1. Fundamental

A computer is an electronic device, which is composed of many circuits inside it. Each circuit has only 2 states: ON and OFF, or 0 and 1, so it can only store binary values. As a result, computers can only store and process sequences of 0 and 1. So all kinds of data has to be converted into binary number before computers can understand them.

The value 0 and 1 are the smallest units of data. They are called bits. 8 bits combine to a byte and 4 bytes combine to a word.

Now, if you don't know how to convert decimal numbers to binary numbers, take a look at this video and do some exercises before continuing.

There are 3 common ways to represent a block of data, which are binary, octal, and hexadecimal. Let's try to represent number 129 in the 3 different ways.

First we convert number 129 from decimal to binary, which is 10000001.

So 129 is represented as 10000001 in binary.

An octal number can represent number from 0 to 7, which is from 000 to 111. So an octal can represent 3 binary digits. To represent 10000001, which is 8 binary digits, we need 3 octal numbers. 10000001 is now splited into 3 parts 010 000 001. After that, we convert each part to octal number, which is 2 0 1. To distinguish it with decimal number, we prefix it with leading 0. So 129 is represented as 0201 in octal.

Hexadecimal number is similar to octal number, but it can represent number from 0 to 15, which is from 0000 to 1111. So an hexadecimal can represent 4 binary digits. Now we're splitting the number 10000001 into 2 parts of 4 bits, which is 1000 and 0001. Then convert each part to hexadecimal, we have the value of 81. To distinguish it with decimal number, we prefix it with 0x. So 129 is represented 0x81 in hexadecimal.

Next we're gonna see how each type of data: integer, rational, and irrational number are stored in memory.

2. Integer

2.1 Unsigned Integer

Unsigned integers are generally stored as simple binary numbers. For example, 1 is stored as 1, 2 is stored as 10, and 129 is stored as 10000001.

2.2 Signed Integer

There are 3 most common formats to store signed integer numbers.

a. Signed Magnitude

Signed Magnitude format uses the left most bit as the sign indication, and treat the remaining bit as they represent unsigned integer.

The convention for the left most bit (most significant bit): 0 for positive, 1 for negative. For example: 100111 represent -7.

The problem with this system is that it does not support binary arithmetic, which is the most important for computers. For example:

100111 // -7

000111 // 7

100111 + 000111 = 101110 // -14, which is not 0 as we expected.

b. One's Complement

One's Complement of a binary number is just another binary number that when we add it to the original number, the result is a binary number with all bits are 1. To obtain binary number, you can simply flip all the bits.

For example, 000111 is 7, then it's one's complement is 111000, which is 56. You can notice that 56 + 7 = 63 = 2^6-1. So one's component of a number is the result of the max value that can be represented with number of bits minus that number itself. With this system, we also have the first bit as the sign indication.

To see which value a negative binary number represents, we just find it's one's complement. For example: 100011 is a negative number, it's one's complement is 011100, which is 28. So 100011 is -28.

We have two 0 values in this system. 000000 which is +0 and 111111 which is -0.

In this system, the binary arithmetic is partially solved. If we add 12 and -12, we get 0 as expected. 12 is 001100, it's one's complement is 110011, when we add these 2 values, we get 111111, which is -0 as expected.

But there are still some special cases: For example, add 3 to -2.

3 // 000011,

-2 // 111101,

000011 + 111101 = 000000 // which is +0, not 1 as we expected.

In this case, we need to add the leftmost (the carry, which is 000001) to the result. After adding the carry, we got 000001 which is 1 as expected.

Let's try another example, add 10 to -4.

10 // 001010

-4 // 111011

001010 + 111011 = 000101 // 5

000101 + 000001 = 000110 // add the carry

After adding the carry, we got 6 as expected.

c. Two's Complement

Contrary with one's complement, a two's complement of a number is another number that when we add it to the original number, we get a number that all bits are 0. To find the two's complement of a number, we first find its one's complement then add 1 to it.

For example, we have 7 which is 000111, it's one's complement is 111000, then it's two's complement is 111000 + 1 = 111001. When we add a number and it's two's complement, we get a number which all bits are 0. In this example, 000111 + 111001 = 000000.

We still use the first bit as sign indicator. To find which value a negative number represents, we just find it's two's complement.

For example, we have a number 100111, it's one's complement is 011000, it's two's complement is 011001, which is 25. Then the original number is -25.

In this system, we have only one representation of 0, which is 000000. and the value of 111111 is -1. and 100000 is -32. Now with 6 bits, we can represent values from -32 to 31. With signed magnitude and one's complement, we can only have values from -31 to 31. Moreover, the binary arithmetic is all solved in two's complement.

2.3 Conclusion

Integer is usually store in 16 bits, 32 bits or 64 bits, depending on how big the number is. They are often called short, int, and long type respectively.

Unsigned integers are stored as simple binary numbers.

Signed integers are often stored as 2's complement so that it can store maximum number of values and leverage arithmetic calculation.

3. Rational number

3.1 What is floating point number?

There are 3 common ways to represent a rational number.

• Decimal representation, such as: 12.34, -0.6, 123.7777
• Fractional representation, such as: 1/2, 5/6, 10/3
• Floating points number, such as: 1.23*10^5, 33.45*10^6

Floating point number is a format to represent rational number with the point (.) can be moved (floating).

3.2 How floating point number is stored?

a. Scientific notation

First, we have to know how to represent a number in scientific notation. It's simple. We rewrite the number in 2 parts. The first part is just the digits with the point right after the first digit. Then multiply it with a number that put the point where it should be. Then rewrite that number in format base^x, which is the second part.

For example, let's write 12345 in scientific notation.

12345 = 1.2345 x 10.000 // so the first part is 1.2345

// decimal number -> the base is 10

10.000 = 10^4 // so the second part is 10^4

=> 1.2345 * 10^4 is scientific notation

Do the same for 101.101, which is a binary number, we have 1.01101 * 2^2 as its scientific notation.

b. IEEE 754

Nearly all hardwares and programming languages use floating point number in the same binary format, which is specified in IEEE Standard for Floating-Point Arithmetic (IEEE 754). Basically, it normalizes the number to the scientific notation that we already learn above, then use a binary number to represent that normalized number.

There are 2 most used formats that are 32 bit (float) and 64 bit (double). They are divided into 3 parts: sign bit, exponent and significand (or mantissa ).

Let's use the number 101.101 as our example. We normalize it to scientific notation first

101.101 = 1.01101x2^2

a. sign bit: The leftmost, or the most significant bit (MSB) is used to indicate the sign of the number, 0 is positive and 1 is negative. In our case, it's a positive number so the sign bit is 0.

b. the exponent: use to indicate the distance between the binary point of the original number and the binary point of the normalized number.

101.101
-.12---
1.01101

In our example, the normalized number is 1.01101x2^2 so the exponent is 2.

To represent both negative and positive exponent, we use a bias to add to the actual exponent. For example: if we use 6 bit to represent number, then we can represent only unsigned number from 000000 to 111111, which is from 0 to 63. But we want to represent negative numbers too, so we find a bias following the IEEE 754 formula.

bias = 2^(n - 1) - 1 = 2^(6 - 1) - 1 = 31

Then we can represent number from (0-31) to (63-31), which is -31 to 32.

In the above number, the actual value of the exponent that is stored is 2 + 31 = 33, which is 100001. In other words, to store the number 2 as exponent, we don't use 000010 but 100001.

c. The mantissa: because in the normalized form (scientific notation form), the integer part is always 1, so we only have to store the fraction part. That fraction part (the digits after the point) is call the mantissa. In the above number: 1.01101x2^2, the mantissa we need to store is 01101.

So the number 101.101 (which is 5.625 in decimal) will be represent as the following using floating point 32 bit:

0	1000 0001	0110 1000 0000 0000 0000 000
sign	exponent	mantissa

• number is positive → bit sign is 0
• next 8 bit is for exponent: 2^2 , the exponent is 2, then we add it to the bias, which is 127 for 8 bit then we get 129 → exponent is 1000 0001
• last 23 bits is for mantissa, as we talk above, the mantissa is 01101

Floating point numbers using 32 bits are called single-precision or single. Floating point numbers using 64 bits are called double-precision or double. It has 3 parts similar to single but it uses different numbers of bits to store the sign bit, exponent, and mantissa.

	sign	exponent	mantissa
single	1 bit	8 bits	23 bits
double	1 bit	11 bits	52 bits

3.3 Why the exponent use bias, not one's complement or two's complement

If we use 1's complement or 2's complement, it will be hard for comparison of numbers.

For example, we want to compare the number 5.625 above with 5. we will have the representation of 5 in 1's complement and 2's complement and in biased

1's complement:	0	1111 1101	0100 0000 0000 0000 0000 000
2's complement:	0	1111 1110	0100 0000 0000 0000 0000 000
biased:	0	1000 0001	0100 0000 0000 0000 0000 000

When looking at 1's complement and 2's complement exponent numbers and compare it to 5.625 above, we can not determine which number is larger without additional computation. On the other hand, with biased exponent, we can easily compare bit by bit from left to right to determine which number is larger. This is faster for computer to compute.

4. Irrational Number

In general, computers cannot store or manipulate all irrational numbers. Most irrational numbers are not computable, and therefore can't be handled or evaluated at all by a computer. Since we cannot compute such numbers, there is never any need to represent them.

5. How numbers are stored in javascript?

Javascript have only one type for numeric values, which is number. There are no different types for integer and real number in javascript. Under the hood, all numeric numbers in javascript are store as double-precision (64 bits) floating-point number, which is described in the section number 3.2b above.

Although Javascript uses 2's complement format (which is described in section 2.2c) to store integer value in order to do some arithmetic operation such as bitwise operation, you should keep in mind that in general, numbers in javascript are floating point number. Because it will sure make you confused sometimes.

0.1 + 0.2; // 0.30000000000000004

Floating-point arithmetic can only produce approximate results, rounding to the nearest representable real number. When you perform a sequence of calculations, these rounding errors can accumulate, leading to less and less accurate results. Rounding also causes surprising deviations from the kind of properties we usually expect of arithmetic. For example, real numbers are associative, meaning that for any real numbers x, y, and z, it’s always the case that (x + y) + z = x + (y + z). But this is not always true of floating-point numbers:

(0.1 + 0.2) + 0.3; // 0.6000000000000001

0.1 + (0.2 + 0.3); // 0.6

Floating-point numbers offer a trade-off between accuracy and performance. When accuracy matters, it's critical to be aware of their limitations.

One useful workaround is to work with integer values wherever possible, since they can be represented without rounding. When doing calculations with money, programmers often scale numbers up to work with the currency’s smallest denomination so that they can compute with whole numbers. For example, if the above calculation were measured in dollars, we could work with whole numbers of cents instead:

(10 + 20) + 30; // 60
10 + (20 + 30); // 60

6. Summary

In short, in javascript, all numbers are 64 bits floating point numbers. After reading this post, I hope you can:

• Understand how computer stores numbers in general.
• Understand how javascript stores its numeric data.
• Be always aware of limitations of precisions in floating point arithmetic when working with javascript.

7. Sources and related reading

• Item 2: "Understand JavaScript’s Floating-Point Numbers" of Effective JavaScript: 68 Specific Ways to Harness the Power of JavaScript by David Herman.
• Computer Memory
• How computers represent negative binary numbers?