## iSlideRule Scales

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Scales supported in the iSlideRule app include the following scales (note that one of the provided scales is a "blank" scale, and is used when less than four scales are desired for a given rule).

### A

The A scale is a scale of squared values, relative to the D scale, and has values from 1 to 100.  It can be used with the D scale to read square or square root values.  This scale is normally located at the bottom of the top rule.

Gauge marks are located at ππ/4, τ, and M (100/π).

Formula:  A = X²

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### AI

The AI scale is an A scale reversed in direction (reciprocal scale).

Gauge marks are located at π and τ.

Formula: AI = 1 / (X²)

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### B

The B scale is identical to the A scale, but is normally located on the top of the slide, rather than on one of the rules.   It is normally associated with the C fundamental scale for reading square roots or squared values.

Gauge marks are located at ππ/4, τ, and M (100/π).

Formula: B = X²

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### BI

The BI scale is a reciprocal B scale, related to the C scale in the same fashion that the AI scale is related to the D scale.

Gauge marks are located at π and τ.

Formula: BI = 1 / (X²)

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### C

The C scale is a fundamental scale, running from 1 to 10, and is normally located at the bottom of the slide.

Gauge marks are located at C (√(4/π)), (dozen), (gross), ρ (π/180), ρ'' (180*60*60)/π),  ρ' (180*60)/π), used in radians to degrees conversions, π, τ, C1 (√(40/π)), ρ⁰ (180/π) and δ (seconds in a radian in French decimal system, (200*100*100)/π ).

Formula: C = X

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### C1

The C1 scale is a ranged fundamental C scale, ranging from 1 to √10.

Formula: C1 = √10 × X

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### C2

The C2 scale is a ranged reciprocal fundamental C scale, ranging from √10 to 1.

Formula: C2 = 1 / ( √10 × X )

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### Celsius

This is a temperature conversion scale, used with its counterpart, the Fahrenheit scale, to convert between degrees Celsius and degrees Fahrenheit.  This scale runs from -23 to 100 degrees C.

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### CF

The CF scale is a folded C scale, with the fold (or wrap) occurring at π (3.14).  This scale expedites the multiplication of values by π.

Gauge marks are located at π and τ.

Formula: CF = π × X

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### CF/m

The CF/m scale is another folded C scale, but this time, the fold is at 2.3 (natural log of 10).  This scale is useful when converting results between base 10 and base e.

Gauge marks are located at π and τ.

Formula: CF/M = ( loge 10 ) × X

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### CI

The CI scale is a reciprocal C fundamental scale, running from 10 to 1.  It is usually located on the slide, like the C scale.

Gauge marks are located at (dozen), (gross), π and τ.

Formula : CI = 1 / X

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### CIF

The CIF scale is a reciprocal folded C scale, folded at π like the CF scale, but reversed as well, like the CI scale.

Gauge marks are located at π and τ.

Formula: CIF = 1 / ( π × X )

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### D

The D fundamental scale is normally positioned as the top scale on bottom rule. It runs from 1 to 10.

Gauge marks are located at C (√(4/π)), (dozen), (gross), ρ (π/180), ρ'' (180*60*60)/π),  ρ' (180*60)/π), used in radians to degrees conversions, π, τ, ρ⁰ (180/π), and C1 (√(40/π)).

Formula: D = X

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### D1

The D1 scale is a ranged fundamental D scale, ranging from 1 to √10.

Formula: D1 = √10 × X

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### D2

The D2 scale is a ranged reciprocal fundamental D scale, ranging from √10 to 1.

Formula: D2 = 1 / ( √10 × X )

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### DF

The DF scale, like the CF scale, is a folded D scale, with the fold at π.  It is found on the lower rule normally, and is used for calculations involving π.

Gauge mark is located at π and τ.

Formula: DF = π × X

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### DF/m

The DF/m scale is a folded D scale like the DF scale, with the fold is at 2.3 (natural log of 10).  This scale is useful when converting results between base 10 and base e.  This scale is usually located on the bottom rule, where the CF/m scale would be located on the slide.

Gauge mark is located at π and τ.

Formula: DF/M = ( loge 10 ) × X

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### DI

The DI scale is a reciprocal D scale, running from 10 to 1.  This scale is normally located on either the top or bottom rule.

Gauge marks are located at (dozen), (gross), π and τ.

Formula: DI = 1 / X

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### DIF

The DIF scale is another reciprocal folded fundamental scale, like the CIF scale.  This scale is normally on the top or bottom rule, and is folded at π, just like the CIF scale.

Gauge marks are located at π and τ.

Formula: DIF = 1 / ( π × X )

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### F

The F scale is a root and power scale, used with the fundamental scale to raise values to the 4th power, or take the 4th root of a value. It runs in values from 1 to 10,000, and is normally located on the top rule.

Formula: F = X⁴

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### F(2π)

The F(2π) scale is used in electronics calculations involving resonant frequency computations.

Gauge marks are located at π and τ.

Formula: F(2π) = X / ( 2π )

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### Fahrenheit

This is a temperature conversion scale, used with its counterpart, the Celsius scale, to convert between degrees Fahrenheit and degrees Celsius.  This scale runs from -10 to 212 °F.

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### FI

The FI scale is a reciprocal root and power scale, used with the fundamental scale to raise values to the 4th power, or take the 4th root of a value. It runs in values from 10,000 to 1, and is normally located on the top rule.

Formula: FI = 1 / X⁴

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### H

The H scale is used in electronics calculations for the resonant frequency of a circuit.  It is effectively a folded BI scale with the fold at 1 / (2π)

Formula: H = 1 / ( 2π² × X² )

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### J

The J scale is a cube scale, raising values from the C scale to the third power, or taking the cube root of a value on the J scale reading the root from the C scale.  Normally, this scale is located on the slide and used with the C scale.

Formula: J = X³

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### JI

The JI scale is reciprocal cube scale, located normally on the slide and used with the C scale.

Formula: JI = 1 / X³

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### K

The K scale is a cube scale, raising values from the D scale to the third power, or taking the cube root of a value on the K scale reading the root from the D scale.  Normally, this scale is located on the top or bottom rule and used with the D scale.

Formula: K = X³

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### KI

The KI scale is reciprocal cube scale, located normally on the slide and used with the D scale.

Formula: KI = 1 / X³

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### KZ

The KZ scale is a folded at 360 fundamental scale, normally found on the top rule.  This scale is used in financial calculations involving compound interest over time.

Gauge marks are located at (dozen), and (gross), π and τ.

Formula: KZ = 360 × X

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### L

The L scale is a linear log scale used to obtain the common (log to base 10) of a number on the fundamental D scale.  This scale is normally on the top or bottom rule.

Formula: L = log10 X
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### LC

The LC scale is used in electronics calculations to compute the resonant frequency of an electrical circuit.  It is similar to the H scale, but using different units.

Formula: LC = 1 / ( 2π² × X² )

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### LL

The LL scale is a log-log scale used to calculate powers.  This scale is similar to the LL1 - LL3 scale set below, covering the majority of the range of those scales in one scale, so results are less precise.  This scale ranges from 1.01 to 10,000.

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### LL0 / LL1 / LL2 / LL3

The LLn scales are used to raise numbers to powers greater than one.  These are used when the normal power scales such as A/B, J/K, and F are unsuitable.  These scales allow raising values to non-integer powers.

Gauge marks are located at π and τ on the LL3 scale.

Formula: LL0 = e0.001X
Formula: LL1 = e0.01X
Formula: LL2 = e0.1X
Formula: LL3 = eX

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### LL/0 / LL/1 / LL/2 / LL/3

The LL/n scales (or log-log scales) are exponential power scales used to raise values to powers less than one.

Formula: LL/0 = e-X
Formula: LL/1 = e-0.1X
Formula: LL/2 = e-0.01X
Formula: LL/3 = e-0.001X

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### Ln

The Ln scale, or natural log scale is use to determine the nature log of numbers from the fundamental scales.

Formula: Ln = loge X

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### M1' / M2' / M3' / M4' / M5'

The Mn (and Mn') scales provide a 5-segment high precision set of scales for performing multiplication with higher precision than would be available with the normal scales.  Running from 1 to 10, the M scales are essentially a longer version of the C or D fundamental scales.  The prime scales have the scale ticks on the top side of the scale, while the non-prime scales have the scale ticks on the bottom of the scale.

Gauge marks are located at π on the M3 scale, and τ on the M4 scale.

Formula: Mn = X

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### P

The P scale (also known as the Pythagorean scale) is normally found on the top or bottom rule, and when used with the D and A scales can calculate the square root of (1 - X²), or with the S scale, a more accurate calculation of the cosine of small angles.  Scale ranges from 0.995 to 0.0

Formula: P = √(1 - X²)

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### Q1' / Q2' / Q3'

The Qn scales are similar to the Mn scales in that they are an extended fundamental scale useful for higher precision results in calculations. The prime scales have the scale ticks on the top side of the scale, while the non-prime scales have the scale ticks on the bottom of the scale.

Gauge marks are located at π on the Q2 scale, and τ on the Q3 scale.

Formula: Q = X

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### R1' / R2'

The R1 and R2 scales, used with the D scale, can rapidly determine square roots. R1 runs from 1.0 to √10, while R2 runs from √10 to 10.0.  Combined, the R1 and R2 scales are like a D scale twice the normal length. The prime scales have the scale ticks on the top side of the scale, while the non-prime scales have the scale ticks on the bottom of the scale.

Gauge marks are located at π on the R1 scale, and τ on the R2 scale.

Formula: Rn = √X

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### S

The S scale is a scale of sines and cosines, running from 5.7 to 90 degrees.  Select the degrees on this scale and read the sine or cosine on the C/D.  The magnitude on the C/D scale will be from 0.1 to 1.0.

Formula: S = sin X.

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### SH1 / SH2

The SH1 and SH2 scales are used in electrical engineering calculations, and calculate the hyperbolic sine function.  SH1 ranges from 0.1 to 0.89, while SH2 ranges from 0.89 to 3.0.

Formula: SHn = sinh X

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### Smann

The Smann scale in iSlideRule is another representation of the ST scale (below), extending the S scale for small angles.  This scale ranges from 0.57 to 5.7 degress.

Formula: Smann = sin, tan X

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### ST

The ST scale extends the S scale for smaller angles between 0.57 and 5.7 degrees.  The values on this scale range from 0.01 to 0.1 read from the D scale.  In this range, sines and tangents are nearly equal so this scale can be used for both.

Formula: ST = sin, tan X

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### T1' / T2'

The T1 scale, also known as the T scale on some rules runs from angle degrees from 5.7 to 45 degrees.  The T1 and T2 scales are normally associated with the C or D scale, depending on whether the T scale is on the rule or the slide.  The T2 scale ranges from 45 to approximately 84.5 degrees.  The tangent or cotangent is read from the C or D scale. The prime scales have the scale ticks on the top side of the scale, while the non-prime scales have the scale ticks on the bottom of the scale.

Formula: T1 = tan X or cot X

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### T2

The T2 scale is a folded at 360 fundamental scale, normally found on the slide.  This scale is used in financial calculations involving compound interest over time.

Gauge marks are located at (dozen), (gross), π, and τ.

Formula: T2 = 360 × X

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### TH

The TH scale is a scale of hyperbolic tangents, and along with the SH scales can be used in electrical engineering calculations.  This scale ranges from 1 to 3.

Formula: TH = tanh X
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### U, U-1, U1/2, V, V-1, V1/2, V2

Available in release 2.0.0, these scales, available only the primary scale set, provide dimensional analysis capability.  The 30 marks on each scale represent physical constants.   For additional information, review Dimensional Analysis.

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