The worked guide

Follow the value,
then follow the remainder.

A base tells you the place values of the digits. Changing it changes the notation. It should not change the rational number you entered.

1. Read the base before reading the digits

In base 10, the integer 21 means two tens and one unit. In base 2, the integer 101 means one four, no twos and one unit:5 in decimal. After a radix point, the powers become negative. The first fractional binary place is one half, the second is one quarter and the third is one eighth. “Radix point” means the separator in any base; it is a decimal point specifically when the base is 10.

For bases above 10, Radixnote uses A=10, B=11 and so on through Z=35. Letters can be entered in either case. The chosen base must include every digit: F is valid in base 16, but not in base 10. Z is valid only in base 36. A digit's position and its value are separate ideas.

2. Convert 101.01 from binary to decimal

The whole part 101 is 5. The fractional digits 01 mean zero halves plus one quarter. The complete value is therefore 5¼, or the reduced ratio 21/4. Decimal notation writes this as 5.25.

An exact parser can remove the radix point and treat the five binary digits 10101 as the integer 21. There were two fractional digits, so the original value is 21 divided by 2². This method uses integer arithmetic throughout. It does not need to estimate the value with a floating-point number.

For a source base b and k fractional digits, the same idea gives the combined digit integer divided by bᵏ. Common factors are then removed from the numerator and denominator. The input 0.100 is allowed; the redundant final zeroes do not create extra precision in an exact arithmetic model.

3. Let the remainder identify a repeating block

To write a fraction in a new base, first separate its integer part. Multiply the remaining fraction's numerator by the target base, divide by its denominator, and keep both the next digit and the new remainder. Continue from that remainder.

For 1/3 in base 2, the first step divides 2 by 3: the next digit is 0 and the remainder is 2. The next step divides 4 by 3: the digit is 1 and the remainder is 1. We have returned to the starting remainder. The digits 01 will therefore repeat forever. Radixnote writes 0.(01).

Parentheses enclose only the repeating block. In decimal,1/6 is 0.1(6): the 1 occurs once and the 6 repeats. The first nonrepeating part may contain zeroes. Decimal 0.1 becomes binary 0.0(0011), exactly 1/10. Removing that initial nonrepeating 0 would change the value.

A positive denominator gives only finitely many possible remainders. Long division must eventually reach zero or return to a previous nonzero remainder. A cycle can still be too long for the chosen display limit. The limit is a work and presentation bound, not evidence against repetition.

An independent check of repeating output

A repeating block of p digits, interpreted as integerR in baseB, has fractional value R/(Bᵖ−1) when it starts immediately after the point. If there arek nonrepeating digits first, divide that repeating contribution byBᵏ and add the finite prefix. This geometric-series identity provides a separate way to check a generated expansion.

4. Distinguish a display prefix from the full value

Try 1/7 with both bases set to 10 and a fractional limit of 4. The display is 0.1428…, a truncated prefix. It is not an exact equality with 1/7. The complete reduced ratio remains 1/7 in its own field and in every report. Increase the limit to 6 and the cycle closes: 0.(142857) is an exact repeating representation.

Truncation is toward zero. For a negative value, the displayed prefix keeps its minus sign but has smaller absolute value than the full number. It is neither rounding to nearest nor a lower-bound guarantee for negative numbers. The ellipsis is a warning that more digits remain, not a digit you can enter.

The integer part is always shown in full. The fractional limit controls digits after the point only. A long integer may still require a large box, so long outputs wrap and can be selected or downloaded locally.

5. Use an explicit and deliberately small input grammar

Choose an input base and enter either a finite numeral or a ratio of two integers in that base. A single leading+ or- is allowed. Examples include -A.F in base 16, .5 in base 10, and -A/F in base 16. The last is−10/15, reducing to−2/3, so its decimal expansion is−0.(6). Both ratio terms are read in the selected input base. The denominator must be positive and nonzero.

A finite numeral can be an integer, an integer followed by a point, a fractional form with a leading point, or digits on both sides of a point. There must be at least one digit overall. Outer whitespace is trimmed, but whitespace inside the number or ratio rejects. No grouping commas, underscores, expressions, mixed numbers or non-ASCII digits are accepted.

Prefixes are not automatically recognised. In base 16, 0xFF rejects because X is not a hexadecimal digit. In base 36, 0X is a legitimate two-digit sequence with value 33; the X is a digit, not a prefix. Similarly, E means digit 14 when the selected base includes it. The text 1e3 is invalid in base 10 and is not interpreted as one thousand. Choose the base first and read every character under that rule.

Repeating parentheses are output notation only. To reuse the complete value, copy its exact integer ratio in the output base and select that base as the new input base. Do not paste a truncated prefix or delete parentheses from repeating digits and expect to preserve the number. Every accepted value's exact target-base ratio fits the stated input limits.

6. Understand the software bounds

Input text is limited to 8192 characters. A finite numeral may contain up to 4096 digits across its whole and fractional parts; each integer term of a ratio may contain up to 4096 digits. After common factors are removed, the absolute numerator and positive denominator must each fit 2048 binary bits. A value that exceeds that canonical bound is rejected rather than silently rounded.

The canonical limit is independent of the selected radix. Its longest integer terms occur in binary, where each has at most 2048 digits. That makes the exact ratio re-enterable in any supported target base. Reducing before checking the bit limit also permits large written common factors that cancel. These are practical browser-software bounds, not mathematical limits on numeral systems.

The fractional display budget is 1–256 digits, with 64 as the initial example setting. Increasing the limit can reveal a later end or repeating cycle. A truncated result never claims a finite or nonrepeating underlying value. Radixnote does not calculate irrational constants or machine encodings such as IEEE floating-point bits, signed bit-width formats or two's complement.

7. Save a useful local record

The text report and CSV include the source input, both bases, digit budget, status and exact reduced ratio. Reports explicitly distinguish a truncated prefix from an exact expansion. Copy may need browser permission; if it is unavailable, use a local text download. A copy operation refers to the value at the time it was requested. Later edits invalidate its pending status message so an old completion cannot claim that a new result was copied.

Calculations run locally without uploading the input. Cloudflare Web Analytics is enabled in this host's settings and its deployment-added beacon supplies aggregate page-view and performance reports when it loads. Advertising and consent-message loaders remain inactive in this release. See the privacy page for the distinction between local calculation, measurement and ordinary hosting requests.

Mathematical references

• Python's official integer documentation describes conventional radix 2–36digit values. Radixnote intentionally accepts a narrower grammar than Python and does not load Python.
• Python's official rational-number documentation distinguishes rational representations and floating input. This site's own arithmetic avoids a floating-point intermediate.
• David Eppstein's university-hosted algorithm discussion explains repeating rational binary expansions and remainder cycles. Radixnote uses its own bounded long-division implementation for the entered target base.

These references support the mathematics. The examples, interface, code and explanations here are original. No external library, conversion service or reference-page code runs in your calculation.