# Definition of Scales

Scales supported in the iSlideRule / iSlideRule HD app include the following eighty different scales (note that one of the eighty scales that is selectable is a "blank" scale, and is used when less than four or five scales are desired for a given rule). The iSlideRule app uses four scales per slide, while iSlideRule HD uses five, and includes (as of release 1.0.2) dimensional analysis scales.

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 π, π/4, and 100/π (M).

Formula:  A = X²

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

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 π, π/4, and 100/π (M).

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.

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), (144) (gross), ρ (π/180), ρ'' (180*60*60)/π),  ρ' (180*60)/π), used in radians to degrees conversions, π, C1 (√(40/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) occuring at π (3.14).  This scale expedites the multiplication of values by π.

Gauge mark is located at π.

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 mark is located at π.

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), (144) (gross), and π.

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 mark is located at π.

Formala: 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), (144) (gross), ρ (π/180), ρ'' (180*60*60)/π),  ρ' (180*60)/π), used in radians to degrees conversions, 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.

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.

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 mark is located at π.

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 mark is located at π.

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 mark is located at π.

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 degrees C.

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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 (144) (gross).

Formula: KZ = 360 × X

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L
The L scale is a linear log scle 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 mark is located at π 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 ( M1' / M2' / M3' / M4' / M5' )
The 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 alternate M' scales have the scales aligned to the top edge of the position, rather than the lower edge.

Gauge mark is located at π on the M3 (and M3') 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 ( 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 alternate Q' scales have the scales aligned to the top edge of the position, rather than the lower edge.

Gauge mark is located at π on the Q2 (and Q2^) scale.

Formula: Q = X

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R1 / R2 ( 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 alternate R' scales have the scales aligned to the top edge of the position, rather than the lower edge.

Gauge mark is located at π on the R1 (and R1') 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.

Gauge mark is located at π.

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 ( 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 alternate T' scales have the scales aligned to the top edge of the position, rather than the lower edge.

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), and (144) (gross).

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

Newly added in release 1.0.2 of iSlideRule HD, these scales, available only the primary scale set, provide dimensional analysis to this app.  The 30 marks on each scale represent physical constants.   Listed below are the marks and the physical property each one represents.  NOTE: Double-tapping the labels to the left of the display normally displays the scale functions and descriptive.  For the dimensional analysis scales, the labels will show the dimensional relationship between a given property, and the fundamental properties of length ( l ), mass ( M ), time ( T ), and charge ( Q ).  Look for information on "dimensional analysis" online.

• a - Acceleration
• B - Magnetic Induction
• C - Capacitance
• D - Electric Displacement
• E - Electric Field
• F - Force
• H - Magnetic Intensity
• Im - Moment of Inertia
• I - Electric Current
• k - Spatial Frequency
• L - Inductance
• Lm - Angular Momentum, Action
• l - Length
• l2 - Area
• l3 - Volume
• M - Mass
• P - Power
• Pm - Linear Momentum
• Q - Charge
• R - Resistance, Reactance
• T - Time, Period
• V - Potential ( Voltage )
• v - Velocity
• W - Work, Energy, Torque
• α - Angular Acceleration
• ε0 - Electric Permitivity
• μ0 - Magentic Permeability
• Φ - Magentic Flux
• ρ - Density
• ω - Frequency, Angular Velocity

Changed in release 2.0.0 (both): Dimensional Analysis scales are now provided in both iSlideRule and iSlideRule HD.