This visual converter helps you understand base conversion through cube representations. For larger numbers or pure calculation needs, use ourFull Base Converterwhich handles unlimited number sizes without visual constraints.
Base (or radix) — the number of distinct digits a number system uses, and the factor by which each place is worth more than the one to its right. Decimal is base 10, binary base 2, octal base 8, hexadecimal base 16.
Digit — one symbol of the system. Base b uses the digits 0 to b−1; above 9 the letters A to Z stand in, so base 16 runs 0 to F and base 36 uses all ten digits and all twenty-six letters.
Place value — the weight of a position: b0,b1,b2,… reading from the right. The digit in a place contributes digit times weight.
Positional notation — writing a number as a string of digits whose meaning depends on position, dk⋯d1d0=∑idibi.
Repeated division — the standard way to convert from decimal to base b: divide by b, record the remainder, repeat with the quotient until it reaches 0, then read the remainders from last to first.
Expansion — the reverse direction: multiply each digit by its place value and add, which converts any base back to decimal.
Getting Started with the Visualizer
Type a number, choose a base, and the tool shows how the number is rebuilt digit by digit in that base.
• Number takes a whole number from 0 to 144, entered as an ordinary decimal • Base takes any base from 2 to 36; binary, octal and hexadecimal are the usual choices, but any value in the range works, and the letters A to Z appear as digits above 9 • The visualization lays out the conversion step by step, so each digit of the result can be traced back to the operation that produced it • Reset clears both fields and the display
The limit of 144 is deliberate: it is 122, large enough for every base in the range to show at least two digits and small enough for every step to fit on screen. Try the same number in bases 2, 8 and 16 in a row; the binary and hexadecimal results line up in groups of four bits, which the grouping section explains.
What a Base Is
Every positional number system works the same way. A base b has b digits, 0 through b−1, and each place is worth b times the place to its right. The string dkdk−1⋯d1d0 stands for
dkbk+dk−1bk−1+⋯+d1b+d0
In decimal, b=10, and 472 means 4⋅100+7⋅10+2. In binary, b=2, and 101 means 1⋅4+0⋅2+1=5. The digits are different, the rule is not.
Decimal is a convention, not a property of numbers. The quantity five is the same whether it is written 5, 1012, 123 or 516; only the notation changes. That is the whole idea the visualizer is built to show: a base conversion changes how a number is written, never what it is.
Two facts follow. A number needs about logbN digits in base b, so small bases give long strings and large bases short ones. And the largest k-digit number in base b is bk−1, all digits at their maximum, just as 999=103−1.
Converting from Decimal: Repeated Division
To write a decimal number N in base b, divide by b and keep the remainder, then divide the quotient by b and keep that remainder, and continue until the quotient is 0. The remainders, read from the last one obtained to the first, are the digits.
The method works because each division strips off the lowest place. The first remainder is the units digit, since N=b⋅q+r leaves r in the b0 place and pushes everything else one place up. The visualizer shows exactly this sequence, which is why the digits appear in reverse order of the steps.
The same steps in base 16 finish sooner: 45÷16=2 remainder 13, then 2÷16=0 remainder 2, so 45=2D16, with D standing for thirteen.
Converting to Decimal: Expansion
The reverse direction needs no division. Multiply each digit by its place value and add.
1011012=1⋅32+0⋅16+1⋅8+1⋅4+0⋅2+1⋅1=45
2D16=2⋅16+13⋅1=45
A faster hand method, Horner's scheme, works left to right: start with the leading digit, and for each further digit multiply the running total by the base and add the digit. For 1011012: 1, then 2⋅1+0=2, then 2⋅2+1=5, then 11, then 22, then 45. It uses one multiplication per digit and never needs the powers of the base written out.
Conversion between two non-decimal bases is done in two hops through decimal: expand the source, then divide into the target. The one shortcut is between bases that are powers of the same number, treated in the grouping section.
Binary, Octal and Hexadecimal
Three bases dominate computing, and they are related by grouping.
Binary, base 2, is the native language of digital hardware: each digit is one bit, on or off. It is also the longest notation; 144 takes eight bits, 100100002.
Octal, base 8, and hexadecimal, base 16, are shorthand for binary. Because 8=23 and 16=24, one octal digit is exactly three bits and one hexadecimal digit is exactly four, so conversion between them and binary needs no arithmetic, only grouping. Split 100100002 into fours from the right, 10010000, and read each group: 9 and 0, so 9016. Split into threes, 10010000, and read 2, 2, 0: 2208.
Hexadecimal won because a byte is eight bits and therefore exactly two hex digits, from 00 to FF. Memory addresses, colour codes and hash values are written in hex for that reason.
Set the visualizer to base 2, then base 16, for the same number and check the grouping by eye. The tool computes both by repeated division, but the results line up as the grouping rule predicts.
Bases Beyond Sixteen
Nothing in the rule stops at 16. Base b needs b digit symbols, and the convention is to continue past 9 with the letters of the alphabet: A=10, B=11, up to Z=35. Base 36 therefore uses every digit and every letter, which is the largest base with a standard set of symbols and the upper limit of the visualizer.
Large bases give short strings. In base 36, 144 is 4036, since 4⋅36+0=144. Short strings of letters and digits are useful wherever a number has to be typed or read by a person: shortened links, licence keys, and identifiers are often base-36 or base-62 encodings of large integers.
Small bases have their uses too. Base 3 appears in balanced ternary and in some logic circuits; base 12 has a long history in measurement because 12 divides evenly by 2, 3, 4 and 6; base 60, inherited from Babylon, still runs our minutes and seconds.
The point of allowing any base from 2 to 36 is that the visualizer never suggests decimal is special. The same steps produce the same number in every base.
Worked Example
Convert 100 to four bases by repeated division.
Base 2.100→50 r 0, 50→25 r 0, 25→12 r 1, 12→6 r 0, 6→3 r 0, 3→1 r 1, 1→0 r 1. Reading the remainders upward: 100=11001002.
Base 8.100→12 r 4, 12→1 r 4, 1→0 r 1. So 100=1448.
Base 16.100→6 r 4, 6→0 r 6. So 100=6416.
Base 36.100→2 r 28, 2→0 r 2. The digit for 28 is S, so 100=2S36.
Check by expansion: 1⋅64+1⋅32+0+0+1⋅4+0+0=100; 1⋅64+4⋅8+4=100; 6⋅16+4=100; 2⋅36+28=100.
Notice the grouping between binary and the two power-of-two bases: 1100100 in threes reads 1, 4, 4, and 1100100 in fours reads 6, 4. Enter 100 in the visualizer with each base to watch the same remainders appear.
Common Mistakes
Base conversion is mechanical, and the errors are mechanical too.
• Reading the remainders in the order they were produced — the first remainder is the units digit, so the digits must be read from the last remainder to the first • Stopping before the quotient reaches zero — the last division, with quotient 0, produces the leading digit; skipping it drops the most significant digit • Using a digit that is too large for the base — base b has digits 0 to b−1 only; there is no digit 2 in binary and no digit 8 in octal • Forgetting that letters are digits — in hexadecimal A to F are the numbers ten to fifteen, not text, and 1A16 is 26 • Grouping bits from the left — grouping into threes or fours must start from the right, the units end, padding with zeros on the left if needed • Treating the conversion as changing the number — 45, 1011012 and 2D16 are one quantity written three ways; arithmetic on any of them gives the same answers
Where Base Conversion Appears
Computing — every number in a processor is binary, and programmers read it as hexadecimal; converting between the two is daily work in debugging, networking and graphics, where colours are three hex bytes.
Data encoding — identifiers, short links and keys are large integers written in base 36 or base 64 so that they are short enough to type.
Digital electronics — octal and binary describe the states of switches and the addresses of memory cells.
Number theory — divisibility rules are facts about base 10; the rule for 9, that a number is divisible by 9 when its digit sum is, works because 10≡1(mod9), and a different base has different rules.
Measurement and time — base 60 for angles and time, base 12 for dozens and inches, base 20 in the French word for eighty; the history of counting is a history of bases.
Teaching place value — conversion is the cleanest way to see what the decimal system actually does, since the mechanism is easier to notice in an unfamiliar base than in the one everyone has used since childhood.
Related Concepts
Place value — the principle every base shares: a digit's contribution is its value times the power of the base at its position.
Powers and exponents — the place values are the powers b0,b1,b2,… of the base.
Division with remainder — the operation repeated division is built from, N=bq+r with 0≤r<b.
Modular arithmetic — the remainder on division by b is Nmodb, the units digit in base b.
Divisibility rules — facts about digits in a particular base, derived from the base's remainders modulo the divisor.
Logarithms — the number of digits of N in base b is about logbN.
Binary arithmetic — addition and multiplication carried out directly in base 2, the next step after conversion.
Base conversion table — the reference table on this site for quick lookup, alongside the numeric base converter for larger inputs.