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# U+2140Double-Struck N-Ary Summation

U+2140 was added in Unicode version 3.2 in 2002. It belongs to the block U+2100 to U+214F Letterlike Symbols in the U+0000 to U+FFFF Basic Multilingual Plane.

This character is a Math Symbol and is commonly used, that is, in no specific script.

The glyph is a font version of the glyph N-Ary Summation. It has no designated width in East Asian texts. In bidirectional text it acts as Other Neutral. When changing direction it is mirrored. The word that U+2140 forms with similar adjacent characters prevents a line break inside it. The glyph can be confused with one other glyph.

In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.

Summations of infinite sequences are called series. They involve the concept of limit, and are not considered in this article.

The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need of parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one element results in this element itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0.

Very often, the elements of a sequence are defined, through a regular pattern, as a function of their place in the sequence. For simple patterns, summation of long sequences may be represented with most summands replaced by ellipses. For example, summation of the first 100 natural numbers may be written as 1 + 2 + 3 + 4 + ⋯ + 99 + 100. Otherwise, summation is denoted by using Σ notation, where $\sum$ is an enlarged capital Greek letter sigma. For example, the sum of the first n natural numbers can be denoted as $\sum _{i=1}^{n}i.$

For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find closed-form expressions for the result. For example,

$\sum _{i=1}^{n}i=\frac{n\left(n+1\right)}{2}.$

Although such formulas do not always exist, many summation formulas have been discovered—with some of the most common and elementary ones being listed in the remainder of this article.

## Representations

System Representation
8512
UTF-8 E2 85 80
UTF-16 21 40
UTF-32 00 00 21 40
URL-Quoted %E2%85%80
HTML hex reference &#x2140;
Wrong windows-1252 Mojibake â…€

## Complete Record

Property Value
Age (age) 3.2 (2002)
Unicode Name (na) DOUBLE-STRUCK N-ARY SUMMATION
Unicode 1 Name (na1)
Block (blk) Letterlike Symbols
General Category (gc) Math Symbol
Script (sc) Common
Bidirectional Category (bc) Other Neutral
Combining Class (ccc) Not Reordered
Decomposition Type (dt) font
Decomposition Mapping (dm) N-Ary Summation
Lowercase (Lower)
Simple Lowercase Mapping (slc) Double-Struck N-Ary Summation
Lowercase Mapping (lc) Double-Struck N-Ary Summation
Uppercase (Upper)
Simple Uppercase Mapping (suc) Double-Struck N-Ary Summation
Uppercase Mapping (uc) Double-Struck N-Ary Summation
Simple Titlecase Mapping (stc) Double-Struck N-Ary Summation
Titlecase Mapping (tc) Double-Struck N-Ary Summation
Case Folding (cf) Double-Struck N-Ary Summation
ASCII Hex Digit (AHex)
Alphabetic (Alpha)
Bidi Control (Bidi_C)
Bidi Mirrored (Bidi_M)
Composition Exclusion (CE)
Case Ignorable (CI)
Changes When Casefolded (CWCF)
Changes When Casemapped (CWCM)
Changes When NFKC Casefolded (CWKCF)
Changes When Lowercased (CWL)
Changes When Titlecased (CWT)
Changes When Uppercased (CWU)
Cased (Cased)
Full Composition Exclusion (Comp_Ex)
Default Ignorable Code Point (DI)
Dash (Dash)
Deprecated (Dep)
Diacritic (Dia)
Emoji Modifier Base (EBase)
Emoji Component (EComp)
Emoji Modifier (EMod)
Emoji Presentation (EPres)
Emoji (Emoji)
Extender (Ext)
Extended Pictographic (ExtPict)
FC NFKC Closure (FC_NFKC) Double-Struck N-Ary Summation
Grapheme Cluster Break (GCB) Any
Grapheme Base (Gr_Base)
Grapheme Extend (Gr_Ext)
Hex Digit (Hex)
Hyphen (Hyphen)
ID Continue (IDC)
ID Start (IDS)
IDS Binary Operator (IDSB)
IDS Trinary Operator and (IDST)
IDSU (IDSU) 0
ID_Compat_Math_Continue (ID_Compat_Math_Continue) 0
ID_Compat_Math_Start (ID_Compat_Math_Start) 0
Ideographic (Ideo)
InCB (InCB) None
Indic Mantra Category (InMC)
Indic Positional Category (InPC) NA
Indic Syllabic Category (InSC) Other
Jamo Short Name (JSN)
Join Control (Join_C)
Logical Order Exception (LOE)
Math (Math)
Noncharacter Code Point (NChar)
NFC Quick Check (NFC_QC) Yes
NFD Quick Check (NFD_QC) Yes
NFKC Casefold (NFKC_CF) N-Ary Summation
NFKC Quick Check (NFKC_QC) No
NFKC_SCF (NFKC_SCF) N-Ary Summation
NFKD Quick Check (NFKD_QC) No
Other Alphabetic (OAlpha)
Other Default Ignorable Code Point (ODI)
Other Grapheme Extend (OGr_Ext)
Other ID Continue (OIDC)
Other ID Start (OIDS)
Other Lowercase (OLower)
Other Math (OMath)
Other Uppercase (OUpper)
Prepended Concatenation Mark (PCM)
Pattern Syntax (Pat_Syn)
Pattern White Space (Pat_WS)
Quotation Mark (QMark)
Regional Indicator (RI)
Sentence Break (SB) Other
Soft Dotted (SD)
Sentence Terminal (STerm)
Terminal Punctuation (Term)
Unified Ideograph (UIdeo)
Variation Selector (VS)
Word Break (WB) Other
White Space (WSpace)
XID Continue (XIDC)
XID Start (XIDS)
Expands On NFC (XO_NFC)
Expands On NFD (XO_NFD)
Expands On NFKC (XO_NFKC)
Expands On NFKD (XO_NFKD)
Bidi Paired Bracket (bpb) Double-Struck N-Ary Summation
Bidi Paired Bracket Type (bpt) None
East Asian Width (ea) neutral
Hangul Syllable Type (hst) Not Applicable
ISO 10646 Comment (isc)
Joining Group (jg) No_Joining_Group
Joining Type (jt) Non Joining
Line Break (lb) Alphabetic
Numeric Type (nt) none
Numeric Value (nv) not a number
Simple Case Folding (scf) Double-Struck N-Ary Summation
Script Extension (scx)
Vertical Orientation (vo) R