Unit Knowledge Management Jonathan Stratford & James H. - - PowerPoint PPT Presentation

Unit Knowledge Management

Jonathan Stratford & James H. Davenport Department of Computer Science University of Bath Bath BA2 7AY England Jonathan.Stratford@alumni.bath.ac.uk J.H.Davenport@bath.ac.uk July 29, 2008

Unit Converters ▼❛♥② ♦♥ t❤❡ ✇❡❜ ◆♦t ❡①t❡♥s✐❜❧❡✱ tr❛♥s♣❛r❡♥t ❡t❝✳ ◆♦t t②♣❡❞ ✭♥♦r♠❛❧❧②✮ ❆❧t❤♦✉❣❤ ❞✐♠❡♥s✐♦♥s t②♣❡ ✉♥✐ts

• ❡♥❡r❛❧❧② ♠♦♥♦❧✐t❤✐❝

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Unit Converters Many on the web ◆♦t ❡①t❡♥s✐❜❧❡✱ tr❛♥s♣❛r❡♥t ❡t❝✳ ◆♦t t②♣❡❞ ✭♥♦r♠❛❧❧②✮ ❆❧t❤♦✉❣❤ ❞✐♠❡♥s✐♦♥s t②♣❡ ✉♥✐ts

• ❡♥❡r❛❧❧② ♠♦♥♦❧✐t❤✐❝

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Unit Converters Many on the web Not extensible, transparent etc. ◆♦t t②♣❡❞ ✭♥♦r♠❛❧❧②✮ ❆❧t❤♦✉❣❤ ❞✐♠❡♥s✐♦♥s t②♣❡ ✉♥✐ts

• ❡♥❡r❛❧❧② ♠♦♥♦❧✐t❤✐❝

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Unit Converters Many on the web Not extensible, transparent etc. Not typed (normally) ❆❧t❤♦✉❣❤ ❞✐♠❡♥s✐♦♥s t②♣❡ ✉♥✐ts

• ❡♥❡r❛❧❧② ♠♦♥♦❧✐t❤✐❝

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Unit Converters Many on the web Not extensible, transparent etc. Not typed (normally) Although dimensions type units

• ❡♥❡r❛❧❧② ♠♦♥♦❧✐t❤✐❝

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Unit Converters Many on the web Not extensible, transparent etc. Not typed (normally) Although dimensions type units Generally monolithic

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Units in OpenMath ✎ ❉❡s❝r✐❜❡❞ ❜② ❉❛✈❡♥♣♦rt ✫ ◆❛②❧♦r ✎ ❯s❡s ❙✐♠♣❧❡ ❚②♣❡ ❙②st❡♠ t♦ ❞♦ ❞✐♠❡♥s✐♦♥s ✎ ❍❛♥❞❧❡s ❙■ ♣r❡☞①✐♥❣

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Units in OpenMath

• Described by Davenport & Naylor

✎ ❯s❡s ❙✐♠♣❧❡ ❚②♣❡ ❙②st❡♠ t♦ ❞♦ ❞✐♠❡♥s✐♦♥s ✎ ❍❛♥❞❧❡s ❙■ ♣r❡☞①✐♥❣

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Units in OpenMath

• Described by Davenport & Naylor
• Uses Simple Type System to do dimensions

✎ ❭❍❛♥❞❧❡s✧ ❙■ ♣r❡☞①✐♥❣

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Units in OpenMath

• Described by Davenport & Naylor
• Uses Simple Type System to do dimensions
• “Handles” SI prefixing

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Typing is Important Distinguishes Between ✎ ❨♦✉ ❝❛♥✬t ❝♦♥✈❡rt ❳ t♦ ❨ ❡✳❣✳ ♠❡tr❡s t♦ ❦✐❧♦❣r❛♠s ✎ ■ ❞♦♥✬t ❦♥♦✇ ❤♦✇ t♦ ❝♦♥✈❡rt ❳ t♦ ❨ ❡✳❣✳ ❞❛②s t♦ ❝❛❧❡♥❞❛r ♠♦♥t❤s ❡✳❣✳ ❡❧❡❝tr♦♥✈♦❧ts t♦ ❥♦✉❧❡s ✭❜② ❡①♣❡r✐♠❡♥t❛❧ ❞❡✲ t❡r♠✐♥❛t✐♦♥✮

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Typing is Important Distinguishes Between

• You can’t convert X to Y

❡✳❣✳ ♠❡tr❡s t♦ ❦✐❧♦❣r❛♠s ✎ ■ ❞♦♥✬t ❦♥♦✇ ❤♦✇ t♦ ❝♦♥✈❡rt ❳ t♦ ❨ ❡✳❣✳ ❞❛②s t♦ ❝❛❧❡♥❞❛r ♠♦♥t❤s ❡✳❣✳ ❡❧❡❝tr♦♥✈♦❧ts t♦ ❥♦✉❧❡s ✭❜② ❡①♣❡r✐♠❡♥t❛❧ ❞❡✲ t❡r♠✐♥❛t✐♦♥✮

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Typing is Important Distinguishes Between

• You can’t convert X to Y

e.g. metres to kilograms ✎ ■ ❞♦♥✬t ❦♥♦✇ ❤♦✇ t♦ ❝♦♥✈❡rt ❳ t♦ ❨ ❡✳❣✳ ❞❛②s t♦ ❝❛❧❡♥❞❛r ♠♦♥t❤s ❡✳❣✳ ❡❧❡❝tr♦♥✈♦❧ts t♦ ❥♦✉❧❡s ✭❜② ❡①♣❡r✐♠❡♥t❛❧ ❞❡✲ t❡r♠✐♥❛t✐♦♥✮

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Typing is Important Distinguishes Between

• You can’t convert X to Y

e.g. metres to kilograms

• I don’t know how to convert X to Y

e.g. days to calendar months ❡✳❣✳ ❡❧❡❝tr♦♥✈♦❧ts t♦ ❥♦✉❧❡s ✭❜② ❡①♣❡r✐♠❡♥t❛❧ ❞❡✲ t❡r♠✐♥❛t✐♦♥✮

14

Typing is Important Distinguishes Between

• You can’t convert X to Y

e.g. metres to kilograms

• I don’t know how to convert X to Y

e.g. days to calendar months e.g. electronvolts to joules (by experimental de- termination)

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Aren’t Intervals the Answer Here? ❭❆♥②♦♥❡ ✇❤♦ t❤✐♥❦s ✐♥t❡r✈❛❧s ❛r❡ t❤❡ ❛♥s✇❡r ❞♦❡s♥✬t ✉♥❞❡rst❛♥❞ ✐♥t❡r✈❛❧s ❛♥❞ ❞♦❡s♥✬t ✉♥✲ ❞❡rst❛♥❞ t❤❡ q✉❡st✐♦♥✧✳ ❬❑❛❤❛♥❄❪ ✎ ❋♦r ♣❤②s✐❝❛❧ ✉♥✐ts✱ ✇❤❛t ✇❡ ❣❡♥❡r❛❧❧② ❤❛✈❡ ✐s ❛ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥✱ ♥♦t ❛♥ ✐♥t❡r✈❛❧ ✎ ❈♦rr❡❧❛t✐♦♥s ❛r❡ ♥♦t ♥♦r♠❛❧❧② r❡❝♦r❞❡❞ ✐♥ ✇❤❛t ✐s ♣✉❜❧✐s❤❡❞ ✎ ❈❛❧❡♥❞r✐❝ t✐♠❡ ✐s ❛ ❝♦♠♣❧❡t❡ ♠❡ss

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Aren’t Intervals the Answer Here? “Anyone who thinks intervals are the answer doesn’t understand intervals and doesn’t un- derstand the question”. [Kahan?] ✎ ❋♦r ♣❤②s✐❝❛❧ ✉♥✐ts✱ ✇❤❛t ✇❡ ❣❡♥❡r❛❧❧② ❤❛✈❡ ✐s ❛ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥✱ ♥♦t ❛♥ ✐♥t❡r✈❛❧ ✎ ❈♦rr❡❧❛t✐♦♥s ❛r❡ ♥♦t ♥♦r♠❛❧❧② r❡❝♦r❞❡❞ ✐♥ ✇❤❛t ✐s ♣✉❜❧✐s❤❡❞ ✎ ❈❛❧❡♥❞r✐❝ t✐♠❡ ✐s ❛ ❝♦♠♣❧❡t❡ ♠❡ss

17

Aren’t Intervals the Answer Here? “Anyone who thinks intervals are the answer doesn’t understand intervals and doesn’t un- derstand the question”. [Kahan?]

• For physical units, what we generally have

is a standard deviation, not an interval ✎ ❈♦rr❡❧❛t✐♦♥s ❛r❡ ♥♦t ♥♦r♠❛❧❧② r❡❝♦r❞❡❞ ✐♥ ✇❤❛t ✐s ♣✉❜❧✐s❤❡❞ ✎ ❈❛❧❡♥❞r✐❝ t✐♠❡ ✐s ❛ ❝♦♠♣❧❡t❡ ♠❡ss

18

Aren’t Intervals the Answer Here? “Anyone who thinks intervals are the answer doesn’t understand intervals and doesn’t un- derstand the question”. [Kahan?]

• For physical units, what we generally have

is a standard deviation, not an interval

• Correlations are not normally recorded in

what is published ✎ ❈❛❧❡♥❞r✐❝ t✐♠❡ ✐s ❛ ❝♦♠♣❧❡t❡ ♠❡ss

19

Aren’t Intervals the Answer Here? “Anyone who thinks intervals are the answer doesn’t understand intervals and doesn’t un- derstand the question”. [Kahan?]

• For physical units, what we generally have

is a standard deviation, not an interval

• Correlations are not normally recorded in

what is published

• Calendric time is a complete mess

20

year month = 12; month day ∈ [28, 31] = {28, 29, 30, 31} but year day ∈ [365, 366] ⊆ [12 ∗ 28, 12 ∗ 31] / ∈ {12 ∗ 28, 12 ∗ 29, 12 ∗ 30, 12 ∗ 31}

21

Convert Months into Days ✶✳ ❚❤❡② ❛r❡ ❜♦t❤ t✐♠❡✱ s♦ t❤❡ ❝♦♥✈❡rs✐♦♥ ✐s ♠❡❛♥✐♥❣❢✉❧✱ ❜✉t ■ ❞♦♥✬t ❤❛✈❡ ❛♥ ❡①❛❝t ❝♦♥✲ ✈❡rs✐♦♥ ❢❛❝t♦r ✭♦✉r s♦❧✉t✐♦♥✮ ✷✳ ❚❤❡r❡ ❛r❡ ✸✵ ❂ ✸✵✿✹✸✻✽✼✺✵✵ ❞❛②s ✐♥ ❛ ♠♦♥t❤✿ ❝♦rr❡❝t ♦♥ ❛✈❡r❛❣❡✱ ❜✉t ❢❛❧s❡ ❢♦r ❡✈❡r② ♠♦♥t❤ ✭●♦♦❣❧❡ ✉s❡s ✸✵✳✹✸✻✽✹✾✾✮✦ ✸✳ ❚❤❡r❡ ❛r❡ ✸✵ ❞❛②s ✐♥ ❛ ♠♦♥t❤✱ ✇❤✐❝❤ ✐s ❭t❤❡ ♥❡❛r❡st✧✱ ❜✉t ♥♦t t❤❡ ♠♦st ❝♦♠♠♦♥✳ ▲❡❛❞s t♦ ❭✶ ❞❡❝❛❞❡ ❂ ✶✷✶ ♠♦♥t❤s✧✳ ✹✳ ■ ❞♦♥✬t ❦♥♦✇ ❛❜♦✉t ♠♦♥t❤s✳

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Convert Months into Days

• 1. They are both time, so the conversion is

meaningful, but I don’t have an exact con- version factor (our solution) ✷✳ ❚❤❡r❡ ❛r❡ ✸✵ ❂ ✸✵✿✹✸✻✽✼✺✵✵ ❞❛②s ✐♥ ❛ ♠♦♥t❤✿ ❝♦rr❡❝t ♦♥ ❛✈❡r❛❣❡✱ ❜✉t ❢❛❧s❡ ❢♦r ❡✈❡r② ♠♦♥t❤ ✭●♦♦❣❧❡ ✉s❡s ✸✵✳✹✸✻✽✹✾✾✮✦ ✸✳ ❚❤❡r❡ ❛r❡ ✸✵ ❞❛②s ✐♥ ❛ ♠♦♥t❤✱ ✇❤✐❝❤ ✐s ❭t❤❡ ♥❡❛r❡st✧✱ ❜✉t ♥♦t t❤❡ ♠♦st ❝♦♠♠♦♥✳ ▲❡❛❞s t♦ ❭✶ ❞❡❝❛❞❡ ❂ ✶✷✶ ♠♦♥t❤s✧✳ ✹✳ ■ ❞♦♥✬t ❦♥♦✇ ❛❜♦✉t ♠♦♥t❤s✳

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Convert Months into Days

• 1. They are both time, so the conversion is

meaningful, but I don’t have an exact con- version factor (our solution)

• 2. There are 30 699

1600 = 30.43687500 days in

a month: correct on average, but false for every month (Google uses 30.4368499)! ✸✳ ❚❤❡r❡ ❛r❡ ✸✵ ❞❛②s ✐♥ ❛ ♠♦♥t❤✱ ✇❤✐❝❤ ✐s ❭t❤❡ ♥❡❛r❡st✧✱ ❜✉t ♥♦t t❤❡ ♠♦st ❝♦♠♠♦♥✳ ▲❡❛❞s t♦ ❭✶ ❞❡❝❛❞❡ ❂ ✶✷✶ ♠♦♥t❤s✧✳ ✹✳ ■ ❞♦♥✬t ❦♥♦✇ ❛❜♦✉t ♠♦♥t❤s✳

24

Convert Months into Days

• 1. They are both time, so the conversion is

meaningful, but I don’t have an exact con- version factor (our solution)

• 2. There are 30 699

1600 = 30.43687500 days in

a month: correct on average, but false for every month (Google uses 30.4368499)!

• 3. There are 30 days in a month, which is

“the nearest”, but not the most common. Leads to “1 decade = 1212

3 months”.

✹✳ ■ ❞♦♥✬t ❦♥♦✇ ❛❜♦✉t ♠♦♥t❤s✳

25

Convert Months into Days

• 1. They are both time, so the conversion is

meaningful, but I don’t have an exact con- version factor (our solution)

• 2. There are 30 699

1600 = 30.43687500 days in

a month: correct on average, but false for every month (Google uses 30.4368499)!

• 3. There are 30 days in a month, which is

“the nearest”, but not the most common. Leads to “1 decade = 1212

3 months”.

• 4. I don’t know about months.

26

“OpenMath handles prefixing” ✰ Pr❡☞①❡s ❞❡☞♥❡❞ ✐♥ ✉♥✐ts❴s✐♣r❡❢✐①✶ ✰ ♣r❡❢✐①✿ ♣r❡☞① ✂ ✉♥✐t ✦ ✉♥✐t ❆❧❧♦✇s ❵♠✐❧❧✐♠✐❝r♦♠❡tr❡✬ ♣r❡☞① ✂ ✉♥✐t ✦ ♣r❡☞①❡❞✉♥✐t

27

“OpenMath handles prefixing” + Prefixes defined in units_siprefix1 ✰ ♣r❡❢✐①✿ ♣r❡☞① ✂ ✉♥✐t ✦ ✉♥✐t ❆❧❧♦✇s ❵♠✐❧❧✐♠✐❝r♦♠❡tr❡✬ ♣r❡☞① ✂ ✉♥✐t ✦ ♣r❡☞①❡❞✉♥✐t

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“OpenMath handles prefixing” + Prefixes defined in units_siprefix1 + prefix: prefix × unit → unit ❆❧❧♦✇s ❵♠✐❧❧✐♠✐❝r♦♠❡tr❡✬ ♣r❡☞① ✂ ✉♥✐t ✦ ♣r❡☞①❡❞✉♥✐t

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“OpenMath handles prefixing” + Prefixes defined in units_siprefix1 + prefix: prefix × unit → unit − Allows ‘millimicrometre’ ♣r❡☞① ✂ ✉♥✐t ✦ ♣r❡☞①❡❞✉♥✐t

30

“OpenMath handles prefixing” + Prefixes defined in units_siprefix1 + prefix: prefix × unit → unit − Allows ‘millimicrometre’ √ prefix × unit → prefixedunit

31

Prefixing the tonne ❚❤❡ t♦♥♥❡ ✐s ♥♦t ❙■✱ ❛♥❞ ♠❡r❡❧② ❛♥ ❛❧✐❛s ❢♦r t❤❡ ♠❡❣❛❣r❛♠♠❡✱ s♦ s❤♦✉❧❞♥✬t t❛❦❡ ♣r❡☞①❡s✳ ■♥ ❵♠❡❣❛t♦♥ ❜♦♠❜✬ ✇❡ ❤❛✈❡ t❤❡ ♠❡❣❛❬t♦♥ ♦❢ ❚◆❚ ❡q✉✐✈❛❧❡♥t❪✱ ✐✳❡✳ ✹✳✶✽✹ ♣❡t❛❥♦✉❧❡s✳ ❍♦✇❡✈❡r ✐♥ ❇❡❧❣✐✉♠✱ ❦✐❧♦t♦♥♥❡✿ ✿ ✿❡①❛t♦♥♥❡ ✭❜✉t ♥♦t ③❡tt❛t♦♥♥❡ ♦r ②♦tt❛t♦♥♥❡✮ ❛r❡ r❡❝♦❣♥✐s❡❞✳ ❇❡❧❣✐✉♠ ❛❧s♦ r❡❝♦❣♥✐s❡s t❤❡ ❝❡♥t✐❛r❡✱ ❛❧✐❛s ♠❡tr❡✷✳

32

Prefixing the tonne The tonne is not SI, and merely an alias for the megagramme, so shouldn’t take prefixes. ■♥ ❵♠❡❣❛t♦♥ ❜♦♠❜✬ ✇❡ ❤❛✈❡ t❤❡ ♠❡❣❛❬t♦♥ ♦❢ ❚◆❚ ❡q✉✐✈❛❧❡♥t❪✱ ✐✳❡✳ ✹✳✶✽✹ ♣❡t❛❥♦✉❧❡s✳ ❍♦✇❡✈❡r ✐♥ ❇❡❧❣✐✉♠✱ ❦✐❧♦t♦♥♥❡✿ ✿ ✿❡①❛t♦♥♥❡ ✭❜✉t ♥♦t ③❡tt❛t♦♥♥❡ ♦r ②♦tt❛t♦♥♥❡✮ ❛r❡ r❡❝♦❣♥✐s❡❞✳ ❇❡❧❣✐✉♠ ❛❧s♦ r❡❝♦❣♥✐s❡s t❤❡ ❝❡♥t✐❛r❡✱ ❛❧✐❛s ♠❡tr❡✷✳

33

Prefixing the tonne The tonne is not SI, and merely an alias for the megagramme, so shouldn’t take prefixes. In ‘megaton bomb’ we have the mega[ton of TNT equivalent], i.e. 4.184 petajoules. ❍♦✇❡✈❡r ✐♥ ❇❡❧❣✐✉♠✱ ❦✐❧♦t♦♥♥❡✿ ✿ ✿❡①❛t♦♥♥❡ ✭❜✉t ♥♦t ③❡tt❛t♦♥♥❡ ♦r ②♦tt❛t♦♥♥❡✮ ❛r❡ r❡❝♦❣♥✐s❡❞✳ ❇❡❧❣✐✉♠ ❛❧s♦ r❡❝♦❣♥✐s❡s t❤❡ ❝❡♥t✐❛r❡✱ ❛❧✐❛s ♠❡tr❡✷✳

34

Prefixing the tonne The tonne is not SI, and merely an alias for the megagramme, so shouldn’t take prefixes. In ‘megaton bomb’ we have the mega[ton of TNT equivalent], i.e. 4.184 petajoules. However in Belgium, kilotonne. . .exatonne (but not zettatonne or yottatonne) are recognised. ❇❡❧❣✐✉♠ ❛❧s♦ r❡❝♦❣♥✐s❡s t❤❡ ❝❡♥t✐❛r❡✱ ❛❧✐❛s ♠❡tr❡✷✳

35

Prefixing the tonne The tonne is not SI, and merely an alias for the megagramme, so shouldn’t take prefixes. In ‘megaton bomb’ we have the mega[ton of TNT equivalent], i.e. 4.184 petajoules. However in Belgium, kilotonne. . .exatonne (but not zettatonne or yottatonne) are recognised. Belgium also recognises the centiare, alias metre2.

36

Prefixes and Abbreviation ✎ Pr❡☞①❡s✱ ❧✐❦❡ ✉♥✐ts✱ ❝❛♥ ❜❡ ❛❜❜r❡✈✐❛t❡❞✿ ❵❦♠✬ ❛s ✇❡❧❧ ❛s ❵❦✐❧♦♠❡tr❡✬ ✎ ❇✉t ♥♦t ❵❦♠❡tr❡✬ ♦r ✬❦✐❧♦♠✬ ✎ ❍♦✇❡✈❡r ❵❦✐❧♦❜❛r✬ ❛♥❞ ❵❦❜❛r✬ s✐♥❝❡ t❤❡ ❜❛r ✐s ✐ts ♦✇♥ ❛❜❜r❡✈✐❛t✐♦♥✳ ■s t❤✐s r❡❛❧❧② ❖♣❡♥▼❛t❤ t❡rr✐t♦r②❄ ❖r ❡✈❡♥ ▼❑▼❄❄

37

Prefixes and Abbreviation

• Prefixes, like units, can be abbreviated: ‘km’

as well as ‘kilometre’ ✎ ❇✉t ♥♦t ❵❦♠❡tr❡✬ ♦r ✬❦✐❧♦♠✬ ✎ ❍♦✇❡✈❡r ❵❦✐❧♦❜❛r✬ ❛♥❞ ❵❦❜❛r✬ s✐♥❝❡ t❤❡ ❜❛r ✐s ✐ts ♦✇♥ ❛❜❜r❡✈✐❛t✐♦♥✳ ■s t❤✐s r❡❛❧❧② ❖♣❡♥▼❛t❤ t❡rr✐t♦r②❄ ❖r ❡✈❡♥ ▼❑▼❄❄

38

Prefixes and Abbreviation

• Prefixes, like units, can be abbreviated: ‘km’

as well as ‘kilometre’

• But not ‘kmetre’ or ’kilom’

✎ ❍♦✇❡✈❡r ❵❦✐❧♦❜❛r✬ ❛♥❞ ❵❦❜❛r✬ s✐♥❝❡ t❤❡ ❜❛r ✐s ✐ts ♦✇♥ ❛❜❜r❡✈✐❛t✐♦♥✳ ■s t❤✐s r❡❛❧❧② ❖♣❡♥▼❛t❤ t❡rr✐t♦r②❄ ❖r ❡✈❡♥ ▼❑▼❄❄

39

Prefixes and Abbreviation

• Prefixes, like units, can be abbreviated: ‘km’

as well as ‘kilometre’

• But not ‘kmetre’ or ’kilom’
• However ‘kilobar’ and ‘kbar’ since the bar

is its own abbreviation. ■s t❤✐s r❡❛❧❧② ❖♣❡♥▼❛t❤ t❡rr✐t♦r②❄ ❖r ❡✈❡♥ ▼❑▼❄❄

40

Prefixes and Abbreviation

• Prefixes, like units, can be abbreviated: ‘km’

as well as ‘kilometre’

• But not ‘kmetre’ or ’kilom’
• However ‘kilobar’ and ‘kbar’ since the bar

is its own abbreviation. Is this really OpenMath territory? Or even MKM??

41

Precision of Conversion There are three types of conversion factors. ❆r❝❤✐t❡❝t❡❞ ▼❡tr✐❝✱ ♦r ❭✸ ❢❡❡t ❂ ✶ ②❛r❞✧✳ ❙t♦r❡❞ ❛s ✭q✉♦t✐❡♥ts ♦❢✮ ❖▼■✳ ❊①♣❡r✐♠❡♥t❛❧ ❙✉❝❤ ❛s s❧✉❣s t♦ ♣♦✉♥❞s✱ ❞❡✲ ♣❡♥❞✐♥❣ ♦♥ ❣✳ ❙t♦r❡❞ ❛s ❖▼❋✳ ❉❡☞♥✐t✐♦♥❛❧ ❲❡r❡ ❡①♣❡r✐♠❡♥t❛❧✱ ♥♦✇ ❢♦r♠❛❧✐s❡❞✱ ❛s ✐♥ ❭✶ ②❛r❞ ❂ ✵✳✾✶✹✹ ♠❡tr❡✧ ♦r ❭✵✍❈❂ ✷✼✸✿✶✺✍❑✧✳ ❲❡r❡ ♥♦r♠❛❧❧② ❖▼❋✱ ❜✉t s❤♦✉❧❞ ❜❡ ✭q✉♦t✐❡♥ts ♦❢✮ ❖▼■✳

42

Precision of Conversion There are three types of conversion factors. Architected Metric, or “3 feet = 1 yard”. Stored as (quotients of) OMI. ❊①♣❡r✐♠❡♥t❛❧ ❙✉❝❤ ❛s s❧✉❣s t♦ ♣♦✉♥❞s✱ ❞❡✲ ♣❡♥❞✐♥❣ ♦♥ ❣✳ ❙t♦r❡❞ ❛s ❖▼❋✳ ❉❡☞♥✐t✐♦♥❛❧ ❲❡r❡ ❡①♣❡r✐♠❡♥t❛❧✱ ♥♦✇ ❢♦r♠❛❧✐s❡❞✱ ❛s ✐♥ ❭✶ ②❛r❞ ❂ ✵✳✾✶✹✹ ♠❡tr❡✧ ♦r ❭✵✍❈❂ ✷✼✸✿✶✺✍❑✧✳ ❲❡r❡ ♥♦r♠❛❧❧② ❖▼❋✱ ❜✉t s❤♦✉❧❞ ❜❡ ✭q✉♦t✐❡♥ts ♦❢✮ ❖▼■✳

43

Precision of Conversion There are three types of conversion factors. Architected Metric, or “3 feet = 1 yard”. Stored as (quotients of) OMI. Experimental Such as slugs to pounds, de- pending on g. Stored as OMF. ❉❡☞♥✐t✐♦♥❛❧ ❲❡r❡ ❡①♣❡r✐♠❡♥t❛❧✱ ♥♦✇ ❢♦r♠❛❧✐s❡❞✱ ❛s ✐♥ ❭✶ ②❛r❞ ❂ ✵✳✾✶✹✹ ♠❡tr❡✧ ♦r ❭✵✍❈❂ ✷✼✸✿✶✺✍❑✧✳ ❲❡r❡ ♥♦r♠❛❧❧② ❖▼❋✱ ❜✉t s❤♦✉❧❞ ❜❡ ✭q✉♦t✐❡♥ts ♦❢✮ ❖▼■✳

44

Precision of Conversion There are three types of conversion factors. Architected Metric, or “3 feet = 1 yard”. Stored as (quotients of) OMI. Experimental Such as slugs to pounds, de- pending on g. Stored as OMF. Definitional Were experimental, now formalised, as in “1 yard = 0.9144 metre” or “0◦C= 273.15◦K”. Were normally OMF, but should be (quotients of) OMI.

45

A User Interface Issue ✎ ❭✶ ♠✐❧❡ ✐♥ ♠❡tr✐❝✧ ⑤ ♣r♦❜❛❜❧② ✶✳✻✵✾✸✹✹❦♠✱ r❛t❤❡r t❤❛♥ ✐♥ ♠❡tr❡s✳ ❄ ❭✶✵✵✵ ♠✐❧❡s ✐♥ ♠❡tr✐❝✧ ⑤ ✶✳✻✵✾✸✹✹▼♠ ✇♦✉❧❞ ❜❡ s✉r♣r✐s✐♥❣✳ ❄❄ ❭✺✵ ♠✐❧❡s ✐♥ ♠❡tr✐❝✧ ⑤ ✵✳✵✽✵✹✻✼✷▼♠ ✇♦✉❧❞ ❜❡ ✈❡r② s✉r♣r✐s✐♥❣✳ ✎ ❭✶✵♠ ✐♥ ✐♠♣❡r✐❛❧✧ ⑤ ❞♦ ✇❡ ✇❛♥t ✶ r♦❞ ✺ ②❛r❞ ✶ ❢♦♦t ✸ ✐♥❝❤ ✼✵✵✳✼✽✼✹✵✶✺✼✹✼✽✼ ♠✐❧❄ ✎ ❭✷ ♣✐♥ts ✐♥ ♠❡tr✐❝✧ ⑤ ❞♦ ✇❡ ✇❛♥t ❧✐tr❡ ♦r ❧✐tr❡❴♣r❡✶✾✻✹❄

46

A User Interface Issue

• “1 mile in metric” — probably 1.609344km,

rather than in metres. ❄ ❭✶✵✵✵ ♠✐❧❡s ✐♥ ♠❡tr✐❝✧ ⑤ ✶✳✻✵✾✸✹✹▼♠ ✇♦✉❧❞ ❜❡ s✉r♣r✐s✐♥❣✳ ❄❄ ❭✺✵ ♠✐❧❡s ✐♥ ♠❡tr✐❝✧ ⑤ ✵✳✵✽✵✹✻✼✷▼♠ ✇♦✉❧❞ ❜❡ ✈❡r② s✉r♣r✐s✐♥❣✳ ✎ ❭✶✵♠ ✐♥ ✐♠♣❡r✐❛❧✧ ⑤ ❞♦ ✇❡ ✇❛♥t ✶ r♦❞ ✺ ②❛r❞ ✶ ❢♦♦t ✸ ✐♥❝❤ ✼✵✵✳✼✽✼✹✵✶✺✼✹✼✽✼ ♠✐❧❄ ✎ ❭✷ ♣✐♥ts ✐♥ ♠❡tr✐❝✧ ⑤ ❞♦ ✇❡ ✇❛♥t ❧✐tr❡ ♦r ❧✐tr❡❴♣r❡✶✾✻✹❄

47

A User Interface Issue

• “1 mile in metric” — probably 1.609344km,

rather than in metres. ? “1000 miles in metric” — 1.609344Mm would be surprising. ❄❄ ❭✺✵ ♠✐❧❡s ✐♥ ♠❡tr✐❝✧ ⑤ ✵✳✵✽✵✹✻✼✷▼♠ ✇♦✉❧❞ ❜❡ ✈❡r② s✉r♣r✐s✐♥❣✳ ✎ ❭✶✵♠ ✐♥ ✐♠♣❡r✐❛❧✧ ⑤ ❞♦ ✇❡ ✇❛♥t ✶ r♦❞ ✺ ②❛r❞ ✶ ❢♦♦t ✸ ✐♥❝❤ ✼✵✵✳✼✽✼✹✵✶✺✼✹✼✽✼ ♠✐❧❄ ✎ ❭✷ ♣✐♥ts ✐♥ ♠❡tr✐❝✧ ⑤ ❞♦ ✇❡ ✇❛♥t ❧✐tr❡ ♦r ❧✐tr❡❴♣r❡✶✾✻✹❄

48

A User Interface Issue

• “1 mile in metric” — probably 1.609344km,

rather than in metres. ? “1000 miles in metric” — 1.609344Mm would be surprising. ?? “50 miles in metric” — 0.0804672Mm would be very surprising. ✎ ❭✶✵♠ ✐♥ ✐♠♣❡r✐❛❧✧ ⑤ ❞♦ ✇❡ ✇❛♥t ✶ r♦❞ ✺ ②❛r❞ ✶ ❢♦♦t ✸ ✐♥❝❤ ✼✵✵✳✼✽✼✹✵✶✺✼✹✼✽✼ ♠✐❧❄ ✎ ❭✷ ♣✐♥ts ✐♥ ♠❡tr✐❝✧ ⑤ ❞♦ ✇❡ ✇❛♥t ❧✐tr❡ ♦r ❧✐tr❡❴♣r❡✶✾✻✹❄

49

A User Interface Issue

• “1 mile in metric” — probably 1.609344km,

rather than in metres. ? “1000 miles in metric” — 1.609344Mm would be surprising. ?? “50 miles in metric” — 0.0804672Mm would be very surprising.

• “10m in imperial” — do we want 1 rod 5

yard 1 foot 3 inch 700.787401574787 mil? ✎ ❭✷ ♣✐♥ts ✐♥ ♠❡tr✐❝✧ ⑤ ❞♦ ✇❡ ✇❛♥t ❧✐tr❡ ♦r ❧✐tr❡❴♣r❡✶✾✻✹❄

50

A User Interface Issue

• “1 mile in metric” — probably 1.609344km,

rather than in metres. ? “1000 miles in metric” — 1.609344Mm would be surprising. ?? “50 miles in metric” — 0.0804672Mm would be very surprising.

• “10m in imperial” — do we want 1 rod 5

yard 1 foot 3 inch 700.787401574787 mil?

• “2 pints in metric” — do we want litre or

litre_pre1964?

51

‘Obsolete’ has two meanings ✎ ❆♥ ❖♣❡♥▼❛t❤ ❈❉ ✐s ♦❜s♦❧❡t❡ ✐❢ t❤❡ ❞❡☞♥✐✲ t✐♦♥s ✐♥ ✐t ❛r❡ ❢♦r ❛r❝❤✐✈❛❧ ♣✉r♣♦s❡s ♦♥❧②✳ ✎ ❆ ✉♥✐t ✐s ♦❜s♦❧❡t❡ ④ ❜② ❢♦r♠❛❧ ❝❤❛♥❣❡ ✭❡✳❣✳ ❧✐tr❡❴♣r❡✶✾✻✹✮ ④ ❜② ✉s❛❣❡ ✭❡✳❣✳ r♦❞ ♦r ❛r❡✮ ◆❡❡❞ ♦❢❢✐❝✐❛❧ ❈❉s ♦❢ ♦❜s♦❧❡t❡ ✉♥✐ts✳

52

‘Obsolete’ has two meanings

• An OpenMath CD is obsolete if the defini-

tions in it are for archival purposes only. ✎ ❆ ✉♥✐t ✐s ♦❜s♦❧❡t❡ ④ ❜② ❢♦r♠❛❧ ❝❤❛♥❣❡ ✭❡✳❣✳ ❧✐tr❡❴♣r❡✶✾✻✹✮ ④ ❜② ✉s❛❣❡ ✭❡✳❣✳ r♦❞ ♦r ❛r❡✮ ◆❡❡❞ ♦❢❢✐❝✐❛❧ ❈❉s ♦❢ ♦❜s♦❧❡t❡ ✉♥✐ts✳

53

‘Obsolete’ has two meanings

• An OpenMath CD is obsolete if the defini-

tions in it are for archival purposes only.

• A unit is obsolete

– by formal change (e.g. litre_pre1964) – by usage (e.g. rod or are) ◆❡❡❞ ♦❢❢✐❝✐❛❧ ❈❉s ♦❢ ♦❜s♦❧❡t❡ ✉♥✐ts✳

54

‘Obsolete’ has two meanings

• An OpenMath CD is obsolete if the defini-

tions in it are for archival purposes only.

• A unit is obsolete

– by formal change (e.g. litre_pre1964) – by usage (e.g. rod or are) Need official CDs of obsolete units.

55

Not all units are monoids ✭✶ ❞❡❣r❡❡ ❈❡❧s✐✉s✮ ♣❧✉s ✭✶ ❞❡❣r❡❡ ❈❡❧s✐✉s✮ ❂ ✷✼✺✳✶✺ ❞❡❣r❡❡s ❈❡❧s✐✉s ✭✲✶✮ ❞❡❣r❡❡ ❈❡❧s✐✉s ❂ ✸✵✳✷ ❞❡❣r❡❡s ❋❛❤r❡♥❤❡✐t ✲✭✶ ❞❡❣r❡❡ ❈❡❧s✐✉s✮ ❂ ✲✾✺✸✳✶✹ ❞❡❣r❡❡s ❋❛❤r❡♥❤❡✐t

56

Not all units are monoids (1 degree Celsius) plus (1 degree Celsius) = 275.15 degrees Celsius ✭✲✶✮ ❞❡❣r❡❡ ❈❡❧s✐✉s ❂ ✸✵✳✷ ❞❡❣r❡❡s ❋❛❤r❡♥❤❡✐t ✲✭✶ ❞❡❣r❡❡ ❈❡❧s✐✉s✮ ❂ ✲✾✺✸✳✶✹ ❞❡❣r❡❡s ❋❛❤r❡♥❤❡✐t

57

Not all units are monoids (1 degree Celsius) plus (1 degree Celsius) = 275.15 degrees Celsius (-1) degree Celsius = 30.2 degrees Fahrenheit ✲✭✶ ❞❡❣r❡❡ ❈❡❧s✐✉s✮ ❂ ✲✾✺✸✳✶✹ ❞❡❣r❡❡s ❋❛❤r❡♥❤❡✐t

58

Not all units are monoids (1 degree Celsius) plus (1 degree Celsius) = 275.15 degrees Celsius (-1) degree Celsius = 30.2 degrees Fahrenheit

• (1 degree Celsius) = -953.14 degrees Fahrenheit

59

Relative vs Absolute ❘❡❧❛t✐✈❡ t❡♠♣❡r❛t✉r❡s ❛r❡ ❛ ♠♦♥♦✐❞ ✶✍❈❂✶✍❑❂ ✍❋ ❆❜s♦❧✉t❡ t❡♠♣❡r❛t✉r❡s ❛r❡ ♥♦t ✶✍❈❂✷✼✹✳✶✺✍❑❂✸✷ ✍❋ ✦ ✶❞❡❣❈

⑤ ④③ ⑥

❛❜s

✰ ✶❞❡❣❑

⑤ ④③ ⑥

r❡❧

❂ ✷❞❡❣❈

⑤ ④③ ⑥

❛❜s

✶❞❡❣❑

⑤ ④③ ⑥

❛❜s

✰ ✶❞❡❣❈

⑤ ④③ ⑥

r❡❧

❂ ✷❞❡❣❑

⑤ ④③ ⑥

❛❜s

❂ ✲✷✼✶✳✶✺❞❡❣❈

⑤ ④③ ⑥

❛❜s

❚❡❧❧✐♥❣ t❤❡ t✇♦ ❛♣❛rt ✐s ❛ ✉s❡r ✐♥t❡r❢❛❝❡ ✐ss✉❡✳

60

Relative vs Absolute Relative temperatures are a monoid 1◦C=1◦K=9

5

• F

❆❜s♦❧✉t❡ t❡♠♣❡r❛t✉r❡s ❛r❡ ♥♦t ✶✍❈❂✷✼✹✳✶✺✍❑❂✸✷ ✍❋ ✦ ✶❞❡❣❈

⑤ ④③ ⑥

❛❜s

✰ ✶❞❡❣❑

⑤ ④③ ⑥

r❡❧

❂ ✷❞❡❣❈

⑤ ④③ ⑥

❛❜s

✶❞❡❣❑

⑤ ④③ ⑥

❛❜s

✰ ✶❞❡❣❈

⑤ ④③ ⑥

r❡❧

❂ ✷❞❡❣❑

⑤ ④③ ⑥

❛❜s

❂ ✲✷✼✶✳✶✺❞❡❣❈

⑤ ④③ ⑥

❛❜s

❚❡❧❧✐♥❣ t❤❡ t✇♦ ❛♣❛rt ✐s ❛ ✉s❡r ✐♥t❡r❢❛❝❡ ✐ss✉❡✳

61

Relative vs Absolute Relative temperatures are a monoid 1◦C=1◦K=9

5

• F

Absolute temperatures are not 1◦C=274.15◦K=329

5

• F

✦ ✶❞❡❣❈

⑤ ④③ ⑥

❛❜s

✰ ✶❞❡❣❑

⑤ ④③ ⑥

r❡❧

❂ ✷❞❡❣❈

⑤ ④③ ⑥

❛❜s

✶❞❡❣❑

⑤ ④③ ⑥

❛❜s

✰ ✶❞❡❣❈

⑤ ④③ ⑥

r❡❧

❂ ✷❞❡❣❑

⑤ ④③ ⑥

❛❜s

❂ ✲✷✼✶✳✶✺❞❡❣❈

⑤ ④③ ⑥

❛❜s

❚❡❧❧✐♥❣ t❤❡ t✇♦ ❛♣❛rt ✐s ❛ ✉s❡r ✐♥t❡r❢❛❝❡ ✐ss✉❡✳

62

Relative vs Absolute Relative temperatures are a monoid 1◦C=1◦K=9

5

• F

Absolute temperatures are not 1◦C=274.15◦K=329

5

• F

! 1degC

• abs

+ 1degK

• rel

= 2degC

• abs

✶❞❡❣❑

⑤ ④③ ⑥

❛❜s

✰ ✶❞❡❣❈

⑤ ④③ ⑥

r❡❧

❂ ✷❞❡❣❑

⑤ ④③ ⑥

❛❜s

❂ ✲✷✼✶✳✶✺❞❡❣❈

⑤ ④③ ⑥

❛❜s

❚❡❧❧✐♥❣ t❤❡ t✇♦ ❛♣❛rt ✐s ❛ ✉s❡r ✐♥t❡r❢❛❝❡ ✐ss✉❡✳

63

Relative vs Absolute Relative temperatures are a monoid 1◦C=1◦K=9

5

• F

Absolute temperatures are not 1◦C=274.15◦K=329

5

• F

! 1degC

• abs

+ 1degK

• rel

= 2degC

• abs

1degK

• abs

+ 1degC

• rel

= 2degK

• abs

= -271.15degC

• abs

❚❡❧❧✐♥❣ t❤❡ t✇♦ ❛♣❛rt ✐s ❛ ✉s❡r ✐♥t❡r❢❛❝❡ ✐ss✉❡✳

64

Relative vs Absolute Relative temperatures are a monoid 1◦C=1◦K=9

5

• F

Absolute temperatures are not 1◦C=274.15◦K=329

5

• F

! 1degC

• abs

+ 1degK

• rel

= 2degC

• abs

1degK

• abs

+ 1degC

• rel

= 2degK

• abs

= -271.15degC

• abs

Telling the two apart is a user interface issue.

65

Conclusions ✎ ❖♣❡♥▼❛t❤ ❞♦❡s ♣r♦✈✐❞❡ ④ ❊①t❡♥s✐❜✐❧✐t② ④ ❉♦❝✉♠❡♥t❛t✐♦♥ ✎ ❖♣❡♥▼❛t❤ ❞♦❡s ♥♦t ♣r♦✈✐❞❡ ④ ◆❡❛t ✉s❡r ✐♥t❡r❢❛❝❡s ✭❜② ✐ts❡❧❢✦✮✳

66

Conclusions

• OpenMath does provide

– Extensibility ④ ❉♦❝✉♠❡♥t❛t✐♦♥ ✎ ❖♣❡♥▼❛t❤ ❞♦❡s ♥♦t ♣r♦✈✐❞❡ ④ ◆❡❛t ✉s❡r ✐♥t❡r❢❛❝❡s ✭❜② ✐ts❡❧❢✦✮✳

67

Conclusions

• OpenMath does provide

– Extensibility – Documentation ✎ ❖♣❡♥▼❛t❤ ❞♦❡s ♥♦t ♣r♦✈✐❞❡ ④ ◆❡❛t ✉s❡r ✐♥t❡r❢❛❝❡s ✭❜② ✐ts❡❧❢✦✮✳

68

Conclusions

• OpenMath does provide

– Extensibility – Documentation

• OpenMath does not provide

④ ◆❡❛t ✉s❡r ✐♥t❡r❢❛❝❡s ✭❜② ✐ts❡❧❢✦✮✳

69

Conclusions

• OpenMath does provide

– Extensibility – Documentation

• OpenMath does not provide

– Neat user interfaces (by itself!).

70