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Scales (Slide Rule): Difference between revisions
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== How to Read Scales == | == How to Read Scales == | ||
The operations of a slide rule are performed by adding physical lengths together between scales to find results. This is best illustrated through simple "linear" scales (such as those found on standard rulers, or the "number line" learned from elementary school). Taking two copies of the number line (of the same size) and laying them next to each other, it is possible to use the physical properties of the scale to add without any extra manipulation. Labeling the two scales "X" and "Y," we can observe an example:< | The operations of a slide rule are performed by adding physical lengths together between scales to find results. This is best illustrated through simple "linear" scales (such as those found on standard rulers, or the "number line" learned from elementary school). Taking two copies of the number line (of the same size) and laying them next to each other, it is possible to use the physical properties of the scale to add without any extra manipulation. Labeling the two scales "X" and "Y," we can observe an example: | ||
<syntaxhighlight> | |||
2 + 3 = 5 | |||
</syntaxhighlight> | |||
On the first scale (X) we find the number 2. We can align the base of scale Y (called the "index") with it so that the 2 on X is next to 0 on Y. We then find the number 3 on scale X, and look for what number on scale Y is aligned with it (which is the number 5). Most operations on slide rules use this method to find various results. | |||
=== Notation === | === Notation === | ||
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==== Example ==== | ==== Example ==== | ||
using the previous example of 2+3=5, we can write the actions as follows: | using the previous example of 2 + 3 = 5, we can write the actions as follows: | ||
<syntaxhighlight> | <syntaxhighlight> | ||
I(Y) | 2(X) → 3(Y) = 5(X) | I(Y) | 2(X) → 3(Y) = 5(X) | ||
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The C and D scales are the primary scales against which most other scales are used. They are two identical scales in a logarithmic arrangement. Just like the above example shows how addition of two numbers can be achieved with linear scales, logarithmic scales can achieve multiplication the same way: | The C and D scales are the primary scales against which most other scales are used. They are two identical scales in a logarithmic arrangement. Just like the above example shows how addition of two numbers can be achieved with linear scales, logarithmic scales can achieve multiplication the same way: | ||
<syntaxhighlight> | <syntaxhighlight> | ||
2 x 3 = 6 | |||
I | 2(D) → 3(C) = 6(D) | I | 2(D) → 3(C) = 6(D) | ||
</syntaxhighlight> | </syntaxhighlight> | ||
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Division is the inverse of subtraction, which means we can simply apply our process in reverse to do division. | Division is the inverse of subtraction, which means we can simply apply our process in reverse to do division. | ||
<syntaxhighlight> | <syntaxhighlight> | ||
3/2=1.5 | 3/2 = 1.5 | ||
3(D) | 2(C) → I = 1.5(D) | 3(D) | 2(C) → I = 1.5(D) | ||
</syntaxhighlight> | </syntaxhighlight> | ||
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Squares and square roots can be calculated using only the C and D scales with some practice, but are more easily performed with the D and A (or C and B) scales. The A and B scales are often called the "square scales" because they are half the length of the C and D scales, which results in any number on D having its square on A (and likewise C with B). This is because of the property of logarithms in which doubling the logarithm of a number (in this case having the scale grow twice as fast) results in the logarithm of its square. To make it more clear, here is an example: | Squares and square roots can be calculated using only the C and D scales with some practice, but are more easily performed with the D and A (or C and B) scales. The A and B scales are often called the "square scales" because they are half the length of the C and D scales, which results in any number on D having its square on A (and likewise C with B). This is because of the property of logarithms in which doubling the logarithm of a number (in this case having the scale grow twice as fast) results in the logarithm of its square. To make it more clear, here is an example: | ||
<syntaxhighlight> | <syntaxhighlight> | ||
3²=9 | 3² = 9 | ||
3(D) = 9(A) | 3(D) = 9(A) | ||
</syntaxhighlight> | </syntaxhighlight> | ||
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When multiplying two numbers, the rules of algebra tell us that AxB=A/(1/B). This means that we can perform our division process using the D and CI scales to achieve the same process as multiplying with C and D, but without any worry about going off scale. | When multiplying two numbers, the rules of algebra tell us that AxB=A/(1/B). This means that we can perform our division process using the D and CI scales to achieve the same process as multiplying with C and D, but without any worry about going off scale. | ||
<syntaxhighlight> | <syntaxhighlight> | ||
9 x 2 = 18 | |||
9(D) | 2(CI) → I = 18(D) | 9(D) | 2(CI) → I = 18(D) | ||
</syntaxhighlight> | </syntaxhighlight> | ||
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A peculiar feature of slide rules is the fact that the scales "wrap around" themselves quite often. This leads to a very nice tool in tracking any extra complications with the decimal point: the slide itself. When performing the standard multiplication procedure (starting at an index), any time the upper index is used, this represents using this "wrapping" feature, which in essence describes adding 1 to the characteristic at the end. With division, the operation is reversed, meaning that any time you reach the upper index, you have to subtract one from the characteristic. The actual rule takes more reasoning, but can be described as follows: if multiplying brings you ''down'' the rule, add one to the characteristic (using either method of multiplying). Likewise, if dividing (using either method) brings you ''up'' the rule, subtract one from the characteristic. This can be seen by using our previous example on scientific notation: | A peculiar feature of slide rules is the fact that the scales "wrap around" themselves quite often. This leads to a very nice tool in tracking any extra complications with the decimal point: the slide itself. When performing the standard multiplication procedure (starting at an index), any time the upper index is used, this represents using this "wrapping" feature, which in essence describes adding 1 to the characteristic at the end. With division, the operation is reversed, meaning that any time you reach the upper index, you have to subtract one from the characteristic. The actual rule takes more reasoning, but can be described as follows: if multiplying brings you ''down'' the rule, add one to the characteristic (using either method of multiplying). Likewise, if dividing (using either method) brings you ''up'' the rule, subtract one from the characteristic. This can be seen by using our previous example on scientific notation: | ||
<syntaxhighlight> | <syntaxhighlight> | ||
431 x 0.00085 = (4.31x10²)x(8.5x10⁻⁴) ⇒ 2 - 4 = -2 | 431 x 0.00085 = (4.31x10²) x (8.5x10⁻⁴) ⇒ 2 - 4 = -2 | ||
4.31(D) | I → 8.5(C) ≈ 3.67(D); (4 > 3) ⇒ Characteristic + 1 (=-1) | 4.31(D) | I → 8.5(C) ≈ 3.67(D); (4 > 3) ⇒ Characteristic + 1 (=-1) | ||
Therefore 3.67x10⁻¹=0.367 | Therefore 3.67x10⁻¹ = 0.367 | ||
</syntaxhighlight> | </syntaxhighlight>Notice here that we move ''down'' the rule from 4.31(D) to 3.67(D), which prompts us to add 1 to the characteristic. | ||
=== Folded and Split Scales === | === Folded and Split Scales === | ||
== List of Common Scales == | == List of Common Scales == | ||