English:Binary Numbers
Binary Numbers
Introduction
Binary numbers are numbers written in the binary numeral system, a base-2 system that uses only the digits 0 and 1. You already use the decimal system every day. Decimal is base 10, so each place is worth ten times the place to its right. Binary works in the same positional way, but each place is worth two times the place to its right.
Binary is especially important in computer science because digital devices can represent two clearly different states, such as off and on, low and high voltage, or false and true. A single binary digit is called a bit. Groups of bits can represent numbers, text, images, sound, instructions, and many other kinds of data.
By the end of this aiMOOC, you should be able to:
- Explain binary place value: Describe why binary uses powers of two.
- Convert numbers: Change small whole numbers between decimal and binary.
- Calculate in binary: Add simple binary numbers and explain carrying.
- Reason about bits: Work out how many patterns can be represented by a given number of bits.
- Connect binary to digital data: Explain how bit patterns can stand for different kinds of information.
Understanding Place Value
Decimal and Binary Place Value
In decimal, the places from right to left are 1, 10, 100, 1000, and so on. These are powers of 10. In binary, the places from right to left are 1, 2, 4, 8, 16, 32, 64, and so on. These are powers of 2.
For example, the binary number 101101 has six places:
| Binary place | 32 | 16 | 8 | 4 | 2 | 1 |
|---|---|---|---|---|---|---|
| Binary digit | 1 | 0 | 1 | 1 | 0 | 1 |
A 1 means "include this place value" and a 0 means "do not include this place value." Therefore 101101 in binary represents 32 + 8 + 4 + 1 = 45 in decimal.
The image above shows another example: decimal 40 can be made from 32 + 8, so its binary form is 101000.
Powers of Two
Powers of two are the key to reading and writing binary numbers. Starting with 2 to the power of zero, the values double each time:
| Power | Value | Binary place |
|---|---|---|
| 2^0 | 1 | rightmost place |
| 2^1 | 2 | second place from the right |
| 2^2 | 4 | third place from the right |
| 2^3 | 8 | fourth place from the right |
| 2^4 | 16 | fifth place from the right |
| 2^5 | 32 | sixth place from the right |
A useful checking question is: Does each place value double as you move left? If not, the place-value row needs correcting.
Converting Between Decimal and Binary
Binary to Decimal
To convert a binary number to decimal, write the place values above the binary digits. Add only the place values that have a 1 underneath them.
Example: 11010
| Place value | 16 | 8 | 4 | 2 | 1 |
|---|---|---|---|---|---|
| Binary digit | 1 | 1 | 0 | 1 | 0 |
The result is 16 + 8 + 2 = 26. A quick estimate can help you check: a five-bit positive number beginning with 1 must be at least 16.
Decimal to Binary
One method is to subtract powers of two. Begin with the largest power of two that does not exceed the decimal number. Put a 1 in that place, subtract its value, and continue with the smaller places. Put 0 in any place you do not need.
Example: convert decimal 29 to binary.
The largest useful place is 16. After using 16, 13 remains. Use 8, leaving 5. Use 4, leaving 1. Skip 2, then use 1. The place values 16, 8, 4, 2, 1 therefore receive the digits 1, 1, 1, 0, 1. So decimal 29 is 11101 in binary.
Another method repeatedly divides by 2 and records the remainders. For Grades 7–8, the place-value method is often easier to see and check, but both methods express the same base-2 structure.
Counting in Binary
Binary counting follows a repeating pattern. Whenever a place would need the digit 2, you reset that place to 0 and carry 1 to the next place on the left.
| Decimal | Binary | Decimal | Binary |
|---|---|---|---|
| 0 | 0000 | 8 | 1000 |
| 1 | 0001 | 9 | 1001 |
| 2 | 0010 | 10 | 1010 |
| 3 | 0011 | 11 | 1011 |
| 4 | 0100 | 12 | 1100 |
| 5 | 0101 | 13 | 1101 |
| 6 | 0110 | 14 | 1110 |
| 7 | 0111 | 15 | 1111 |
Notice the patterns. The rightmost bit changes every step. The next bit changes every two steps. The next changes every four steps. These patterns come directly from powers of two.
Bits, Bytes, and Range
A bit is one binary digit. With one bit, there are two possible patterns: 0 and 1. With two bits, there are four patterns. With three bits, there are eight patterns. In general, n bits can form 2^n different patterns.
A byte is commonly a group of eight bits. Eight bits can form 2^8 = 256 different patterns. If all eight bits are used for an unsigned whole number, the values can run from 0 through 255. The highest value is one less than the number of patterns because counting starts at zero.
Lights are a useful model. If each light can be either off or on, each light can act like one bit. A row of lights can therefore represent a binary number.
Why More Bits Matter
Adding one more bit doubles the number of available patterns. Four bits provide 16 patterns, five bits provide 32 patterns, and eight bits provide 256 patterns. More available patterns allow a system to distinguish more values.
This also explains overflow. If a calculation produces a value outside the range available to a fixed number of bits, the system cannot store that result in the same unsigned format without an additional rule or a larger number of bits.
Binary Addition
Binary addition uses the same idea of place value and carrying that you know from decimal addition, but the base is 2.
| Addition | Result | Explanation |
|---|---|---|
| 0 + 0 | 0 | Nothing to carry |
| 0 + 1 | 1 | One in the current place |
| 1 + 0 | 1 | One in the current place |
| 1 + 1 | 10 | Write 0 and carry 1 |
| 1 + 1 + 1 | 11 | Write 1 and carry 1 |
Example: 0101 + 0011 = 1000. In decimal, that is 5 + 3 = 8, so converting the result back to decimal is a good way to check your work.
From Binary Numbers to Digital Data
A bit pattern does not explain its own meaning. The same sequence of zeros and ones can represent a number, a letter, part of an image, a sound sample, or an instruction. The meaning depends on the rules used to interpret the pattern. Those rules are called an encoding or a data format.
For text, a character encoding assigns bit patterns to characters. For images, numbers can describe the brightness or color of pixels. For sound, numbers can describe measurements taken from a changing sound wave. Binary is therefore not a special "computer language" that replaces all other information. It is a compact way for digital systems to represent many kinds of information using two states.
Binary and Digital Images
A simple black-and-white image can be modeled with one bit per pixel if one state stands for black and the other stands for white. Color images need more information per pixel because the system must distinguish many colors.
The image above is a color chart produced with a 9-bit RGB palette. It illustrates a central idea: when more bits are available for a coded property, more distinct combinations can be represented.
A Short History
Binary ideas are much older than modern electronic computers. The mathematician and philosopher Gottfried Wilhelm Leibniz described binary arithmetic using 0 and 1 in a work published in 1703. His table showed decimal values alongside their binary forms.
This history is useful because it shows that binary is first a mathematical numeral system. Electronic computers later made base 2 especially practical because hardware can be designed around components with two distinguishable states.
Common Mistakes and Checking Strategies
A common mistake is to read a binary string as if it were a decimal number. For example, binary 1010 is not "one thousand and ten"; it represents decimal 10. Another mistake is to assign place values 1, 2, 3, 4 instead of 1, 2, 4, 8. Remember that binary place values double.
When converting from binary to decimal, check that every digit is either 0 or 1 and that the place values are powers of two. When converting from decimal to binary, convert your answer back to decimal. When adding binary numbers, convert both the original numbers and your result to decimal as an independent check.
Interactive Tasks
Quiz: Test Your Knowledge
Which base does the binary numeral system use? (Base two) (!Base eight) (!Base ten) (!Base sixteen)
What is binary 1010 in decimal? (Ten) (!Eight) (!Twelve) (!Fourteen)
Which binary number represents decimal 13? (1101) (!1011) (!1110) (!1001)
What is a single binary digit called? (Bit) (!Byte) (!Pixel) (!Codeword)
How many different patterns can three bits represent? (Eight) (!Three) (!Six) (!Nine)
What is the place value immediately to the left of the fours place in binary? (Eight) (!Five) (!Six) (!Ten)
What is binary 0101 plus binary 0011? (1000) (!0110) (!1010) (!1111)
What is the greatest unsigned decimal value that four bits can represent? (Fifteen) (!Four) (!Eight) (!Sixteen)
Why can the same bit pattern represent a number or a letter? (Its meaning depends on the encoding rules) (!Every bit pattern has one universal meaning) (!Letters are stored without binary) (!Numbers cannot be encoded with bits)
Which scholar published a well-known explanation of binary arithmetic in 1703? (Gottfried Wilhelm Leibniz) (!Isaac Newton) (!Alan Turing) (!Ada Lovelace)
Memory Game
| Bit | A single binary digit |
| Byte | A common group of eight bits |
| Place value | The value assigned to a digit position |
| Power of two | A value in the sequence formed by repeated doubling |
| Overflow | A result that lies outside the available fixed range |
| Encoding | Rules that assign meaning to stored patterns |
Drag and Drop
| Match the correct terms. | Topic |
|---|---|
| Most significant bit | Leftmost position with the highest place value |
| Least significant bit | Rightmost position with the lowest place value |
| Binary addition | Calculation that uses base-two carrying rules |
| Decimal conversion | Rewriting a value in another numeral system |
| Data encoding | Assigning agreed meaning to bit patterns |
...
Crossword Puzzle
| Binary | Which numeral system uses only zero and one as digits? |
| Decimal | Which everyday numeral system uses ten digits? |
| Digit | What is one written symbol in a numeral system called? |
| Byte | What is a common group of eight bits called? |
| Overflow | What happens when a result exceeds the available fixed range? |
| Leibniz | Which scholar published an explanation of binary arithmetic in 1703? |
LearningApps
Cloze Text
Open-Ended Tasks
Easy
- Binary scavenger hunt: Find five examples of two-state choices in everyday life, photograph or sketch them, and explain how each could model a binary bit.
- Place-value poster: Create a clear poster showing the binary place values from one through one hundred twenty-eight and include three worked conversions.
- Binary card game: Make place-value cards and use them with a partner to represent ten teacher-chosen decimal numbers; record the binary results.
- Human binary counter: With classmates, act as bits that switch between zero and one while counting upward, then write a short explanation of the pattern you observed.
Standard
- Binary interview: Interview someone who works with computers, electronics, or digital media and ask where binary representation appears in their work; summarize the answers in a short report.
- Pixel image encoding: Design a small black-and-white pixel picture, choose a rule for zero and one, encode the image as a bit string, and give the code to a classmate to reconstruct.
- Binary addition tutorial: Produce a one-minute video or illustrated guide that teaches binary addition with at least two examples and a decimal check.
- Computer lab investigation: Visit a school computer lab or technology classroom, identify devices that store or process digital data, and explain how bits are relevant even when users never see binary numbers.
Advanced
- Bit range experiment: Build a table for bit widths from one through eight, determine the number of patterns and greatest unsigned value for each, then describe the mathematical relationship you discover.
- Binary clock design: Design a paper, spreadsheet, or programmable binary clock display and write instructions that another learner can use to read the time.
- Encoding comparison project: Choose a short word and investigate how a standard character encoding represents it; present the characters, numeric codes, and binary patterns while clearly naming the encoding used.
- Overflow investigation: Create several fixed-width binary addition examples that do and do not overflow, test them with a calculator or simple program, and explain a general rule for predicting overflow in unsigned arithmetic.
Learning Assessment
- Conversion reasoning: Convert decimal 73 to binary, convert your result back to decimal, and explain how the reverse conversion checks your work.
- Error analysis: A learner says that binary 10110 equals decimal 16 because there are sixteen visible symbols in the place-value row; identify the misunderstanding and give a correct place-value explanation.
- Pattern transfer: Without listing every number first, predict which bit positions change when a four-bit counter moves from 0111 to the next value and justify your prediction.
- Range application: A sensor must distinguish 100 different levels; determine the smallest whole number of bits that can provide at least 100 different patterns and explain your reasoning.
- Binary addition proof: Add two binary numbers of your choice, verify the result in decimal, and explain why carrying in base 2 works in the same structural way as carrying in base 10.
- Data representation explanation: Explain how one bit pattern could mean different things in different systems and give two realistic examples that use different interpretation rules.
Evidence of Learning
Strong evidence of learning combines knowledge with explanation, accurate calculation, communication, and transfer to new situations.
| Area | Evidence |
|---|---|
| Knowledge | You correctly explain base 2, powers of two, bits, bytes, range, carrying, encoding, and overflow. |
| Skills | You convert between decimal and binary, add binary values, calculate pattern counts, and check results using a second method. |
| Products | Your posters, encoded images, videos, reports, tables, or designs communicate binary ideas accurately to another learner. |
| Reasoning | You explain why procedures work instead of only giving answers, and you can diagnose common mistakes. |
| Transfer | You connect binary place value to digital images, text, devices, sensors, or other unfamiliar applications. |
OERs on the Topic
The English Wikipedia article on binary numbers gives a broad reference for the numeral system, its notation, arithmetic, and history.
Linked Learning Areas
Binary numbers connect mathematics with computing, electronics, and digital media. Place value links binary to other numeral systems. Powers of two connect the topic to exponentiation and patterns. Bits and bytes connect number representation to data representation, while binary addition links arithmetic to the way digital circuits process information.