Other Numeral Systems
Your cell phone stores photos as zeros and ones. The Maya of Chichén Itzá wrote their numbers with dots and bars. On the facade of an old theater you see the year carved as MCMXCVIII. Our decimal system (base $10$) is not the only one: the same number can be written in many ways depending on the base you choose. By the end of this page you will be able to read all of these notations.
What a base is and how to read a binary number
In base $10$ we use ten symbols ($0$ through $9$) and each position is worth $10$ times the one before it. In base $2$ (binary) there are only two symbols, $0$ and $1$, and each position is worth twice the one before it.
Try the interactive: turn the bits on and off and watch how the decimal number changes.
From decimal to binary: successive divisions
For the reverse direction, divide the number by $2$ over and over until the quotient is $0$. The remainders, read from last to first, form the binary number.
Shifting digits is multiplying by powers of 2
In binary, adding a zero at the end shifts the digits one position to the left and doubles the number, just as in decimal adding a zero multiplies by $10$. Shifting $n$ positions is equivalent to multiplying by $2^n$. For example, $101_2 = 5$ and $1010_2 = 10$.
Computers use the binary system (base 2) to process information, using only the digits $0$ and $1$. To understand how a machine "thinks", convert the decimal number $13$…
Computers use the binary system (base 2) to process information, using only the digits $0$ and $1$. To understand how a machine "thinks", convert the decimal number $13$ to binary code using the method of successive divisions by $2$.
The diagram shows the divisions you need to do. In each row, the quotient from the previous row is divided by $2$, and you continue until the quotient is $0$.
- a) Fill in the boxes of the diagram: the quotient and remainder of each division, until you reach a quotient of $0$.
- b) Read the remainders from bottom to top (from the last to the first) and write the number $13$ in base 2.
In computer science, the smallest unit of information is the bit, which can only take the values $0$ or $1$ (binary system). An old processor in a retro console uses…
In computer science, the smallest unit of information is the bit, which can only take the values $0$ or $1$ (binary system). An old processor in a retro console uses 8-bit registers. If a character has an energy level represented by the decimal number $43$, how would that value appear in the console's binary code?
In a computer system, multiplying a binary number by $2^n$ is equivalent to shifting its digits $n$ positions to the left (adding zeroes at the end). Given the decimal…
In a computer system, multiplying a binary number by $2^n$ is equivalent to shifting its digits $n$ positions to the left (adding zeroes at the end).
Given the decimal number $21$:
- a) Express it in the binary system.
- b) Perform the operation of shifting its digits $3$ positions to the left.
- c) Convert the resulting number back to the decimal system and check that the result is equivalent to multiplying $21 \times 2^3$.
Powers of 2 and units of information
When you store photos or videos on your cell phone, memory is measured with powers of $2$. A bit is a $0$ or a $1$. A byte is $8$ bits ($8 = 2^3$).
| Unit | Equals | Power of 2 |
|---|---|---|
| 1 byte (B) | 8 bits | $2^3$ bits |
| 1 KB | 1,024 B | $2^{10}$ B |
| 1 MB | 1,024 KB | $2^{20}$ B |
| 1 GB | 1,024 MB | $2^{30}$ B |
A file server for an esports gaming community has a total capacity of $2^{35}$ bits. Currently, the server contains three folders with game screenshots: folder A…
A file server for an esports gaming community has a total capacity of $2^{35}$ bits. Currently, the server contains three folders with game screenshots: folder A occupies $1.2$ GB, folder B occupies $950$ MB and folder C contains $2^{30}$ bytes. Given that $1$ GB = $1024$ MB and $1$ MB = $1024$ KB, determine whether it is possible to save an additional video of $1.5$ GB without exceeding the server's total capacity. Express all calculations as powers of base 2 where possible.
Roman numerals
Roman numerals are not positional: each symbol has a fixed value and they are combined by adding or subtracting depending on the order.
To do operations with Roman numerals, the easiest way is to convert everything to decimal, calculate, and convert back to Roman numerals. For numbers from $4{,}000$ on, the letter $M$ keeps being repeated: $6000 = MMMMMM$.
On the façade of an old theater in your city, the year of its opening is carved in Roman numerals: MCMXCVIII. Convert this number to our decimal system to find out which…
On the façade of an old theater in your city, the year of its opening is carved in Roman numerals: MCMXCVIII. Convert this number to our decimal system to find out which year the theater opened.
Calculate the result of the following operation with Roman numerals, expressing the final value both in Roman numerals and in the decimal number system:…
Calculate the result of the following operation with Roman numerals, expressing the final value both in Roman numerals and in the decimal number system:
$(MMCDLXIV - MDCCCLXXXIX) \times XII$
Other bases: Mayan, base 4, base 5
The Maya used base $20$ (vigesimal): they wrote with dots ($1$) and bars ($5$), in vertical levels. The bottom level is worth $20^0 = 1$, the next $20^1 = 20$, the next $20^2 = 400$.
This works for any base. For example, in base 4 the digits are $0, 1, 2, 3$ and the positions are worth $1, 4, 16, 64, \dots$ In base 5 the digits are $0$ through $4$ and the positions are worth $1, 5, 25, 125, \dots$
The Mayan number system was vigesimal (base 20) and vertically positional. Look at the following number represented by its levels and convert it into our decimal system:…
The Mayan number system was vigesimal (base 20) and vertically positional. Look at the following number represented by its levels and convert it into our decimal system:
-
Level 3 (top): One dot.
-
Level 2 (middle): Two bars and three dots.
-
Level 1 (bottom): One bar.
(Remember that in the Mayan system, the dot is worth $1$ and the bar is worth $5$).
An intergalactic explorer has arrived on a planet where the natives use a base 5 numeral system. You are given the following equivalences to understand their language:…
An intergalactic explorer has arrived on a planet where the natives use a base 5 numeral system. You are given the following equivalences to understand their language:
$1_{5} = \text{I}$, $2_{5} = \text{V}$, $3_{5} = \text{X}$, $4_{5} = \text{L}$.
If they tell you that there are $\text{XXL}$ hours left until the mothership takes off:
- a) Write the number of hours in the decimal system.
- b) Calculate how many minutes that time is equivalent to.
In an archaeological dig, an inscription with an unknown number system was found. After analyzing it, experts determined that it is a base-4 system, where the symbols…
In an archaeological dig, an inscription with an unknown number system was found. After analyzing it, experts determined that it is a base-4 system, where the symbols are: $\triangle = 0$, $\square = 1$, $\bigcirc = 2$ and $\star = 3$.
If the treasure is located at a distance of $\bigcirc \star \square \triangle$ meters from the main entrance, calculate this distance in our decimal system. Then, express the result obtained in the binary system.
Back to the beginning: the cell phone stores in base $2$, the Maya counted in base $20$, and the theater uses a non-positional system with letters. Now you know how to read and translate all three notations.