Note: We are here ignoring discrete mathematics that involve only integers.

A slide rule (not slide ruler) was once used to do manual computations.

Whenever a slide rule is used, it is very evident that any computation involving real numbers is an approximation.

What are the values of the following (using a computer program)?

Note: These are rational numbers, not irrational like √2 or transcendental numbers like π or e.

Here is the mathematics result expressed as real (rational) numbers.

Here is a simple Lua program to add values of 1/10 as 0.1.
sum1 = 0.0 for count1=1,10 do sum1 = sum1 + 0.1 print(string.format("%2d/10 = %1.17f",count1,sum1)) end
The 17 places in the output is important to being exactly precise.

Here is the result from a computation point of view as the output of the above program.
1/10 = 0.10000000000000001 2/10 = 0.20000000000000001 3/10 = 0.30000000000000004 4/10 = 0.40000000000000002 5/10 = 0.50000000000000000 6/10 = 0.59999999999999998 7/10 = 0.69999999999999996 8/10 = 0.79999999999999993 9/10 = 0.89999999999999991 10/10 = 0.99999999999999989
Even when symbolic math can solve a problem, any attempt to compute real answers involves the same approximations.

The difference is small at each step but the difference can add up to a significant amount over many iterations.

Note: One can fix this specific instance by using a fixed decimal notation, but that only works, in base 10, for numbers and increments that are multiples of 2 or 5, the prime factors of 10.

Rounding errors need to be addressed in fields of computer science such as numerical analysis.

Fallacy: Dollars and cents should be represented using floating point variables.
The general rule, sometimes taught, that if the number has a decimal point, it should be represented as a floating point variable does not hold for dollars and cents.

• The amount $123.56 is not a floating point value. It an integer value. That is 12345¢ cents.

Rule: Convert dollars and cents to cents, do the arithmetic, then convert back into dollars and cents (for display purposes). This distinction needs to be carefully handled in languages such as JavaScript and Lua which do not have integer variables.

The approximation issue involves floating point division so every such division needs to result in an integer value.

Fallacy: Social security numbers and phone numbers should be represented using integers.
How should social security numbers and phone numbers be represented? They appear to be numbers. That is, integers.

Are you ever going to add, subtract, multiply or divide these values? If not, use text.

• SSN: 999-99-9999 (1 billion values, originally geographically located)

Ask your self the following question.

• Are you ever going to add, subtract, multiply or divide these values?

If not, represent them using text. Note:

• Leading zeros are lost using integers.
• If an ordering is present, care must be taken when sorting lists of such values.