Approximation, Error, and Precedence Help

By — McGraw-Hill Professional
Updated on Oct 3, 2011

Truncation and Rounding

In trigonometry, the numbers we work with are rarely exact values. We must almost always settle for an approximation. There are two ways of doing this: truncation (easy but not very accurate) and rounding (a little trickier, but more accurate).


The process of truncation deletes all the numerals to the right of a certain point in the decimal part of an expression. Some electronic calculators use this process to fit numbers within their displays. For example, the number 3.830175692803 can be shortened in steps as follows:

Scientific Notation Approximation, Error, and Precedence Truncation


Rounding is the preferred method of rendering numbers in shortened form. In this process, when a given digit (call it r) is deleted at the right-hand extreme of an expression, the digit q to its left (which becomes the new r after the old r is deleted) is not changed if 0 ≤ r ≤ 4. If 5 ≤ r ≤ 9, then q is increased by 1 (“rounded up”). The better electronic calculators use rounding rather than truncation. If rounding is used, the number 3.830175692803 can be shortened in steps as follows:

Scientific Notation Approximation, Error, and Precedence Rounding


When physical quantities are measured, exactness is impossible. Errors occur because of imperfections in the instruments, and in some cases because of human observational shortcomings or outright mistakes.

Suppose x a represents the actual value of a quantity to be measured. Let x m represent the measured value of that quantity, in the same units as x a . Then the absolute error, D a (in the same units as x a ), is given by:

D a = x mx a

The proportional error, D p , is equal to the absolute error divided by the actual value of the quantity:

D p = ( x mx a )/ x a

The percentage error, D % , is equal to 100 times the proportional error expressed as a ratio:

D % = 100 (x m – x a )/x a

Error values and percentages are positive if x m > x a , and negative if x m < x a . That means that if the measured value is too large, the error is positive, and if the measured value is too small, the error is negative. Sometimes the possible error or uncertainty in a situation is expressed in terms of “plus or minus” a certain number of units or percent. This is indicated by a plus-or-minus sign (±).

Note the denominators above that contain x a , the actual value of the quantity under scrutiny, a quantity we don’t exactly know because our measurement is imperfect! How can we calculate an error based on formulas containing a quantity subject to the very error in question? The common practice is to derive a theoretical or “ideal” value of x a from scientific equations, and then compare the observed value to the theoretical value. Sometimes the observed value is obtained by taking numerous measurements, each with its own value x m1 , x m2 , x m3 , and so on, and then averaging them all.


Mathematicians, scientists, and engineers have all agreed on a certain order in which operations should be performed when they appear together in an expression. This prevents confusion and ambiguity. When various operations such as addition, subtraction, multiplication, division, and exponentiation appear in an expression, and if you need to simplify that expression, perform the operations in the following sequence:

  • Simplify all expressions within parentheses, brackets, and braces from the inside out
  • Perform all exponential operations, proceeding from left to right
  • Perform all products and quotients, proceeding from left to right
  • Perform all sums and differences, proceeding from left to right

Here are two examples of expressions simplified according to the above rules of precedence. Note that the order of the numerals and operations is the same in each case, but the groupings differ.

Scientific Notation Approximation, Error, and Precedence Precedence

Suppose you’re given a complicated expression and there are no parentheses in it? This does not have to be ambiguous, as long as the abovementioned rules are followed. Consider this example:

z = –3x 3 + 4x 2 y – 12 xy 2 – 5 y 3

If this were written with parentheses, brackets, and braces to emphasize the rules of precedence, it would look like this:

z = [–3(x 3 )] + {4[(x 2 ) y ]} – {12[x(y 2 )]} – [5(y 3 )]

Because we have agreed on the rules of precedence, we can do without the parentheses, brackets, and braces.

There is a certain elegance in minimizing the number of parentheses, brackets, and braces in mathematical expressions. But extra ones do no harm if they’re placed correctly. You’re better off to use a couple of unnecessary markings than to risk having someone interpret an expression the wrong way.

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