The Language of a Light Switch

Imagine your entire world of communication was reduced to a single light switch. You can turn it ON or OFF. How could you possibly convey complex ideas? This is the fundamental reality for a computer's processor. At its core, it's just a mind-bogglingly vast collection of microscopic, lightning-fast switches called transistorsA tiny electronic switch that can be turned on or off very quickly. Modern CPUs have billions of them.. An "ON" state can represent a 1, and an "OFF" state can represent a 0. This single unit of information—a 0 or a 1—is called a bit, short for "binary digit."

But how do you get from a simple ON/OFF to the rich, complex data we see on our screens? You group them together. Just like how we group letters to form words, computers group bits to form numbers, letters, and everything else. This system of representing information with only two symbols is called the binary, or base-2, number system.

This entire article is a deep dive into that binary world. We will explore how computers cleverly manipulate these simple 0s and 1s to represent every number you can imagine—from simple integers to complex decimals—and perform the arithmetic that powers our digital lives.

Representing Integers: The World of Whole Numbers

The simplest numbers to represent are integers (..., -2, -1, 0, 1, 2, ...). We'll start with the positive ones and then tackle the challenge of representing negative signs.

Unsigned Integers: Simple Binary Counting

An unsigned integer is a whole number that can only be positive. Representing them is a straightforward process of counting in base-2. In our familiar base-10 system, each digit's place represents a power of 10 (1s, 10s, 100s, etc.). In binary, each place represents a power of 2 (1s, 2s, 4s, 8s, 16s, etc.).

The Problem of Negativity: Representing Signed Integers

This is all well and good for positive numbers, but how does a computer represent -42? It has no "-" symbol. It only has 0s and 1s. Over the years, computer scientists developed several methods, but one proved to be vastly superior.

The most elegant and universally adopted solution is called Two's Complement. It's a clever system that not only represents negative numbers seamlessly but also makes computer arithmetic incredibly efficient. The magic of two's complement is that the operation A - B becomes identical to A + (-B) at the circuit level. This means the computer doesn't need separate hardware for subtraction; it just re-uses the same adder circuit!

To find the two's complement of a positive number (i.e., to make it negative):

1. Flip all the bits (change 0s to 1s and 1s to 0s). This is the "one's complement".
2. Add one to the result.

Representing Real Numbers: The Floating-Point Standard

Representing integers is one thing, but how does a computer store numbers with fractions, like 9.75, or incredibly large and small numbers, like the mass of a planet or an atom? This requires a system analogous to scientific notation, and it's called floating-point representation.

The universal standard for this is called **IEEE 754**. It defines a way to pack a real number into a finite number of bits (usually 32 or 64) by breaking it into three parts:

• The Sign Bit (S): The simplest part. 1 bit that's 0 for positive and 1 for negative.
• The Exponent (E): This part determines the number's magnitude (its range). It says how far to shift the binary point, allowing for both very large and very small numbers.
• The Mantissa (M) or Significand: This part holds the actual digits of the number—its precision.

This system is incredibly efficient, but it's also the source of the round-off errorThe error introduced when a number with too many digits to store is rounded to the nearest value that the computer can actually represent. we discussed earlier. Because the mantissa can only hold a finite number of digits, any number that can't be perfectly represented in binary (like 0.1) must be rounded.

Solving Numerical Problems: A Step-by-Step Guide

Let's apply these concepts to solve some typical problems.

Problem 1: Binary Arithmetic

Question: Using 8-bit registers, calculate 50 - 20 by converting to binary, finding the two's complement of 20, and adding the result to 50.

  00110010  (50)
+ 11101100  (-20)
----------
1 00011110  (30)

Problem 2: Decimal to IEEE 754 Floating Point

Question: Find the 32-bit single-precision representation for the number -12.625.

Test Your Intuition!