Introduction

The JavaScript Number Type Format is not separated into Integer and Float as is with other programming languages. An integer or a float number is of the 64-bit Floating Point International IEEE 754 Standard, in JavsScript. There are some documents concerning the 64-bit Floating Point International IEEE 754 Standard, online (Internet) and off-line. However, when many people (including computer programmers) read those documents, they do not really understand.

There are three objectives for this article. The first is to explain the 64-bit Floating Point International IEEE 754 Standard in a simplified way, so that many people (including any computer programmer) can understand. The second objective is to explain, still in a simplified way, how this standard is applicable to the JavaScript Number Data Type. The third objective is to do some explanation of the JavaScript BigInt data type. After reading this article, the reader will understand the nature and limits of the JavaScript Number Data Type, and will be able to prevent certain number errors. The reader will also be able to handle the BigInt data type.

Actually, JavaScript has eight different data types, which are: String, Number, BigInt, Boolean, Undefined, Null, Symbol, Object. The focus in this article is on the Number Type. The BigInt type will be addressed towards the end of this article.

The other online and offline documents on this topic are either not well explained, or do not go deep enough. This article overcomes all that.

After reading this article, the reader will have confidence in using the JavaScript Number BigInt type.

Floating Point Number Format

A number without a decimal part is an integer. The number 36 is an integer. 36.375 is not an integer. It is a decimal number with a decimal part. The decimal part, .375 is a fraction, which is less than 1. Such a fraction is called a proper fraction.

36.375 is interpreted in decimal form as:

=>

Now,

=>

So, 100100₂ = 36₁₀, which is the whole number part of 36.375₁₀ .

Now,

      +2⁰ x (1 + 2^-52)  approximately  1.0000000000000002

The number of mixed fractions between two consecutive integers on the number line, is infinite. So, no format (e.g. 32-bit or 64-bit) can provide all the mixed fractions between any two consecutive integers (whole numbers). The smaller the gap (interval) between two consecutive integers, provided by a format (e.g. 32-bit or 64-bit), the greater the number of mixed fractions between the consecutive integers (for the number line).

The reasons why the 64-bit format is described as double or higher precision, compared to the 32-bit format, are that the interval between two consecutive mixed fractions, bounded by two consecutive integers, for the 64-bit format, is smaller than the corresponding 32-bit format interval; also, there are more possible mixed fractions between two bounded integers, for the 64-bit format, than there are correspondingly, for the 32-bit format.

Representing the number, 0.0 does not really follow the above arguments, because of the un-indicated "1." . The representation for 0.0 is declared and has to be learned as such. To represent 0.0, all the cells for the significand are 0 and all the cells for the exponent are also zero. The sign bit can only either be 0 or 1. Unfortunately this gives rise to positive 0 and negative 0, as follows:

+ve zero: 0 00000000000 00000000000000000000 00000000000 000000000000000000000₂

-ve zero: 1 00000000000 00000000000000000000 00000000000 000000000000000000000₂

In real life there is only one zero. Positive 0 and negative 0 do not exist. However, 0 is usually considered as positive, in practice. Positive 0 and negative 0 exist here, because of this particular format description. The number line can also have +0 and -0 in theory, but only one zero exists.

The reader has to know the conversion of a number from base 10 to base 2 and vice-versa. That comes next.

Conversion of Integer from Base 10 to Base 2

The conversion method is continuous division of the decimal number (in base 10), by 2; and then reading the result from the bottom, as the following table illustrates, for the decimal number (integer), 36:

Read from the bottom, the answer is, 100100. For any division step, there is the dividend divided by the divisor, to give the quotient. The quotient always has a whole number and a remainder. The remainder may be zero. In the table, the remainder for a quotient, is one row higher than the whole number. When converting to base 2, the last quotient is always, zero remainder 1.

Converting Decimal Part (fraction) of Decimal Number to Binary Part

36.375 is a decimal number with the decimal part, ".375". The decimal part ".375" is a fraction between zero and one. 0.5 in base ten is the same in value, as 1/2 in base two. 0.5₁₀ expressed with the base two expansion is:

It is not 0.101₂, which means 0.625₁₀. The decimal part of a decimal number has its equivalent binary part for the corresponding binary number. So, to convert a decimal number like 36.375₁₀, to base two, convert 36 to binary and then convert .375 also to binary. Then join both results with the binary point. The methods to convert the two sections are different. How to convert a decimal integer to base 2, has been explained above.

To convert the decimal fraction to binary fraction, follow the following steps:

• Multiply the decimal fraction (decimal part) by 2. The integer resulting from this is the first binary digit.

• Repeat the above step with the fractional decimal result, to get the next binary digit.

• Keep repeating the above step until the decimal fractional result is .0000 - - - .

Example: Convert the fractional part of 36.375₁₀, to the equivalent fractional part in base two.

Solution:

Note that in the third step, .500 was multiplied by 2 and not 1.500 (with the whole number). The binary corresponding fraction, is read at the last column, from the top. And so,

.375₁₀ = .011₂

Converting an Integer (whole number Part) from Base 2 to Base 10

How to convert a number from base 2 to base 10 is shown in this section. The computer works basically in base 2.

Since everybody appreciates the value of a number in base 10, this section explains the conversion of a base 2 number, to base 10. To convert a base 2 number to base 10, multiply each digit in the base 2 number, by the base 2, raised to the index of its position, then add the answers.

Each digit for any number in any base, has an index position, beginning from 0 and from the right end of the number, moving leftwards. The following table show the digit index positions of 100100₂

Index - >	5	4	3	2	1	0
Digit ->	1	0	0	1	0	02

Converting 100100₂ to base 10 is as follows:

      1 x 2⁵ + 0 x 2⁴ + 0 x 2³ + 1 x 2² + 0 x 2¹ + 0 x 2⁰

Note: any digit raised to the index 0, becomes 1. Also,

      2³ = 2 x 2 x 2;

      2² = 2 x 2

      2¹ = 2

      2⁰ = 1

Note as well in mathematics, that =>, means "this implies that" and ∴ means, therefore.

In a mathematical expression, all the multiplications must be done first before addition; this is from the sequence, BODMAS (Brackets first, followed by Of, which is still multiplication, followed by Division, followed by Multiplication, followed by Addition, and followed by Subtraction). So,

      1x2⁵+0x2⁴ +0x2³+1x2²+0x2¹+0x2⁰ = 1x2x2x2x2x2 + 0x2x2x2x2 + 0x2x2x2 + 1x2x2 + 0x2 + 0x1

=> 1 x 2⁵ + 0 x 2⁴ + 0 x 2³ + 1 x 2² + 0 x 2¹ + 0 x 2⁰ = 1 x 32 + 0 x 16 + 0 x 8 + 1 x 4 + 0 x 2 + 0 x 1

=> 1 x 2⁵ + 0 x 2⁴ + 0 x 2³ + 1 x 2² + 0 x 2¹ + 0 x 2⁰ = 32 + 0 + 0 + 4 + 0 + 0

=> 1 x 2⁵ + 0 x 2⁴ + 0 x 2³ + 1 x 2² + 0 x 2¹ + 0 x 2⁰ = 32 + 4

=> 1 x 2⁵ + 0 x 2⁴ + 0 x 2³ + 1 x 2² + 0 x 2¹ + 0 x 2⁰ = 36 (as expected)

Converting Binary Part (fraction) of Binary Number to Decimal Part

To achieve this, expand the binary fraction in reciprocal powers of 2.

Example: Convert the fractional part of 100100.011₂, to the equivalent fractional part in base ten.

Solution:

=>

=>

Therefore

Positive and Negative Zeros in Base 2 Standard Form

The 64-bit format for the positive and negative zeros again are: