Logarithmic Scale and Rounding A logarithmic scale is a nonlinear scale* used when there is a large range of quantities. Common uses include earthquake strength, sound loudness, light intensity, and pH of solutions. It is based on orders of magnitude, rather than a standard linear scale, so the value represented by each equidistant mark on the scale is the value at the previous mark multiplied by a constant. Logarithmic scales are also used in slide rulesΨ for multiplying or dividing numbers by adding or subtracting lengths on the scales. (* Non-linear: Which is uneven or not equal is known as non-linear. So a non-linear scale consists of unequally space divisions, sections or proportions. This means that the lines, grids or markings you see are not equal or constant in value or size.) (Ψ Slide Rulers are the rulers with a sliding central strip, marked with logarithmic scales and used for making rapid calculations, especially multiplication and division.)

The two logarithmic scales of a slide rule Rounding a numerical means replacing it by another value that is approximately equal but has a shorter, simpler value , or more explicit representation; for example, replacing 67.1462 with 67.15or the fraction 312/937 with 1/3, or the expression √2 with 1.414. A wavy equals sign (≈ approximately equal to) is sometimes used to indicate rounding of exact numbers, e.g., 9.98 ≈ 10. This sign was introduced by Alfred George Greenhill in 1892.

Advantages:  Rounding is often done to obtain a value that is easier to report and communicate than the original.  Rounding can also be important to avoid misleadingly precise reporting of a computed number, measurement or estimate; for example, a quantity that was computed as 123,456 but is known to be accurate only to within a few hundred units is better stated as "about 123,500". Limitations:  On the other hand, rounding of exact numbers will introduce some round-off error in the reported result.  Rounding is almost unavoidable when reporting many computations — especially when dividing two numbers in integer or fixed-point arithmetic; when computing mathematical functions such as square roots, logarithms, and sines; or when using a floating-point representation with a fixed number of significant digits. In a sequence of calculations, these rounding errors generally accumulate, and in certain ill-conditioned cases they may make the result meaningless.  Accurate rounding of transcendental mathematical functions is difficult because the number of extra digits that need to be calculated to resolve whether to round up or down cannot be known in advance. This problem is known as "the table-maker's dilemma".  Rounding has many similarities to the quantization that occurs when physical quantities must be encoded by numbers or digital signals. **************