How many bytes are needed to store a number? The answer depends on the size of the number and the data type used. In computing, numbers are typically stored in fixed sizes like 8-bit, 16-bit, 32-bit, 64-bit etc. The number of bits determines the range of values that can be represented. More bits allow a wider range of numbers. Let’s take a quick look at some common data types and how many bytes they require:

– Integers (whole numbers) are commonly stored in 1, 2, 4 or 8 bytes. A 1-byte integer can store values from 0 to 255. A 2-byte integer can store 0 to 65,535. And so on.

– Floating point numbers (decimals) commonly use 4 or 8 bytes. This allows storing fractional values.

– Complex numbers (a + bi) require 2 floating point values, so 8 or 16 bytes total.

– Characters typically require 1 byte (8 bits). This covers ASCII characters. Unicode characters may require 2 or more bytes.

So in summary, the number of bytes needed depends on:

- The data type – integer, float, complex, character etc.
- The size of the data type – 8-bit, 16-bit, 32-bit etc.
- Whether it’s a simple or compound data type.

Small integers may use only 1 byte. Typical modern systems use 8 byte integers and floats for efficiency. Characters and booleans use 1 byte. And large numbers may use 16 bytes or more.

## Number of Bytes Needed for Different Data Types

Let’s look at some common data types and the typical number of bytes needed:

### Integers

Data type | Typical bytes | Range |
---|---|---|

int8_t | 1 | 0 to 255 |

int16_t | 2 | 0 to 65,535 |

int32_t | 4 | -2 billion to +2 billion |

int64_t | 8 | Large range of 19 digits |

As you can see, the number of bytes doubles as the data size doubles, which exponentially increases the range of values available.

### Floating Point Numbers

Data type | Typical bytes | Range |
---|---|---|

float | 4 | 6-7 decimal digits |

double | 8 | 15-16 decimal digits |

Floating point representation allows storing fractional values by encoding the mantissa and exponent. More bytes allow greater precision.

### Characters and Booleans

Data type | Typical bytes |
---|---|

char | 1 |

bool | 1 |

Characters and booleans (true/false) commonly use a single byte. Characters are encoded using ASCII, Unicode etc.

### Complex Numbers

Complex numbers have a real and imaginary part. They require 2 floating point values:

Data type | Typical bytes |
---|---|

complex float | 8 (4 bytes each part) |

complex double | 16 (8 bytes each part) |

So complex numbers require double the space of their float/double components.

## Number of Bytes Needed Based on Size

Instead of specific data types, we can look at the number of bytes needed based on the maximum size of the number:

Maximum number size | Typical bytes |
---|---|

0 to 255 | 1 |

0 to 65,535 | 2 |

Hundreds to tens of thousands | 2 |

Millions | 3-4 |

Billions | 4-8 |

Trillions | 8+ |

For very small integers, only 1 byte may be needed. For most practical uses, 4 or 8 bytes can store integers efficiently. Larger numbers will require more bytes.

## Factors That Determine Bytes Needed

The key factors that determine how many bytes are needed to store a number are:

- The data type – integer, float, complex etc.
- The size of the data type – 8-bit, 16-bit, 32-bit etc.
- Whether compression is used. Some formats compress better than others.
- Required precision. More digits require more bytes.
- Required value range. Small ranges need less bytes.
- Integer or floating point representation. Integers need less space for a given value.
- Simple or compound data type. Complex numbers require space for real and imaginary parts.

By carefully selecting the appropriate data type and size, numbers can be efficiently stored using the minimum required bytes. Compression can help reduce storage but decreases precision.

## Number of Bytes Used by Programming Languages

The number of bytes to store a number also depends on the programming language and platform used. Here are some typical defaults:

### C/C++

- char – 1 byte
- short – 2 bytes
- int – 4 bytes
- long – 8 bytes
- long long – 8 bytes
- float – 4 bytes
- double – 8 bytes

C/C++ has built-in support for 8-bit, 16-bit, 32-bit and 64-bit data types.

### Java

- byte – 1 byte
- short – 2 bytes
- int – 4 bytes
- long – 8 bytes
- float – 4 bytes
- double – 8 bytes

Java also uses standard data sizes similar to C/C++. Minimum 1 byte to 8 byte primitives.

### Python

Python integers have unlimited precision by default:

- Small integers use 1-2 bytes
- Larger integers use more bytes as needed
- float – 8 bytes
- complex – 16 bytes

Python can optionally use fixed sized primitives for performance.

### JavaScript

- Number – 8 bytes (double precision float)
- BigInt – Arbitrary precision integers

JavaScript only has Number and BigInt as numeric types.

So in summary, the most common sizes used today are 1, 2, 4, 8 bytes depending on data type, precision and programming language.

## Converting Between Numerical Bases

The number of digits and bytes required also depends on the numerical base:

- Binary – Only 0-1. Most compact but not human readable.
- Decimal – 0-9 digits. Used for human readable numbers.
- Hexadecimal – 0-F digits. Compact representation of binary.

As a rule of thumb:

- 1 hex digit = 4 binary digits = log
_{16}(2^{4}) = 2 nibbles - 1 byte = 8 bits = 2 hex digits
- To convert between bases, divide by base successively

So hexadecimal provides a compact way to visualize bytes and nibbles compared to binary.

Here’s an example conversion of the decimal number 152 to binary and hex:

Division | Quotient | Remainder |
---|---|---|

152/2 | 76 | 0 |

76/2 | 38 | 0 |

38/2 | 19 | 0 |

19/2 | 9 | 1 |

9/2 | 4 | 1 |

4/2 | 2 | 0 |

2/2 | 1 | 0 |

1/2 | 0 | 1 |

Reading remainders from bottom => Binary: 10011000

Hex: 0x98

This shows how decimal 152 is represented as binary 10011000 or hexadecimal 0x98.

## Storing Integers Efficiently

There are some techniques we can use to store integers efficiently using minimum bytes:

### Pick the Smallest Data Type

Use the smallest data type needed rather than defaulting to 4 bytes or 8 bytes. For example, if the range is 0 to 127, use an int8 rather than int32.

### Use Variable Length Encoding

Rather than fixed size, store a preamble indicating the integer size. This allows optimizing based on actual value.

### Split into Bytes/Nibbles

We can split an integer into lower order bytes or nibbles (4 bits). This allows compact storage for small values.

### Bit Packing

Store several small integer values together by packing their bits. Use bit masks to extract each value.

### Differential Encoding

Instead of absolute values, store increments from previous value. Good for sequences of small deltas.

### Compression Algorithms

Use general purpose compression like LZW, Huffman, delta encoding etc. applied to integer sequences.

### Sparse/Dense Representation

Use sparse matrix for mostly small values with few large outliers. Or dense for large continuous ranges.

By analyzing our data and range requirements, we can pick optimized representations. The simplest methods work for many cases.

## Storing Floating Point Numbers Efficiently

Some techniques for optimizing floating point storage:

### Reduce Precision Bits

Use the smallest float or double precision needed rather than default doubles.

### Quantize to Fixed Point

Multiply by a power of 2 to convert range to integer. Then store as integer.

### Split Exponent and Mantissa

Separate components for more efficient compression. Exponent needs fewer bits.

### Differential Coding

Store deltas between values instead of absolute values. Good for smooth curves.

### Avoid Denormals

Flush denormal numbers tiny magnitudes to zero. Range reduction.

### Variable Length

Only use bytes needed for a given value’s precision.

### Compression Algorithms

Use LZW, delta encoding, Huffman etc. tailored to typical FP patterns.

Analyzing the data characteristics helps pick optimal representations beyond just float/double.

## Storing Complex Numbers Efficiently

Some techniques for optimizing complex number storage:

### Real and Imaginary Parts Separately

Store real and imaginary arrays separately using above float/double techniques.

### Magnitude and Phase Components

Use polar form – magnitude and phase angle. Phase often smaller range.

### Differential Coding

Store real and imaginary deltas rather than absolute values.

### Variable Precision

Use fewer bits for imaginary if not same range as real.

### Requantize to Fixed Point

Convert real and imaginary parts to integers by multiplying.

### Compression Algorithms

Exploit correlation between real and imaginary parts.

Representing magnitude/phase or separate real/imag arrays can provide more efficient storage.

## Conclusion

To summarize key points:

- Common integer sizes are 8-bit, 16-bit, 32-bit and 64-bit requiring 1, 2, 4 and 8 bytes respectively.
- Floating point uses 32-bit float (4 bytes) or 64-bit double (8 bytes).
- Characters and booleans use 1 byte.
- Complex numbers require 16 bytes for two doubles generally.
- Size depends on data type, precision and programming language.
- Converting between number bases shows storage needs.
- Many techniques like compression allow reducing storage requirements.

Carefully selecting data representations and storage techniques allows optimizing the number of bytes required for a given numerical application based on the range, precision and performance needs. In most cases, standard data types provide good efficiency but application-specific tuning can help further.