 175 successively by its smallest prime factors: 175 ÷ 5 = 35, 35 ÷ 5 = 7, and 7 is prime.[9] The positive divisors of 175 are all products of these prime powers: 1, 5, 7, 25, 35, and 175.[10] The sum of the divisors function, denoted σ(175), equals 248, calculated as the product of the sums of the exponents in the prime factorization plus one: (1 + 5 + 25)(1 + 7) = 31 × 8 = 248.[10] The sum of the proper divisors (excluding 175 itself) is therefore 73.[11] Since this value is less than 175, the number is classified as deficient.[1] 175 is an odd composite number, as it is odd (not divisible by 2) and has more than two distinct positive divisors.[1] It is not a sphenic number, which requires exactly three distinct prime factors; here, one prime (5) is repeated.[12]Figurate and polyhedral numbers
Figurate numbers, also known as polygonal numbers, represent the number of dots or units arranged in the shape of a regular polygon, extending the concepts of triangular and square numbers to higher-sided polygons. Polyhedral numbers build on this by forming three-dimensional structures, such as pyramids or other polyhedra, where layers of polygonal numbers are stacked. The number 175 is the seventh decagonal number, corresponding to the arrangement of dots forming a decagon with side length 7, given by the formula where .[4][13] Additionally, 175 is the fifth decagonal pyramidal number, representing the total units in a pyramid with a decagonal base and height 5, and it is the smallest integer greater than 1 that is both a decagonal number and a decagonal pyramidal number.[14] 175 is also the fourth rhombic dodecahedral number, which counts the points in a three-dimensional figurate arrangement based on the rhombic dodecahedron, a space-filling polyhedron dual to the cuboctahedron.[15][16]Special sums and identities
175 is a disarium number, a type of integer where the sum of its digits each raised to the power of its position (counting from the left, starting at 1) equals the original number itself.[17] For 175, this holds as .[17] Disarium numbers are relatively rare; among positive integers, there are 19 such numbers, all with at most 20 digits, including single-digit numbers and others like 89, 135, and 518.[18] This property positions 175 as a narcissistic-like number, akin to Armstrong numbers but using varying exponents based on digit positions rather than a fixed exponent equal to the number of digits.[17] Additionally, 175 can be expressed as the difference of consecutive fourth powers: .[19] This representation appears in historical computations of power differences, such as those by Thomas Harriot in the early 17th century, where 175 is listed among the first differences of fourth powers (1, 15, 65, 175, 369).[19]Representations in numeral systems
Verbal and symbolic notations
In English, the cardinal number 175 is denoted verbally as "one hundred seventy-five," following standard conventions for composing hundreds, tens, and units in the language.[20] The corresponding ordinal form is "one hundred seventy-fifth," often abbreviated as 175th in written contexts to indicate position or sequence.[21] The Roman numeral system, which originated in ancient Rome during the 8th century BCE and derived from earlier Etruscan notations, represents 175 as CLXXV; this breaks down to C for 100, L for 50, XX for 20 (two instances of X), and V for 5, using additive principles with subtractive notation where applicable.[22][23] Ancient Greeks employed two primary numeral systems: the earlier Attic (acrophonic) system from the first millennium BCE, based on initial letters of number words, and the later Ionic (alphabetic) system, standardized around the 4th century BCE, which assigned values to letters of the alphabet.[24] In the Attic system, 175 is represented as Η𐅄ΔΔΔΠ (Η for 100, 𐅄 for 50, three Δ for 30, and Π for 5). In the Ionic system, 175 is written as ΡΟΕ´, where Ρ denotes 100, Ο represents 70, Ε signifies 5, and the overline (´) marks the symbols as numerals rather than letters.[25]Positional numeral systems
In positional numeral systems, numbers like 175 are expressed as sums of powers of the chosen base, where each digit represents a coefficient from 0 to base-1, positioned to indicate the power it multiplies. This allows compact representation in various computational and historical contexts, with the decimal (base-10) form 175 serving as the standard reference for conversions. The conversion process involves repeated division of the number by the base, recording remainders as digits from least to most significant, until the quotient reaches zero.[26] The following table summarizes 175's representations in selected positional bases, focusing on common systems used in computing and mathematics:| Base | Common Name | Representation | Digits Used |
|---|---|---|---|
| 2 | Binary | 10101111₂ | 0, 1 |
| 3 | Ternary | 20111₃ | 0, 1, 2 |
| 6 | Senary | 451₆ | 0–5 |
| 8 | Octal | 257₈ | 0–7 |
| 12 | Duodecimal | 127₁₂ | 0–9, A (for 10), B (for 11) |
| 16 | Hexadecimal | AF₁₆ | 0–9, A–F |