Binary numbers: counting in base two

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Why Computers Use Binary

Computers use binary because their electronic circuits are two-state devices (on/off).
Binary uses only two symbols: 0 and 1.
This simplicity makes binary noise-immune and easy to implement with logic gates.
Almost all modern computers and devices use the binary system.

Computers use binary because a switch has two reliable states, easy to detect and hard to confuse.

Place Value in Base Two

Binary is a positional notation with a radix (base) of 2.
Each digit in a binary number is called a bit (binary digit).
In base two, each position represents a power of 2 (from right to left: 1, 2, 4, 8, 16, ...).
The rightmost bit is the least significant bit (2⁰ = 1).
The leftmost bit is the most significant bit (highest power of 2).

In an 8-bit number each position is a power of two, from 128 down to 1.

Counting in Binary

Counting in binary follows the same pattern as decimal, but only uses digits 0 and 1.
When you reach 1, the next number is 10 (binary for 2).
Binary sequence: 0, 1, 10, 11, 100, 101, 110, 111, ...
Each time you add 1, you carry over to the next place when a bit becomes 2 (which is not allowed).
With n bits, you can represent numbers from 0 to 2ⁿ - 1.

Bits, Nibbles, and Bytes

A bit is a single binary digit (0 or 1).
A nibble is a group of 4 bits (can represent 16 values, 0-15).
A byte is a group of 8 bits (can represent 256 values, 0-255).
Bytes are the standard unit for storing data in computers.

One binary digit is a bit, four bits are a nibble, and eight bits are a byte.

Converting Binary to Decimal

To convert binary to decimal, multiply each bit by its place value (power of 2) and sum the results.
Example: 1011₂ = 1×8 + 0×4 + 1×2 + 1×1 = 11₁₀.
Start from the rightmost bit (2⁰) and move left, doubling the place value each time.
Practice with small numbers: 1101₂ = 13₁₀, 10000₂ = 16₁₀.

Denary, binary and hexadecimal are three ways to count: the same idea in a different base.

Historical Context

The modern binary system was studied by Gottfried Leibniz in the 17th century.
Earlier cultures used binary-like systems: ancient Egypt (Horus-Eye fractions), China (I Ching), India (Pingala's prosody).
Ancient Egyptian multiplication used a method related to binary numbers.
The I Ching uses trigrams and hexagrams that resemble binary numerals.

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1.A binary number system uses only the digits 0 and 1.
Easy
True or false?
2.In the binary system, each digit is called a 'byte'.
Easy
True or false?
3.Why do computers use the binary system?
Easy
- ABecause it is easier for humans to read
- BBecause it is simple to implement in digital electronic circuitry
- CBecause it uses fewer digits than decimal
- DBecause it was invented by Charles Babbage
4.In binary, the place values are powers of 2 (1, 2, 4, 8, ...).
Easy
True or false?
5.Which of the following is the correct binary representation of the decimal number 5?
Medium
- A101
- B110
- C100
- D011
6.In binary, the number 1000 is equivalent to the decimal number 8.
Medium
True or false?
7.Arrange the following binary numbers in increasing order of their decimal values.
Easy
- 101
- 11
- 1000
- 110
8.Match each binary number to its decimal equivalent.
Medium
- 1010
- 1100
- 1001
- 9
- 10
- 12

Soal lainnya

9.Which of the following are valid binary numbers? (Select all that apply)
Medium
- A101
- B102
- C1101
- D2001
- E0110
10.The binary number 1010 is equal to the decimal number 10.
Easy
True or false?
11.What is the decimal value of the binary number 10101?
Medium
- A21
- B17
- C13
- D25
12.Which of the following binary numbers are equivalent to decimal numbers greater than 10? (Select all that apply)
Medium
- A1010
- B1011
- C1100
- D1001
- E1110
13.Arrange the following decimal numbers in increasing order of their 4-bit binary representations when read as binary strings.
Medium
- 7
- 3
- 12
- 5
14.In the ancient Egyptian Horus-Eye fraction system, a hekat was expressed as a sum of binary fractions. Which of the following fractions is NOT part of the Horus-Eye system?
Easy
- A1/2
- B1/4
- C1/8
- D1/3
15.In the I Ching, eight trigrams are analogous to which binary numeral system?
Medium
- A1-bit
- B2-bit
- C3-bit
- D4-bit
16.In Pingala's ancient Indian binary system for describing meters, the first pattern '0000' corresponds to the value one. If the system uses four short syllables (laghu) and long syllables (guru), how many distinct patterns are possible?
Medium
- A4
- B8
- C16
- D32
17.The binary number system uses only the digits 0 and 1.
Easy
True or false?
18.Which of the following is the binary representation of the decimal number 13?
Medium
- A1101
- B1011
- C1110
- D1100
19.Which of the following binary numbers are less than the decimal number 10? (Select all that apply)
Medium
- A1001
- B1010
- C0110
- D1100
- E0011
20.What is the largest decimal number that can be represented with a single byte?
Medium
- A127
- B255
- C256
- D128
21.The binary number 1010 is equivalent to the decimal number 10.
Easy
True or false?
22.Which of the following binary numbers are equivalent to decimal numbers greater than 20? (Select all that apply)
Medium
- A10010
- B11000
- C10111
- D11111
- E10001
23.A computer stores a number using 8 bits. If the number is 11111111, what is its decimal value?
Hard
- A255
- B256
- C128
- D511
24.A binary digit is also called a 'bit'.
Easy
True or false?
25.The binary number system uses the digits 0, 1, and 2.
Easy
True or false?
26.What is the name for a group of 4 bits?
Easy
- Anibble
- Bbyte
- Cword
- Dbit
27.In binary, the number 10 (one-zero) represents the decimal number ten.
Easy
True or false?
28.Which of the following is the binary representation of the decimal number 2?
Easy
- A10
- B11
- C01
- D100
29.The binary number 1011 is greater than the decimal number 10.
Medium
True or false?
30.Which of the following binary numbers is equivalent to the decimal number 7?
Medium
- A111
- B110
- C101
- D1000
31.Put the following binary numbers in order of increasing decimal value (smallest first).
Medium
- 101
- 11
- 1000
- 110
32.Which of the following binary numbers are equivalent to decimal numbers between 5 and 10 inclusive? (Select all that apply)
Medium
- A101
- B1100
- C111
- D1001
- E1010
33.The binary number 1010 represents the decimal number ten.
Easy
True or false?
34.What is the largest decimal number that can be represented with 4 bits?
Medium
- A8
- B15
- C16
- D31
35.How many different values can be represented with a single byte?
Medium
- A8
- B16
- C128
- D256
36.In ancient Egyptian multiplication, the Rhind Mathematical Papyrus uses a method related to binary numbers. To multiply 13 by 11 using this method, which of the following steps is correct?
Hard
- ADouble 13 to get 26, then add 13 to get 39, then double 39 to get 78, then add 13 to get 91
- BDouble 13 to get 26, double again to get 52, then add 13 to get 65, then add 13 again to get 78
- CDouble 13 to get 26, add 13 to get 39, double to get 78, then add 13 to get 91
- DDouble 13 to get 26, double to get 52, then add 13 to get 65, then double to get 130
37.In Pingala's ancient Indian binary system, the pattern '0000' represents the number zero.
Medium
True or false?
38.Which of the following statements about binary number systems in different cultures are TRUE? (Select all that apply)
Medium
- AThe I Ching uses 64 hexagrams, analogous to 6-bit binary numbers.
- BThe Ifá system has 256 signs, which equals 2⁸.
- CPingala's system starts counting from zero.
- DHorus-Eye fractions include the fraction 1/64.
- EThe Mangarevan system was purely binary.
39.Match each cultural binary system to its number of distinct symbols or signs.
Medium
- I Ching hexagrams
- Ifá signs
- Horus-Eye fractions
- 64
- 256
- 6
40.Arrange the following binary numbers from smallest to largest by their decimal values: 1001, 1100, 1011, 1110.
Medium
- 1001
- 1011
- 1100
- 1110
41.What is the name for a single binary digit?
Easy
42.Complete the sentence.
Easy
A group of 8 bits is called a ____.
43.Convert the binary number 1010 to decimal.
Medium
44.What is the decimal value of the binary number 111?
Easy
45.Complete the sequence.
Easy
Counting in binary: 0, 1, ____, ____, 100, ...
46.How many different values can be represented with 4 bits?
Medium
47.What is the decimal value of the binary number 1101?
Easy
48.How many different values can be represented with 3 bits?
Medium
49.Fill in the blank: In binary, the place values are powers of ___. (Write the number)
Easy
In binary, the place values are powers of ____.
50.Fill in the blanks with the correct binary digits to represent the decimal number 5 in 4-bit binary.
Easy
The 4-bit binary representation of 5 is ____ ____ ____ ____.
51.Fill in the blank: In the I Ching, the set of 64 hexagrams is analogous to ___ -bit binary numerals.
Easy
In the I Ching, the set of 64 hexagrams is analogous to ____ -bit binary numerals.
52.The Ifá divination system has 256 possible signs (Odú). How many bits are needed to represent all these signs?
Medium
53.Explain why computers use binary rather than decimal.
Medium
54.Explain how to convert the binary number 1101 to decimal using place values.
Hard
55.Convert the binary number 1010 to decimal.
Medium
- A10
- B11
- C9
- D12
56.How many different values can be represented with 4 bits?
Medium
- A16
- B17
- C18
- D15
57.How many different values can be represented with 3 bits?
Medium
- A8
- B7
- C9
- D10
58.The Ifá divination system has 256 possible signs (Odú). How many bits are needed to represent all these signs?
Medium
- A10 bits
- B9 bits
- C8 bits
- D7 bits

Catatan ulasan

## Binary Numbers: Counting in Base Two ### Why Binary? - Computers use binary because they are built from **two-state devices** (switches that are either ON or OFF). - Binary uses only two symbols: **0** and **1**. - It is simple and has good noise immunity in electronic circuits. ### Place Value in Base Two - Binary is a **positional notation** with a **radix (base) of 2**. - Each digit is called a **bit** (binary digit). - The place values are powers of 2: ... 8, 4, 2, 1 (from right to left). - For example, the binary number 1011 means: - 1×8 + 0×4 + 1×2 + 1×1 = 8 + 0 + 2 + 1 = 11 in decimal. ### Counting in Binary - Start at 0, then 1, then 10 (which is 2 in decimal), then 11 (3), then 100 (4), etc. - Each time you reach 1 in a place, you reset it to 0 and carry 1 to the next place to the left. ### Bits, Nibbles, and Bytes - **Bit**: a single binary digit (0 or 1). - **Nibble**: 4 bits (can represent 16 different values, 0–15). - **Byte**: 8 bits (can represent 256 different values, 0–255). ### Converting Binary to Decimal - Write the binary number, then multiply each bit by its place value (powers of 2) and sum the results. - Example: Convert 1101 to decimal: - 1×8 + 1×4 + 0×2 + 1×1 = 8 + 4 + 0 + 1 = 13. ### Quick Reference: Powers of 2 - 2^0 = 1 - 2^1 = 2 - 2^2 = 4 - 2^3 = 8 - 2^4 = 16 - 2^5 = 32 - 2^6 = 64 - 2^7 = 128 ### Practice - Convert these binary numbers to decimal: 101, 1111, 10000. - Answers: 5, 15, 16. Remember: binary is just another way to write numbers, using only 0s and 1s, with place values that double each time.

Binary number: A number expressed in the base-2 numeral system, using only the symbols 0 and 1.
Bit: A single binary digit (0 or 1).
Nibble: A group of 4 bits.
Byte: A group of 8 bits.
Place value: The value of a digit based on its position in a number; in binary, each place value is a power of 2.
Radix: The base of a number system; for binary, the radix is 2.
Positional notation: A system of writing numbers where the position of a digit determines its value.

Kartu flash

Why do computers use binary?

Computers use binary because digital electronic circuitry (logic gates) has two states, making binary straightforward to implement and providing noise immunity.

What is a binary number?

A number expressed in the base-2 numeral system, using only the symbols 0 and 1.

What is the radix of the binary system?

The radix (base) is 2.

What is a bit?

A bit is a binary digit, either 0 or 1.

What is a nibble?

A nibble is a group of 4 bits.

What is a byte?

A byte is a group of 8 bits.

How many different values can a nibble represent?

A nibble (4 bits) can represent 16 different values (2⁴ = 16).

How many different values can a byte represent?

A byte (8 bits) can represent 256 different values (2⁸ = 256).

What is the place value of the rightmost bit in a binary number?

The rightmost bit has a place value of 1 (2⁰).

What are the place values for the first four bits from right to left?

The place values are 1, 2, 4, and 8 (2⁰, 2¹, 2², 2³).

Convert binary 1010 to decimal.

1010 in binary = 1×8 + 0×4 + 1×2 + 0×1 = 10 in decimal.

Convert binary 1111 to decimal.

1111 in binary = 8+4+2+1 = 15 in decimal.

Convert decimal 5 to binary.

5 in binary is 101 (4+1).

Convert decimal 12 to binary.

12 in binary is 1100 (8+4).

What is the binary representation of zero?

Zero is represented as 0 in binary.

Count from 0 to 5 in binary.

0, 1, 10, 11, 100, 101.

What is the largest decimal number you can represent with 4 bits?

With 4 bits, the largest decimal number is 15 (1111).

What is the largest decimal number you can represent with 8 bits?

With 8 bits, the largest decimal number is 255 (11111111).

What does 'base two' mean?

Base two means each digit's place value is a power of 2, and only two digits (0 and 1) are used.

What is positional notation?

Positional notation is a system where the value of a digit depends on its position, each position having a place value (e.g., powers of 2 in binary).

Who is credited with the invention of the modern binary number system?

Gottfried Leibniz is credited with the invention, though Thomas Harriot also studied it earlier.

Give an example of an early culture that used binary-like systems.

Ancient Egyptians used Horus-Eye fractions (binary fractions) and binary multiplication; the I Ching in China used binary-like trigrams and hexagrams.

What is the binary representation of decimal 2?

Decimal 2 is 10 in binary.

What is the binary representation of decimal 7?

Decimal 7 is 111 in binary.

What is the binary representation of decimal 16?

Decimal 16 is 10000 in binary (1×16).

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