1 | # Units
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2 |
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3 | Math.js supports units. Units can be used to do calculations and to perform
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4 | conversions.
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5 |
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6 | ## Usage
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7 |
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8 | Units can be created using the function `math.unit`. This function accepts
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9 | either a single string argument containing a value and unit, or two arguments,
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10 | the first being a numeric value and the second a string containing a unit.
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11 | Most units support prefixes like `k` or `kilo`, and many units have both a
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12 | full name and an abbreviation. The returned object is a `Unit`.
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13 |
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14 | Syntax:
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15 |
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16 | ```js
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17 | math.unit(value: number, name: string) : Unit
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18 | math.unit(unit: string) : Unit
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19 | math.unit(unit: Unit) : Unit
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20 | ```
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21 |
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22 | Example usage:
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23 |
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24 | ```js
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25 | const a = math.unit(45, 'cm') // Unit 450 mm
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26 | const b = math.unit('0.1 kilogram') // Unit 100 gram
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27 | const c = math.unit('2 inch') // Unit 2 inch
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28 | const d = math.unit('90 km/h') // Unit 90 km/h
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29 | const e = math.unit('101325 kg/(m s^2)') // Unit 101325 kg / (m s^2)
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30 |
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31 | const d = c.to('cm') // Unit 5.08 cm
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32 | b.toNumber('gram') // Number 100
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33 | math.number(b, 'gram') // Number 100
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34 |
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35 | c.equals(a) // false
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36 | c.equals(d) // true
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37 | c.equalBase(a) // true
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38 | c.equalBase(b) // false
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39 |
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40 | d.toString() // String "5.08 cm"
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41 | ```
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42 |
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43 | Use care when creating a unit with multiple terms in the denominator. Implicit multiplication has the same operator precedence as explicit multiplication and division, which means these three expressions are identical:
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44 |
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45 | ```js
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46 | // These three are identical
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47 | const correct1 = math.unit('8.314 m^3 Pa / mol / K') // Unit 8.314 (m^3 Pa) / (mol K)
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48 | const correct2 = math.unit('8.314 (m^3 Pa) / (mol K)') // Unit 8.314 (m^3 Pa) / (mol K)
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49 | const correct3 = math.unit('8.314 (m^3 * Pa) / (mol * K)') // Unit 8.314 (m^3 Pa) / (mol K)
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50 | ```
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51 | But this expression, which omits the second `/` between `mol` and `K`, results in the wrong value:
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52 |
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53 | ```js
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54 | // Missing the second '/' between 'mol' and 'K'
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55 | const incorrect = math.unit('8.314 m^3 Pa / mol K') // Unit 8.314 (m^3 Pa K) / mol
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56 | ```
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57 |
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58 | ## Calculations
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59 |
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60 | The operations that support units are `add`, `subtract`, `multiply`, `divide`, `pow`, `abs`, `sqrt`, `square`, `cube`, and `sign`.
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61 | Trigonometric functions like `cos` are also supported when the argument is an angle.
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62 |
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63 | ```js
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64 | const a = math.unit(45, 'cm') // Unit 450 mm
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65 | const b = math.unit('0.1m') // Unit 100 mm
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66 | math.add(a, b) // Unit 0.65 m
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67 | math.multiply(b, 2) // Unit 200 mm
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68 |
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69 | const c = math.unit(45, 'deg') // Unit 45 deg
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70 | math.cos(c) // Number 0.7071067811865476
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71 |
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72 | // Kinetic energy of average sedan on highway
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73 | const d = math.unit('80 mi/h') // Unit 80 mi/h
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74 | const e = math.unit('2 tonne') // Unit 2 tonne
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75 | const f = math.multiply(0.5, math.multipy(math.pow(d, 2), e))
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76 | // 1.2790064742399996 MJ
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77 | ```
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78 |
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79 | Operations with arrays are supported too:
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80 |
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81 | ```js
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82 | // Force on a charged particle moving through a magnetic field
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83 | const B = math.evaluate('[1, 0, 0] T') // [1 T, 0 T, 0 T]
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84 | const v = math.evaluate('[0, 1, 0] m/s') // [0 m / s, 1 m / s, 0 m / s]
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85 | const q = math.evaluate('1 C') // 1 C
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86 |
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87 | const F = math.multiply(q, math.cross(v, B)) // [0 N, 0 N, -1 N]
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88 | ```
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89 |
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90 | All arithmetic operators act on the value of the unit as it is represented in SI units.
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91 | This may lead to surprising behavior when working with temperature scales like `celsius` (or `degC`) and `fahrenheit` (or `degF`).
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92 | In general you should avoid calculations using `celsius` and `fahrenheit`. Rather, use `kelvin` (or `K`) and `rankine` (or `R`) instead.
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93 | This example highlights some problems when using `celsius` and `fahrenheit` in calculations:
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94 |
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95 | ```js
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96 | const T_14F = math.unit('14 degF') // Unit 14 degF (263.15 K)
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97 | const T_28F = math.multiply(T1, 2) // Unit 487.67 degF (526.3 K), not 28 degF
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98 |
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99 | const Tnegative = math.unit(-13, 'degF') // Unit -13 degF (248.15 K)
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100 | const Tpositive = math.abs(T1) // Unit -13 degF (248.15 K), not 13 degF
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101 |
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102 | const Trate1 = math.evaluate('5 (degC/hour)') // Unit 5 degC/hour
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103 | const Trate2 = math.evaluate('(5 degC)/hour') // Unit 278.15 degC/hour
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104 | ```
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105 |
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106 | The expression parser supports units too. This is described in the section about
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107 | units on the page [Syntax](../expressions/syntax.md#units).
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108 |
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109 | ## User-Defined Units
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110 |
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111 | You can add your own units to Math.js using the `math.createUnit` function. The following example defines a new unit `furlong`, then uses the user-defined unit in a calculation:
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112 |
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113 | ```js
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114 | math.createUnit('furlong', '220 yards')
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115 | math.evaluate('1 mile to furlong') // 8 furlong
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116 | ```
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117 |
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118 | If you cannot express the new unit in terms of any existing unit, then the second argument can be omitted. In this case, a new *base unit* is created:
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119 |
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120 | ```js
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121 | // A 'foo' cannot be expressed in terms of any other unit.
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122 | math.createUnit('foo')
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123 | math.evaluate('8 foo * 4 feet') // 32 foo feet
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124 | ```
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125 |
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126 | The second argument to `createUnit` can also be a configuration object consisting of the following properties:
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127 |
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128 | * **definition** A `string` or `Unit` which defines the user-defined unit in terms of existing built-in or user-defined units. If omitted, a new base unit is created.
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129 | * **prefixes** A `string` indicating which prefixes math.js should use with the new unit. Possible values are `'none'`, `'short'`, `'long'`, `'binary_short'`, or `'binary_long'`. Default is `'none'`.
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130 | * **offset** A value applied when converting to the unit. This is very helpful for temperature scales that do not share a zero with the absolute temperature scale. For example, if we were defining fahrenheit for the first time, we would use: `math.createUnit('fahrenheit', {definition: '0.555556 kelvin', offset: 459.67})`
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131 | * **aliases** An array of strings to alias the new unit. Example: `math.createUnit('knot', {definition: '0.514444 m/s', aliases: ['knots', 'kt', 'kts']})`
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132 | * **baseName** A `string` that specifies the name of the new dimension in case one needs to be created. Every unit in math.js has a dimension: length, time, velocity, etc. If the unit's `definition` doesn't match any existing dimension, or it is a new base unit, then `createUnit` will create a new dimension with the name `baseName` and assign it to the new unit. The default is to append `'_STUFF'` to the unit's name. If the unit already matches an existing dimension, this option has no effect.
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133 |
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134 | An optional `options` object can also be supplied as the last argument to `createUnits`. Currently only the `override` option is supported:
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135 |
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136 | ```js
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137 | // Redefine the mile (would not be the first time in history)
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138 | math.createUnit('mile', '1609.347218694', {override: true}})
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139 | ```
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140 | Base units created without specifying a definition cannot be overridden.
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141 |
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142 | ### Create several units at once
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143 | Multiple units can defined using a single call to `createUnit` by passing an object map as the first argument, where each key in the object is the name of a new unit and the value is either a string defining the unit, or an object with the configuration properties listed above. If the value is an empty string or an object lacking a definition property, a new base unit is created.
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144 |
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145 | For example:
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146 |
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147 | ```js
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148 | math.createUnit( {
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149 | foo: {
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150 | prefixes: 'long',
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151 | baseName: 'essence-of-foo'
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152 | },
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153 | bar: '40 foo',
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154 | baz: {
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155 | definition: '1 bar/hour',
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156 | prefixes: 'long'
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157 | }
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158 | },
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159 | {
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160 | override: true
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161 | })
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162 | math.evaluate('50000 kilofoo/s') // 4.5 gigabaz
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163 | ```
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164 |
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165 | ### Return Value
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166 | `createUnit` returns the created unit, or, when multiple units are created, the last unit created. Since `createUnit` is also compatible with the expression parser, this allows you to do things like this:
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167 |
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168 | ```js
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169 | math.evaluate('45 mile/hour to createUnit("knot", "0.514444m/s")')
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170 | // 39.103964668651976 knot
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171 | ```
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172 |
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173 | ### Support of custom characters in unit names
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174 | Per default, the name of a new unit:
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175 | - should start by a latin (A-Z or a-z) character
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176 | - should contain only numeric (0-9) or latin characters
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177 |
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178 | It is possible to allow the usage of special characters (such as Greek alphabet, cyrillic alphabet, any Unicode symbols, etc.) by overriding the `Unit.isValidAlpha` static method. For example:
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179 | ```js
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180 | const isAlphaOriginal = math.Unit.isValidAlpha
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181 | const isGreekLowercaseChar = function (c) {
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182 | const charCode = c.charCodeAt(0)
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183 | return charCode > 944 && charCode < 970
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184 | }
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185 | math.Unit.isValidAlpha = function (c) {
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186 | return isAlphaOriginal(c) || isGreekLowercaseChar(c)
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187 | }
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188 |
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189 | math.createUnit('θ', '1 rad')
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190 | math.evaluate('1θ + 3 deg').toNumber('deg') // 60.29577951308232
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191 | ```
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192 |
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193 | ## API
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194 | A `Unit` object contains the following functions:
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195 |
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196 | ### unit.clone()
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197 | Clone the unit, returns a new unit with the same parameters.
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198 |
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199 | ### unit.equalBase(unit)
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200 | Test whether a unit has the same base as an other unit:
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201 | length, mass, etc.
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202 |
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203 | ### unit.equals(unit)
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204 | Test whether a unit equals an other unit. Units are equal
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205 | when they have the same base and same value when normalized to SI units.
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206 |
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207 | ### unit.format([options])
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208 | Get a string representation of the unit. The function
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209 | will determine the best fitting prefix for the unit. See the [Format](../reference/functions/format.md)
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210 | page for available options.
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211 |
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212 | ### unit.fromJSON(json)
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213 | Revive a unit from a JSON object. Accepts
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214 | An object `{mathjs: 'Unit', value: number, unit: string, fixPrefix: boolean}`,
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215 | where the property `mathjs` and `fixPrefix` are optional.
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216 | Used when deserializing a unit, see [Serialization](../core/serialization.md).
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217 |
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218 | ### unit.splitUnit(parts)
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219 | Split a unit into the specified parts. For example:
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220 |
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221 | ```js
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222 | const u = math.unit(1, 'm')
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223 | u.splitUnit(['ft', 'in']) // 3 feet,3.3700787401574765 inch
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224 | ```
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225 |
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226 | ### unit.to(unitName)
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227 | Convert the unit to a specific unit name. Returns a clone of
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228 | the unit with a fixed prefix and unit.
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229 |
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230 | ### unit.toJSON()
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231 | Returns a JSON representation of the unit, with signature
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232 | `{mathjs: 'Unit', value: number, unit: string, fixPrefix: boolean}`.
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233 | Used when serializing a unit, see [Serialization](../core/serialization.md).
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234 |
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235 | ### unit.toNumber(unitName)
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236 | Get the value of a unit when converted to the
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237 | specified unit (a unit with optional prefix but without value).
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238 | The type of the returned value is always `number`.
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239 |
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240 | ### unit.toNumeric(unitName)
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241 | Get the value of a unit when converted to the
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242 | specified unit (a unit with optional prefix but without value).
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243 | The type of the returned value depends on how the unit was created and
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244 | can be `number`, `Fraction`, or `BigNumber`.
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245 |
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246 | ### unit.toSI()
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247 | Returns a clone of a unit represented in SI units. Works with units with or without a value.
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248 |
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249 | ### unit.toString()
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250 | Get a string representation of the unit. The function will
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251 | determine the best fitting prefix for the unit.
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252 |
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253 | ## Unit reference
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254 |
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255 | This section lists all available units, prefixes, and physical constants. These can be used via the Unit object, or via `math.evaluate()`.
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256 |
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257 | ## Reference
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258 |
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259 | Math.js comes with the following built-in units.
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260 |
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261 | Base | Unit
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262 | ------------------- | ---
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263 | Length | meter (m), inch (in), foot (ft), yard (yd), mile (mi), link (li), rod (rd), chain (ch), angstrom, mil
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264 | Surface area | m2, sqin, sqft, sqyd, sqmi, sqrd, sqch, sqmil, acre, hectare
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265 | Volume | m3, litre (l, L, lt, liter), cc, cuin, cuft, cuyd, teaspoon, tablespoon
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266 | Liquid volume | minim (min), fluiddram (fldr), fluidounce (floz), gill (gi), cup (cp), pint (pt), quart (qt), gallon (gal), beerbarrel (bbl), oilbarrel (obl), hogshead, drop (gtt)
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267 | Angles | rad (radian), deg (degree), grad (gradian), cycle, arcsec (arcsecond), arcmin (arcminute)
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268 | Time | second (s, secs, seconds), minute (mins, minutes), hour (h, hr, hrs, hours), day (days), week (weeks), month (months), year (years), decade (decades), century (centuries), millennium (millennia)
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269 | Frequency | hertz (Hz)
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270 | Mass | gram(g), tonne, ton, grain (gr), dram (dr), ounce (oz), poundmass (lbm, lb, lbs), hundredweight (cwt), stick, stone
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271 | Electric current | ampere (A)
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272 | Temperature | kelvin (K), celsius (degC), fahrenheit (degF), rankine (degR)
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273 | Amount of substance | mole (mol)
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274 | Luminous intensity | candela (cd)
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275 | Force | newton (N), dyne (dyn), poundforce (lbf), kip
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276 | Energy | joule (J), erg, Wh, BTU, electronvolt (eV)
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277 | Power | watt (W), hp
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278 | Pressure | Pa, psi, atm, torr, bar, mmHg, mmH2O, cmH2O
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279 | Electricity and magnetism | ampere (A), coulomb (C), watt (W), volt (V), ohm, farad (F), weber (Wb), tesla (T), henry (H), siemens (S), electronvolt (eV)
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280 | Binary | bits (b), bytes (B)
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281 |
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282 | Note: all time units are based on the Julian year, with one month being 1/12th of a Julian year, a year being one Julian year, a decade being 10 Julian years, a century being 100, and a millennium being 1000.
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283 |
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284 | Note that all relevant units can also be written in plural form, for example `5 meters` instead of `5 meter` or `10 seconds` instead of `10 second`.
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285 |
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286 | Surface and volume units can alternatively be expressed in terms of length units raised to a power, for example `100 in^2` instead of `100 sqin`.
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287 |
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288 | ### Prefixes
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289 |
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290 | The following decimal prefixes are available.
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291 |
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292 | Name | Abbreviation | Value
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293 | ------- | ------------- | -----
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294 | deca | da | 1e1
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295 | hecto | h | 1e2
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296 | kilo | k | 1e3
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297 | mega | M | 1e6
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298 | giga | G | 1e9
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299 | tera | T | 1e12
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300 | peta | P | 1e15
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301 | exa | E | 1e18
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302 | zetta | Z | 1e21
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303 | yotta | Y | 1e24
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304 |
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305 | Name | Abbreviation | Value
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306 | ------ | ------------- | -----
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307 | deci | d | 1e-1
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308 | centi | c | 1e-2
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309 | milli | m | 1e-3
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310 | micro | u | 1e-6
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311 | nano | n | 1e-9
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312 | pico | p | 1e-12
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313 | femto | f | 1e-15
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314 | atto | a | 1e-18
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315 | zepto | z | 1e-21
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316 | yocto | y | 1e-24
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317 |
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318 | The following binary prefixes are available.
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319 | They can be used with units `bits` (`b`) and `bytes` (`B`).
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320 |
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321 | Name | Abbreviation | Value
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322 | ---- | ------------ | -----
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323 | kibi | Ki | 1024
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324 | mebi | Mi | 1024^2
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325 | gibi | Gi | 1024^3
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326 | tebi | Ti | 1024^4
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327 | pebi | Pi | 1024^5
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328 | exi | Ei | 1024^6
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329 | zebi | Zi | 1024^7
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330 | yobi | Yi | 1024^8
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331 |
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332 | Name | Abbreviation | Value
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333 | ----- | ------------ | -----
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334 | kilo | k | 1e3
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335 | mega | M | 1e6
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336 | giga | G | 1e9
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337 | tera | T | 1e12
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338 | peta | P | 1e15
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339 | exa | E | 1e18
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340 | zetta | Z | 1e21
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341 | yotta | Y | 1e24
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342 |
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343 |
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344 | ### Physical Constants
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345 |
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346 | Math.js includes the following physical constants. See [Wikipedia](https://en.wikipedia.org/wiki/Physical_constants) for more information.
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347 |
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348 |
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349 | #### Universal constants
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350 |
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351 | Name | Symbol | Value | Unit
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352 | ----------------------|--------------------------------------------------------|-------------------|-------------------------------------------------------
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353 | speedOfLight | <i>c</i> | 299792458 | m · s<sup>-1</sup>
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354 | gravitationConstant | <i>G</i> | 6.6738480e-11 | m<sup>3</sup> · kg<sup>-1</sup> · s<sup>-2</sup>
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355 | planckConstant | <i>h</i> | 6.626069311e-34 | J · s
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356 | reducedPlanckConstant | <i><span style="text-decoration:overline">h</span></i> | 1.05457172647e-34 | J · s
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357 |
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358 |
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359 | #### Electromagnetic constants
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360 |
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361 | Name | Symbol | Value | Unit
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362 | --------------------------|--------------------------------------------------|-----------------------|----------------------------------------
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363 | magneticConstant | <i>μ<sub>0</sub></i> | 1.2566370614e-6 | N · A<sup>-2</sup>
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364 | electricConstant | <i>ε<sub>0</sub></i> | 8.854187817e-12 | F · m<sup>-1</sup>
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365 | vacuumImpedance | <i>Z<sub>0</sub></i> | 376.730313461 | Ω
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366 | coulomb | <i>κ</i> | 8.9875517873681764e9 | N · m<sup>2</sup> · C<sup>-2</sup>
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367 | elementaryCharge | <i>e</i> | 1.60217656535e-19 | C
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368 | bohrMagneton | <i>μ<sub>B</sub></i> | 9.2740096820e-24 | J · T<sup>-1</sup>
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369 | conductanceQuantum | <i>G<sub>0</sub></i> | 7.748091734625e-5 | S
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370 | inverseConductanceQuantum | <i>G<sub>0</sub><sup>-1</sup></i> | 12906.403721742 | Ω
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371 | magneticFluxQuantum | <i><font face="Symbol">f</font><sub>0</sub></i> | 2.06783375846e-15 | Wb
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372 | nuclearMagneton | <i>μ<sub>N</sub></i> | 5.0507835311e-27 | J · T<sup>-1</sup>
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373 | klitzing | <i>R<sub>K</sub></i> | 25812.807443484 | Ω
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374 |
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375 |
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376 | josephson | <i>K<sub>J</sub></i> | 4.8359787011e-14 | Hz · V<sup>-1</sup>
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377 | -->
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378 |
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379 |
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380 | #### Atomic and nuclear constants
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381 |
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382 | Name | Symbol | Value | Unit
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383 | ------------------------|------------------------------|-----------------------|----------------------------------
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384 | bohrRadius | <i>a<sub>0</sub></i> | 5.291772109217e-11 | m
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385 | classicalElectronRadius | <i>r<sub>e</sub></i> | 2.817940326727e-15 | m
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386 | electronMass | <i>m<sub>e</sub></i> | 9.1093829140e-31 | kg
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387 | fermiCoupling | <i>G<sub>F</sub></i> | 1.1663645e-5 | GeV<sup>-2</sup>
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388 | fineStructure | <i>α</i> | 7.297352569824e-3 | -
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389 | hartreeEnergy | <i>E<abbr>h</abbr> </i> | 4.3597443419e-18 | J
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390 | protonMass | <i>m<sub>p</sub></i> | 1.67262177774e-27 | kg
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391 | deuteronMass | <i>m<sub>d</sub></i> | 3.3435830926e-27 | kg
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392 | neutronMass | <i>m<sub>n</sub></i> | 1.6749271613e-27 | kg
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393 | quantumOfCirculation | <i>h / (2m<sub>e</sub>)</i> | 3.636947552024e-4 | m<sup>2</sup> · s<sup>-1</sup>
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394 | rydberg | <i>R<sub>∞</sub></i> | 10973731.56853955 | m<sup>-1</sup>
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395 | thomsonCrossSection | | 6.65245873413e-29 | m<sup>2</sup>
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396 | weakMixingAngle | | 0.222321 | -
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397 | efimovFactor | | 22.7 | -
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398 |
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399 |
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400 | #### Physico-chemical constants
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401 |
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402 | Name | Symbol | Value | Unit
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403 | --------------------|------------------------------|---------------------|--------------------------------------------
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404 | atomicMass | <i>m<sub>u</sub></i> | 1.66053892173e-27 | kg
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405 | avogadro | <i>N<sub>A</sub></i> | 6.0221412927e23 | mol<sup>-1</sup>
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406 | boltzmann | <i>k</i> | 1.380648813e-23 | J · K<sup>-1</sup>
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407 | faraday | <i>F</i> | 96485.336521 | C · mol<sup>-1</sup>
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408 | firstRadiation | <i>c<sub>1</sub></i> | 3.7417715317e-16 | W · m<sup>2</sup>
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409 | loschmidt | <i>n<sub>0</sub></i> | 2.686780524e25 | m<sup>-3</sup>
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410 | gasConstant | <i>R</i> | 8.314462175 | J · K<sup>-1</sup> · mol<sup>-1</sup>
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411 | molarPlanckConstant | <i>N<sub>A</sub> · h</i>| 3.990312717628e-10| J · s · mol<sup>-1</sup>
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412 | molarVolume | <i>V<sub>m</sub></i> | 2.241396820e-10 | m<sup>3</sup> · mol<sup>-1</sup>
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413 | sackurTetrode | | -1.164870823 | -
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414 | secondRadiation | <i>c<sub>2</sub></i> | 1.438777013e-2 | m · K
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415 | stefanBoltzmann | <i>σ</i> | 5.67037321e-8 | W · m<sup>-2</sup> · K<sup>-4</sup>
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416 | wienDisplacement | <i>b</i> | 2.897772126e-3 | m · K
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417 |
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418 |
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419 | spectralRadiance | <i>c<sub>1L</sub></i> | 1.19104286953e-16 | W · m<sup>2</sup> · sr<sup>-1</sup>
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420 | -->
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421 |
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422 | Note that the values of `loschmidt` and `molarVolume` are at `T = 273.15 K` and `p = 101.325 kPa`.
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423 | The value of `sackurTetrode` is at `T = 1 K` and `p = 101.325 kPa`.
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424 |
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425 |
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426 | #### Adopted values
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427 |
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428 | Name | Symbol | Value | Unit
|
429 | --------------|------------------------------|---------|-------------------------
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430 | molarMass | <i>M<sub>u</sub></i> | 1e-3 | kg · mol<sup>-1</sup>
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431 | molarMassC12 | <i>M(<sub>12</sub>C)</i> | 1.2e-2 | kg · mol<sup>-1</sup>
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432 | gravity | <i>g<sub>n</sub></i> | 9.80665 | m · s<sup>-2</sup>
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433 | atm | <i>atm</i> | 101325 | Pa
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434 |
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435 |
|
436 | #### Natural units
|
437 |
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438 | Name | Symbol | Value | Unit
|
439 | ------------------|-----------------------|--------------------|-----
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440 | planckLength | <i>l<sub>P</sub></i> | 1.61619997e-35 | m
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441 | planckMass | <i>m<sub>P</sub></i> | 2.1765113e-8 | kg
|
442 | planckTime | <i>t<sub>P</sub></i> | 5.3910632e-44 | s
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443 | planckCharge | <i>q<sub>P</sub></i> | 1.87554595641e-18 | C
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444 | planckTemperature | <i>T<sub>P</sub></i> | 1.41683385e+32 | K
|