Understanding Number Limits, Overflow, and Roundoff in Computers

When computer programs store numbers, they need to represent those numbers in the computer's memory. This process varies depending on whether the number is an integer or a fraction. Due to memory limitations, we sometimes encounter issues like roundoff errors, overflow, or precision problems.

Integer Representation

An integer is a whole number without any fractions. Examples of integers include 1, 2, -3, and 0. Computers store all data using bits, which are tiny units represented by 0s and 1s.

Imagine a simple computer that uses only 4 bits to represent integers. The first bit indicates the sign (positive or negative), and the remaining 3 bits represent the actual number. Here's how it works:

• Positive number 1 in 4 bits: 0 0 0 1
  • The first bit 0 indicates a positive number.
  • The last three bits 001 represent the number 1.

What's the largest number we can represent with 4 bits? Let's fill all the value bits with 1s:

• 0 1 1 1
  • This represents the number 7.

So, the largest number our 4-bit system can represent is 7.

Overflow

Overflow occurs when a number exceeds the maximum value a system can represent. For example:

javascript

var x = 7;
var y = x + 1;

Here, y would be 8, but our 4-bit system can't store 8. The computer might show an "overflow error" or give a strange result, like wrapping around to 0.

Most modern computers use 64 bits, allowing them to store very large numbers. In JavaScript, the largest safe integer is 9,007,199,254,740,992. Beyond this value, calculations can become unreliable.

Floating-Point Representation

Things get trickier with numbers that have fractions, like 3.14 or 0.75. Computers use "floating-point" representation for these numbers, similar to scientific notation you might know from school. In floating-point, a number is written as:

number * base^exponent

But instead of using base 10 like in decimal, computers use base 2 (binary).

For example, the number 8 in floating-point might look like:

1.0 * 2^3

Numbers that are not exact powers of 2, like 0.1, are harder to represent precisely in binary, leading to small errors called "roundoff errors."

Roundoff Errors

Roundoff errors happen when a number cannot be represented exactly in binary and must be rounded. For example, the fraction 1/3 in decimal is 0.333... (repeating forever). In binary, some fractions also repeat forever and need to be rounded.

Imagine trying to add 0.1 + 0.2 in a computer. You might expect the result to be 0.3, but because of roundoff errors, it might be a tiny bit off. This is usually fine for most calculations, but can be a problem in very precise situations, like scientific calculations or financial transactions.

Understanding how computers store and handle numbers helps us write better programs and avoid errors. While modern computers are powerful and can handle large numbers and complex calculations, knowing their limits is crucial, especially when precision matters, like in scientific research, engineering, or finance.