Bits, Bytes, and Why Two Symbols Are Enough

A bit is a binary digit: a single value that is either 0 or 1. It is the smallest unit of data a computer stores. A byte is a group of 8 bits. Computers represent all data — numbers, text, images, and sound — as sequences of bits. A photo, a song, and this sentence are all, underneath, just long strings of 0s and 1s. The meaning comes from how a program agrees to interpret those bits, not from the bits themselves.

Two symbols are enough because you combine them in a place-value system, the same idea you already use in base ten. In decimal you have ten digits and each place is worth ten times the one to its right. Binary is base 2: only two digits, and each place is worth twice the one to its right. Add more bits and you can count higher, exactly the way adding another decimal digit lets you count past 9.

Binary Place Values and Converting to Decimal

Because binary is base 2, the place values are the powers of 2. Reading right to left they are 1, 2, 4, 8, 16, 32, 64, 128, and so on, each one double the last. To convert a binary number to decimal, write the place values above the bits and add up the place values wherever a 1 appears.

In the row above, 1s sit in the 8, 2, and 1 columns, so the value is 8 + 2 + 1 = 11. That is how you read the binary number 1011 as decimal 11. The reverse direction, decimal to binary, is covered next.

Converting Decimal to Binary

Two reliable methods give the same answer. Pick whichever you trust under time pressure.

• Subtract the largest power of 2. Find the biggest power of 2 that fits, put a 1 there, subtract it, and repeat with what is left. For 37: 32 fits, leaving 5; 4 fits, leaving 1; 1 fits, leaving 0. Ones go in the 32, 4, and 1 columns, giving 100101.
• Repeated division by 2. Divide by 2 and record each remainder. The remainders, read from bottom to top, are the binary digits. Dividing 37 gives remainders 1, 0, 1, 0, 0, 1 from last to first, which reads as 100101.

Check your work by converting back: 32 + 4 + 1 = 37. If the place values you marked do not add to the original decimal number, one of the bits is wrong.

How Many Values Can n Bits Hold?

With n bits you can represent 2 to the n distinct values. Each bit doubles the count, because every existing pattern can now end in either a 0 or a 1. So 1 bit gives 2 values, 2 bits give 4, 3 bits give 8, and 8 bits (one byte) give 2^8 = 256 values. When those 256 values are used for whole numbers starting at zero, the range is 0 to 255, not 1 to 256.

This rule is the backbone of many exam questions. Adding a single bit does not add one more value, it doubles the number of values you can store.

Analog vs Digital

Analog data changes smoothly and continuously, with infinitely many possible values between any two points — think of a dimmer knob sliding, a real sound wave, or a shadow lengthening through the afternoon. Digital data uses discrete values: a finite set of separate steps, stored as bits. To put analog information into a computer you take samples, measuring the signal at many moments and recording each measurement as a number.

A digital version is an approximation of the analog original, like a staircase tracing a ramp. The more samples you take and the more bits you use per sample, the closer the digital copy gets to the smooth original — but with a finite number of bits it can never be a perfect match.

Limits of Fixed Bit-Width: Overflow and Roundoff

Because a computer sets aside a fixed number of bits for a value, some numbers simply do not fit, and the framework names two specific errors that result.

• Overflow error: a value is too large to represent in the available bits. With 8 bits the largest whole number is 255; try to store 256 and there is no room, so the result is wrong.
• Roundoff error: a value cannot be stored exactly and must be approximated or rounded. Many real numbers, such as the result of 1 / 3, would need infinitely many digits, so a fixed number of bits can only hold a rounded version.

The clean split: overflow is about a number being too big to fit at all; roundoff is about a number being impossible to store precisely, so it is stored close-but-not-exact. Both come from the same root cause, a finite number of bits.

Many Representations, Real Tradeoffs

The same information can be represented in more than one way, and the choice has consequences. The number eleven can be written as decimal 11 or binary 1011; both mean the same quantity. Using more bits per value widens the range you can store and reduces overflow and roundoff, but it also uses more memory and storage. Representations are chosen by weighing these tradeoffs, which is a theme you will see again in data compression.

How Binary Is Tested on the Exam

Binary number representation is multiple-choice material, not a skill you demonstrate on the Create Performance Task. You will not hand-convert numbers inside your Create Task program; instead the end-of-course multiple-choice exam asks you to convert between binary and decimal, apply the 2 to the n rule, and tell overflow apart from roundoff. Treat this as fast, high-confidence points.

Quick strategy for the exam room:

• Write the place values first. Jot 128 64 32 16 8 4 2 1 across the top and line the bits up underneath. For binary to decimal, add the place values under each 1.
• For decimal to binary, subtract the largest power of 2 that fits, mark a 1, and repeat with the remainder.
• Always check by converting back. Add your marked place values; they must equal the original decimal number.
• Sort the error type by keyword. "Too large to store" is overflow; "cannot be stored exactly" is roundoff.