0.7 Phy1040: scientific notation and significant figures  (Page 3/8)

Diminishing returns

If the measuring stick were graduated in tenth-inch units, however, that may or may not lead to a more precise measurement. That would beapproaching the point of diminishing returns where your inability to take full advantage of the more precise graduations and the inability of the subject toalways stand with the same degree of rigidity might come into play.

Scientific notation

According to Wikipedia , scientific notation is a way of writing numbers that accommodates values too large or small to be conveniently written in standard decimal notation. The notation has a number of useful properties and is commonly used by scientists and engineers.A variation of scientific notation is also used internally by computers.

Scientific notation format

Numbers in scientific notation are written using the following format:

x * 10^y

which can be read as the value x multiplied by ten raised to the y power where y is an integer and x is any real number. (Constraints are placed on the value of x when using the normalized form of scientific notation which I will explain below.)

The values for x and y can be either positive or negative.

The term referred to as x is often called the significand or the mantissa (not to be confused with the term mantissa used with common logarithms).

The computer display version of scientific notation

Because of difficulties involved in displaying superscripts in the output of computer programs, a typical display of a number in scientific notation by acomputer program might look something like the following example:

-3.141592653589793e+1

where

• Either numeric value can be positive, negative, or zero.
• The number of digits in the numeric value to the left of the e (the mantissa) may rangefrom a few to many.
• The e may be either upper-case or lower-case depending on the computer and the program.

A power of ten is understood

In this format, it is understood that the number consists of the value to the left of the e (the mantissa) multiplied by ten raised to a power given by the value to the right of the e (the exponent).

For example, in JavaScript exponential format, the value -10*Math.PI is displayed as

-3.141592653589793e+1

The value Math.PI/10 is displayed as

3.141592653589793e-1

The value Math.PI is displayed as

3.141592653589793e+0

The value 0 is displayed as

0e+0

The normalized form of scientific notation

Using general scientific notation, the number -65700 could be written in several different ways including the following:

• -6.57 * 10^4
• -65.7 * 10^3
• -657 * 10^2

In normalized scientific notation, the exponent is chosen such that the absolute value of the mantissa is at least one but less than ten. For example, -65700 is written:

-6.57 * 10^4

In normalized notation the exponent is negative for a number with absolute value between 0 and 1. For example, the value 0.00657 would be written:

6.57 * 10^(-3)

The 10 and the exponent are usually omitted when the exponent is 0.

Significant figures

According to Wikipedia , The significant figures of a number are those digits that carry meaning contributing to its precision. This includes all digits except: