## Introduction:

You should not worry about using all comparison
operators with floating-point numbers (

, should the expression

**float**,**double**, and**decimal**). The**==**

, **<**,**>**,**<=**, >=, and**!=**

operators work just fine with these
numbers. But, it is important to remember that they are floating-point numbers,
rather than real numbers or rational numbers or any other such thing.
In
pure (real) math, every decimal has an equivalent binary. In floating-point
math, this is just not true! Consider the following example, let:

**double d = 0.1;**

**float f = 0.1;**

**f > d**return**true**or**false**? Let’s analyze the answer to this question during the remaining part of this article.## 0.1 in Binary:

Many
new programmers become aware of binary floating-point after seeing their
programs give odd results:

“Why
does my program print

**0.10000000000000001**when I enter**0.1**?”
“Why
does

**0.3 + 0.6 = 0.89999999999999991**?”
“Why
does

**6 * 0.1**not equal**0.6**?”
Questions
like these are asked every day, on online forums like stackoverflow.com.

The answer is that most decimals have infinite representations in binary.
Take **0.1**for example. It’s one of the simplest decimals you can think of, and yet it looks so complicated in binary:

**Decimal 0.1 In Binary ( To 1369 Places)**

**- Photo by [2]**

The bits go on forever; no matter how many of those
bits you store in a computer, you will

__never__end up with the binary equivalent of decimal**0.1**.**0.1**is one-tenth, or

**1/10**. To show it in binary, divide binary

**1**by binary

**1010**, using binary long division:

**Computing One-Tenth In Binary - Photo by [2]**

**100**re-appear as the working portion of the dividend. Recognizing this, we can abort the division and write the answer in repeating bicimal notation, as

**0.00011**.

When working with
floating-point numbers, it is important to remember that they are
floating-point numbers, rather than real numbers or rational numbers or any
other such thing. You have to take into account their properties and not the
properties everyone wants them to have. Do this and you automatically avoid
most of the commonly-cited "pitfalls" of working with floating-point
numbers.

## Floating Binary Point Types :

**Float**

and **Double**

are floating *binary*point types. In other words, they represent a number like this:

**10001.10010110011**

.**Decimal**

is a floating *decimal*point type. In other words, they represent a number like this:

**12345.65789**

.
Precision is the main difference:

**Float**

is 7 digits (32 bit), **Double**

is 15:16 digits (64 bit), and **Decimal**

is 28:29 significant digits (128 bit).
Decimals have much higher
precision and are usually used within financial applications that require a
high degree of accuracy. Decimals are much slower (up to 20X times in some
tests [4]) than a double/float. Decimals versus Floats/Doubles cannot be
compared without a cast whereas

__Floats versus Doubles can__.## Question Answer :

As

**0.1**

cannot be perfectly represented in
binary, while **double**

has `15`

to `16`

decimal digits of precision, and **float**

has only `7`

. So, they both are less than **0.1**

.
I'd say the answer depends on the
rounding mode when converting the

In binary,

**double**

to **float**

. **float**

has 24 *binary*bits of precision, and**double**

has 53.In binary,

**0.1**

is:**0.1₁₀ = 0.0001100110011001100110011001100110011001100110011…₂**

^ ^ ^ ^

1 10 20 24

^ ^ ^ ^

1 10 20 24

So if we

__round__at the 24th digit, we'll get:*up***0.1₁₀ ~ 0.000110011001100110011001101**

^ ^ ^ ^

1 10 20 24

^ ^ ^ ^

1 10 20 24

, which is greater than both of the
exact value and the more precise approximation at 53 digits.

So,

__yes 0.1 float is greater than 0.1 double__. This expression returns true!## Examples :

It’s important to note that some
decimals with terminating bicimals don’t exist in floating-point either. This
happens when there are more bits than the precision allows for. For example,

**0.500000000000000**

**166533453693773481063544750213623046875**

converts to :

**0.1000000000000000000000000000000000000000000000000000**

**11**

, but that’s 54 bits. Rounded to 53
bits it becomes :

**0.100000000000000000000000000000000000000000000000000**

**1**

, which in decimal is :

**0.500000000000000**

**2220446049250313080847263336181640625**

**.**

Such precisely specified numbers
are not likely to be used in real programs, so this is not an issue that’s
likely to come up.

Interesting fact:

**1/3**is a repeating decimal =**0.333333333333333333333**……....
But in Ternary (The base-3 numeral
system) it’s only

**0.1**!**References:**

1-
Hesham
Eraqi, http://stackoverflow.com/questions/19292283

## 2 comments:

So, 0.1 Float > 0.1 Double. Good to Know !

Thanks for sharing such a helpful, and understandable blog. I really enjoyed reading it.

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