# 2.5 — Floating point numbers

Integers are great for counting whole numbers, but sometimes we need to store very large numbers, or numbers with a fractional component. A floating point type variable is a variable that can hold a real number, such as 4320.0, -3.33, or 0.01226. The floating part of the name floating point refers to the fact that the decimal point can “float”; that is, it can support a variable number of digits before and after the decimal point.

There are three different floating point data types: float, double, and long double. As with integers, C++ does not define the size of these types. On modern architectures, floating point representation almost always follows IEEE 754 binary format. In this format, a float is 4 bytes, a double is 8, and a long double can be equivalent to a double (8 bytes), 80-bits (often padded to 12 bytes), or 16 bytes.

Floating point data types are always signed (can hold positive and negative values).

Category Type Minimum Size Typical Size
floating point float 4 bytes 4 bytes
double 8 bytes 8 bytes
long double 8 bytes 8, 12, or 16 bytes

Here are some definitions of floating point numbers:

When using floating point literals, it is convention to always include at least one decimal place. This helps distinguish floating point values from integer values.

Note that by default, floating point literals default to type double. An f suffix is used to denote a literal of type float. When assigning a literal value to a variable, always make sure the type of the literal matches the type of the variable.

Printing floating point numbers

Now consider this simple program:

The results of this seemingly simple program may surprise you:

```5
6.7
9.87654e+06
```

In the first case, the std::cout printed 5, even though we typed in 5.0. By default, std::cout will not print the fractional part of a number if the fractional part is 0.
In the second case, the number prints as we expect.
In the third case, what the heck? It turns out that by default, when printing some floating point numbers (we’ll cover which number later in the lesson), they are printed in scientific notation.

Scientific notation

How floating point variables store information is beyond the scope of this tutorial, but it is very similar to how numbers are written in scientific notation. Scientific notation is a useful shorthand for writing lengthy numbers in a concise manner. And although scientific notation may seem foreign at first, understanding scientific notation will help you understand how floating point numbers work, and more importantly, what their limitations are.

Numbers in scientific notation take the following form: significand x 10exponent. For example, in the scientific notation `1.2 x 104`, `1.2` is the significand and `4` is the exponent. This number evaluates to 12,000.

By convention, numbers in scientific notation are written with one digit before the decimal, and the rest of the digits afterward.

Consider the mass of the Earth. In decimal notation, we’d write this as `5973600000000000000000000 kg`. That’s a really large number (too big to fit even in an 8 byte integer). It’s also hard to read (is that 19 or 20 zeros?). In scientific notation, this would be written as `5.9736 x 1024 kg`, which is much easier to read. Scientific notation has the added benefit of making it easier to compare the magnitude of two really large or really small numbers simply by comparing the exponent.

Because it can be hard to type or display exponents in C++, we use the letter ‘e’ or ‘E’ to represent the “times 10 to the power of” part of the equation. For example, `1.2 x 104` would be written as `1.2e4`, and `5.9736 x 1024` would be written as `5.9736e24`.

For numbers smaller than 1, the exponent can be negative. The number `5e-2` is equivalent to `5 * 10-2`, which is `5 / 102`, or `0.05`. The mass of an electron is `9.1093822e-31 kg`.

In fact, we can use scientific notation to assign values to floating point variables.

How to convert numbers to scientific notation

Use the following procedure:

• Your exponent starts at zero.
• Slide the decimal so there is only one non-zero digit to the left of the decimal.
• Each place you slide the decimal to the left increases the exponent by 1.
• Each place you slide the decimal to the right decreases the exponent by 1.
• Trim off any leading zeros (on the left end)
• Trim off any trailing zeros (on the right end) only if the original number had no decimal point. We’re assuming they’re not significant unless otherwise specified.

Here’s some examples:

```Start with: 42030
Slide decimal left 4 spaces: 4.2030e4
No leading zeros to trim: 4.2030e4
Trim trailing zeros: 4.203e4 (4 significant digits)
```
```Start with: 0.0078900
Slide decimal right 3 spaces: 0007.8900e-3
Don't trim trailing zeros: 7.8900e-3 (5 significant digits)
```
```Start with: 600.410
Slide decimal left 2 spaces: 6.00410e2
No leading zeros to trim: 6.00410e2
Don't trim trailing zeros: 6.00410e2 (6 significant digits)
```

Here’s the most important thing to understand: The digits in the significand (the part before the E) are called the significant digits. The number of significant digits defines a number’s precision. The more digits in the significand, the more precise a number is.

Precision and trailing zeros after the decimal

Consider the case where we ask two lab assistants each to weigh the same apple. One returns and says the apple weighs 87 grams. The other returns and says the apple weighs 87.000 grams. Assuming the weighings were correct, in the former case, we know the apple actually weighs somewhere between 86.50 and 87.49 grams. Maybe the scale was only precise to the nearest gram. Or maybe our assistant rounded a bit. In the latter case, we are confident about the actual weight of the apple to a much higher degree (it weighs between 86.9950 and 87.0049 grams, which has much less variability).

So in scientific notation, we prefer to keep trailing zeros after a decimal, because those digits impart useful information about the precision of the number.

However, in C++, 87 and 87.000 are treated exactly the same, and the compiler will store the same value for each. There’s no technical reason why we should prefer one over the other (though there might be scientific reasons, if you’re using the source code as documentation).

Precision and range

Consider the fraction 1/3. The decimal representation of this number is 0.33333333333333… with 3’s going out to infinity. An infinite length number would require infinite memory to store, and we typically only have 4 or 8 bytes. Floating point numbers can only store a certain number of significant digits, and the rest are lost. The precision of a floating point number defines how many significant digits it can represent without information loss.

When outputting floating point numbers, std::cout has a default precision of 6 -- that is, it assumes all floating point variables are only significant to 6 digits, and hence it will truncate anything after that.

The following program shows std::cout truncating to 6 digits:

This program outputs:

```9.87654
987.654
987654
9.87654e+006
9.87654e-005
```

Note that each of these is only 6 significant digits.

Also note that cout will switch to outputting numbers in scientific notation in some cases. Depending on the compiler, the exponent will typically be padded to a minimum number of digits. Fear not, 9.87654e+006 is the same as 9.87654e6, just with some padding 0’s. The minimum number of exponent digits displayed is compiler-specific (Visual Studio uses 3, some others use 2 as per the C99 standard).

However, we can override the default precision that cout shows by using the std::setprecision() function that is defined in a header file called iomanip.

Outputs:

```3.333333253860474
3.333333333333334
```

Because we set the precision to 16 digits, each of the above numbers is printed with 16 digits. But, as you can see, the numbers certainly aren’t precise to 16 digits!

The number of digits of precision a floating point variable has depends on both the size (floats have less precision than doubles) and the particular value being stored (some values have more precision than others). Float values have between 6 and 9 digits of precision, with most float values having at least 7 significant digits (which is why everything after that many digits in our answer above is junk). Double values have between 15 and 18 digits of precision, with most double values having at least 16 significant digits. Long double has a minimum precision of 15, 18, or 33 significant digits depending on how many bytes it occupies.

Precision issues don’t just impact fractional numbers, they impact any number with too many significant digits. Let’s consider a big number:

Output:

```123456792
```

123456792 is greater than 123456789. The value 123456789.0 has 10 significant digits, but float values typically have 7 digits of precision. We lost some precision!

Consequently, one has to be careful when using floating point numbers that require more precision than the variables can hold.

Assuming IEEE 754 representation:

Size Range Precision
4 bytes ±1.18 x 10-38 to ±3.4 x 1038 6-9 significant digits, typically 7
8 bytes ±2.23 x 10-308 to ±1.80 x 10308 15-18 significant digits, typically 16
80-bits (12 bytes) ±3.36 x 10-4932 to ±1.18 x 104932 18-21 significant digits
16 bytes ±3.36 x 10-4932 to ±1.18 x 104932 33-36 significant digits

It may seem a little odd that the 12-byte floating point number has the same range as the 16-byte floating point number. This is because they have the same number of bits dedicated to the exponent -- however, the 16-byte number offers a much higher precision.

Rule: Favor double over float unless space is at a premium, as the lack of precision in a float will often lead to inaccuracies.

Rounding errors

One of the reasons floating point numbers can be tricky is due to non-obvious differences between binary (how data is stored) and decimal (how we think) numbers. Consider the fraction 1/10. In decimal, this is easily represented as 0.1, and we are used to thinking of 0.1 as an easily representable number. However, in binary, 0.1 is represented by the infinite sequence: 0.00011001100110011… Because of this, when we assign 0.1 to a floating point number, we’ll run into precision problems.

You can see the effects of this in the following program:

This outputs:

```0.1
0.10000000000000001
```

On the top line, cout prints 0.1, as we expect.

On the bottom line, where we have cout show us 17 digits of precision, we see that d is actually not quite 0.1! This is because the double had to truncate the approximation due to its limited memory, which resulted in a number that is not exactly 0.1. This is called a rounding error.

Rounding errors can have unexpected consequences:

```1
0.99999999999999989
```

Although we might expect that d1 and d2 should be equal, we see that they are not. If we were to compare d1 and d2 in a program, the program would probably not perform as expected. We discuss this more in section 3.5 -- Relational operators (comparisons).

One last note on rounding errors: mathematical operations (such as addition and multiplication) tend to make rounding errors grow. So even though 0.1 has a rounding error in the 17th significant digit, when we add 0.1 ten times, the rounding error has crept into the 16th significant digit.

NaN and Inf

There are two special categories of floating point numbers. The first is Inf, which represents infinity. Inf can be positive or negative. The second is NaN, which stands for “Not a Number”. There are several different kinds of NaN (which we won’t discuss here).

Here’s a program showing all three:

And the results using Visual Studio 2008 on Windows:

```1.#INF
-1.#INF
1.#IND
```

INF stands for infinity, and IND stands for indeterminate. Note that the results of printing Inf and NaN are platform specific, so your results may vary.

Conclusion

To summarize, the two things you should remember about floating point numbers:

1) Floating point numbers are great for storing very large or very small numbers, including those with fractional components, so long as they have a limited number of significant digits (precision).

2) Floating point numbers often have small rounding errors, even when the number has fewer significant digits than the precision. Many times these go unnoticed because they are so small, and because the numbers are truncated for output. Consequently, comparisons of floating point numbers may not give the expected results. Performing mathematical operations on these values will cause the rounding errors to grow larger.

Quiz

1) Convert the following numbers to C++ style scientific notation (using an e to represent the exponent) and determine how many significant digits each has (keep trailing zeros after the decimal):
a) 34.50
b) 0.004000
c) 123.005
d) 146000
e) 146000.001
f) 0.0000000008
g) 34500.0

### 276 comments to 2.5 — Floating point numbers

• Sarah Gunn

I wanted to ask about the division by zero.  While I am sure you understand the mathematics behind why it is undefined, here in your code it indicates that it will return infinity.  Is it the double having a decimal that causes this to happen rather than just giving an error instead?  I ask because I want to understand what the code is doing under the hood so that I never mistakenly fall into this trap.  Thanks and great set of lessons!

• Alex

The short answer is that it works that way because that's how IEEE 754 (the standard to which floating point number implementations adhere) defines it that way. See http://grouper.ieee.org/groups/754/faq.html#exceptions.

• Sarah Gunn

Thanks!  I will bear this in mind when I wish to reprogram my coffee pot!  😉

It does make sense though from that perspective, hardware can be stupid and this has to be adaptable to that.

• waqas

Dear Alex, Is there a way to compare integer value with float type. Like If I want to write a code which takes input number from user and finds out if its a prime/even/odd number? Consider if for prime, division by any number other than 1 and the entered integer, would result in a fraction Now if i had taken input from user in Int and it just got converted to a fraction (which should be stored in a Float variable) how would I knoe if any thing like that has happened? OfCourse one can use as manu If else statements to compare the division or even the inserted number but I want to write smalest code for frist 100 or 200 prime numbers.

• Alex

Hi waqas,

> Now if i had taken input from user in Int and it just got converted to a fraction

I'm having a hard time understanding what you're asking.

Instead of doing a floating point division on an integer to see if the result is a fraction or not, you'd be better off using the modulus operator (%) to see if doing an integer division yields a modulus. You can use a for loop to loop through all of the possible divisors to see if any of them have a modulus of 0. Modulus is covered in chapter 3, and loops are covered in chapter 5.

• Compiler says:
"error: 'setprecision' was not declared in this scope."
but when I delete "cout" from "std::cout << setprecision(x) " (e.g. std::setprecision(x)), it works as expected. Pretty confusing for beginners like me.

• Alex

I updated the lesson to make it more clear that std::setprecision() lives in the std:: namespace.

• Odgarig

f = 9876549.21f in this case would scientific notation be 9.87655e006 because of after 6th specific digit there is 9? Mathematically 0.49 can be equal to 5.So could it be same in c++ ?

• Alex

The precise scientific notation would be 9.87654921e006.

However, because std::cout rounds to 6 significant digits, 9.87654921e006 would be rounded to 9.87655e006.

This is something that is easy to verify yourself with a simple program:

• Jeff

This is driving me crazy! Do floats have 6 or 7 digits of precision?

"When outputting floating point numbers, cout has a default precision of 6 -- that is, it assumes all variables are only significant to 6 digits, and hence it will truncate anything after that."

"The value 123456789.0 has 9 significant digits, but float only supports about 7 digits of precision"

"4 bytes    ±1.18 x 10-38 to ±3.4 x 1038    7 significant digits"

• Alex

The answer to this question is complicated due to the way floating point numbers are stored. Most floats have (at least) 7 digits of precision, however there is a subset of the floating point numbers that only have 6 digits of precision. I'll update the article to indicate 6 instead of 7.

• Big_Jiraffe

Outputs:

0.10000000000000001
0.99999999999999989

In my machine. So, when I use double without setprecission(17), It shows the correct results: 0.1 and 1
So my question is, without using precission(17), do they store the same value? Or double without precission stores 0.10000000000000001 but somehow does a rounding to 0.1?
Greaat tutorials by the way! You are helping a lott!!

• Alex

Cout rounds the numbers for display based on the precision. Internally, they're stored as 0.10000000000000001 and 0.99999999999999989.

• I have to point this out, for the sake of correct math:

First off, infinity is not a number, it's a concept.

Second, x/0 is not and never will be infinity or -infinity.

Numberphile did a good video that explains the division by 0 problem: https://www.youtube.com/watch?v=BRRolKTlF6Q

• Darren

Ah mathemagicians and their concepts. Almost as bad as physicist.

• Jimmy

I understood everything perfectly except one thing: why are we leaving trailing zeroes untrimmed when the original number has a decimal point?

• Alex

Good question. If we write a large number with no decimal point, e.g. 21,000,000,000, it's not clear whether we mean "exactly 21 billion", or "somewhere around 21 billion". More often than not, we mean "Somewhere around 21 billion" (because in most cases, the difference between 21,000,000,000 and 21,000,000,001 is insignificant).

So when we convert 21,000,000,000, we assume that the trailing zeros are not significant.

However, when the number has a decimal point, e.g. 21,000,000,000.01, or even 21,000,000,000.0, it's clear that this number is exact, otherwise we wouldn't have provided the decimal point. So in the case where a decimal is provided, we assume all of the trailing zeros are significant.

Make sense?

Your explanation makes sense, but does the compiler care?  What does it do differently? Or is this just to let the code document what we know about the number?

• Alex

I'm guessing that the compiler would consider 21,000,000,000.0 to be equivalent to 21,000,000,000. So although we would probably say one has a high precision and one doesn't, the compiler likely treats them both as low-precision numbers.

• Tito

In your 0.1 example you forgot to add "#include <iostream>" at the top

• Alex

Fixed.

• Metereo_Rex

So... Just to clarify, the varying results seen in the above examples are caused by varying error(s) in rounding?

Eg, each time you round to the hundreds/hundredths place, it will be different from rounding to the tens/tenths place. And with that, the higher you the place you round to, the more accurate the answer will be (As with standard math). Is this correct?

Also, I'm getting slightly different results than what's presented above. For instance, what's 3.333333333333334 to you is 3.333333333333333 on my machine, despite rounding to the same place. Just thinking about that makes me uncomfortable!

Edit: it's also worth noting that despite the fact that the comment box says we can use certain HTML tags, it automatically encodes the brackets.

• Alex

I'm not sure which examples you're referring to. But generally speaking, the fact that floating point numbers have limited precision leads to rounding errors, especially in cases where a number is represented as an infinite sequence in binary (like 0.1).

re: The HTML tags issue, I used to allow HTML on the site, but I've installed a plugin to treat all text literally so that people who post code snippets won't have lines such as "#include " treated as invalid HTML. I'll see if I can remove/disable the misleading text.

• Hello, I'm not sure if you'll read this but I'm confused about some of the outputs:
9.87654e+006
9.87654e-005

Why is 006 and 005 after the e?
How does that make sense?
I don't get what 9.87654e+006 means... I thought you could only have numbers like 6 or -3 and stuff after the e not numbers like 006.
I'm confused now... Shouldn't it just be 9.87654321e6 ? Why is it 006 ?

PS: I noticed your site is built on WordPress and I think it would be really helpful if you got a plugin that would email commenters when someone replies, would save me a lot of time, lol.

Anyway, thanks a lot, I'm finding this site really helpful 🙂

• Alex

9.87654e+006 is the same as 9.87654e6. Visual Studio always prints exponents with at least 3 digits. Other compilers print exponents with at least 2 (as required by C99). In other words, it's implementation specific.

• Oh right, Ok I understand thanks a lot!

SIR PLZ TEL ME HOW TO PRINT ON OUTPUT SCREEN 14 ND 0.5 IF THE GIVEN VALUE IS 14.5 BASICALLY I WANT TO PRINT IT SEPARATELY ON OT PUT SCREEN BY FLOAT ND INT DATA TYPE

• Alex

Casting your float to an int via static_cast will drop the remainder, allowing you to print the integer. You can then subtract the integer from the floating point value to get just the remainder.

• Kevin

whats the difference between:

cout<<precision(17)

&

cout.setprecision(17)

• Alex

With Visual Studio, cout << setprecision() calls cout.precision() internally. So they essentially do the same thing. Calling cout.precision() is probably slightly more efficient, but unless you're calling it hundreds of times, it won't really matter.

• noobcaxe

I'm perplexed! I'm working through Bjarne S's book and have done one of the drills but I don't understand what's going on with varying results.

Here's some output;

9.99
10
smaller is a 9.99  the larger is b 10
result 0.01
they are almost equal

99.99
100
smaller is a 99.99  the larger is b 100
result 0.01

199.99
200
smaller is a 199.99  the larger is b 200
result 0.01
they are almost equal

Why doesn't entering 99.99 and 100 give the message "they are almost equal"?

• Alex

This is an interesting example of rounding errors.

10 - 9.99 = 0.01, but due to rounding error, C++ is representing this as 0.0099999999999997868
100 - 99.99 = 0.01, but due to rounding error, C++ is representing this as 0.0100000000000005116

One of this is larger than 0.01, and one is smaller.

• Noobcaxe

Argggh! I almost went insane yesterday evening try to see where I'd gone wrong.
How does one avoid rounding errors? I can think of any number of applications where rounding errors even at the level of precision in my example might be disastrous.

• Alex

There are a couple of ways to "avoid" rounding errors:
1) Avoid use of floating point numbers altogether (sometimes this is possible, sometimes it isn't).
2) Don't do raw comparisons like you're doing. In section 3.5 -- Relational operators (comparisons), we discuss how to tell if floating point numbers are equal. These can be extended to handle less then/greater than cases. This would help avoid the case you see above.
3) Ensure that when you use floating point numbers, you only treat them as accurate to a certain level of precision.

• Billy

I don't know why all outputs are same.
Please someone help me.(VS2015,64 bit Windows)
Thank you!!

• Alex

Your float doesn't have enough precision to store the entire number, so it's being truncated. If you change f to a double (and remove the f suffix on the literals) then you'll see that the numbers print differently.

• Billy

Sorry, I still feel confused.
So even i use setprecision, i still cannot change precision.
Is my understanding correct?

• Alex

setprecision can't show more significant digits than the underlying number has. A float only has 6 to 9 digits of precision, so you're generally only going to get between 6 and 9 digits of precision even if you ask for more.

The number you picked (9876543.21f) has 9 significant digits, but because you tried to put it in a float, it got truncated to 7 significant digits (9876543) internally. So regardless of whether you ask for 7, 8, or 9 significant digits of precision, you're only going to get 7 because that's all the underlying number that got stored can offer.

• Billy

Got it.
Thank you Alex!

• sajibfinix

How to control the numbers after decimal point in C++?

Like in C language if i take a floating variable f = 123.4567 and i want to show only 2 numbers after decimal point than i will use printf("%0.2f").Then it will show 123.45.

In C++ i have to use setprecision().But It determines total numbers not just numbers after decimal point.So, it makes problem.Like if i dont know that what numbers my floating variable will contain after calculation it can contain 123.123 or 1234.123 so if i set precision to 5 for first case it will show 123.12 and for second case it will show 1234.1!But i always want to show 2 numbers after decimal points for every case.How can i do that in c++?

• duyquyen1710

i have same question like you

• rameye

Use the `std::fixed` stream manipulator and the member function `std::precision`.

For example if you want to display with 2 decimal places:

``` double pi=3.14159; std::cout.precision(2); std::cout << "Today's price for a slice of pi is \$" << std::fixed << pi << std::endl; ```

and it should print:

`Today's price for a slice of pi is \$3.14`

• mohibouki

Hi people,
I am new to C++ so please don't flame me 🙂

I wrote a simple prog. to test this course but something isn't really working well and I can't figure out why...

``` #include<iostream> #include<string> #include <iomanip> // for setprecision() using namespace std;```

``` main() { cout<<setprecision(7); //7 decimals float v = 1; float j = 3; float cc; cc = v/j; //TEST with FLOAT NUMMERS float ff = 0.3333333; // 7 decimals as set in "setprecusion(7);" if(cc<ff) { cout<<"cc is smaller then ff"<<endl; } else if(cc>ff) { cout<<"cc is bigger then ff"<<endl; } else{ cout<<"cc equals to ff"<<endl; } cout<<cc<<" = cc"<<endl; cout<<ff<<" = ff"<<endl; ```

```return 0; } ```

The output gives me that cc is bigger than ff...
I don't understand why as I set precision to 7 and my var ff has also 7 decimals.
They should both be equal.

Any suggestions where I made an error?
Thanks!!

• sajibfinix

No, You didn't make any wrong.
I think setprecision() function is only for setting the precision at time of showing your variable when you use cout.
I mean setprecision() cant change your variable.Like in your code cc = v/j so cc is stored as 0.33333333333333333333333333.......... & setprecision() cant change this.You stored ff as 0.3333333.
0.3333333333333333........ is greater than 0.3333333 isnt it?So your code show cc is bigger than ff.

Read the "Comparison of floating point numbers" part of this tutorial.It dosent say that you can use setprecision() for Comparison of floating point numbers.
Thank You

• Sergey

This is a very good article on the floating-point computation issue: "Microsoft Visual C++ Floating-Point Optimization", by Eric Fleegal, MSDN, 2004
http://msdn.microsoft.com/en-us/library/aa289157(v=vs.71).aspx

• jimboshack

When I run this code, I get z = 0.333333 and q = 0

float x = 1;
float y = 3;
float z = (x/y);
float q = (1/3);

Can someone explain why? I realize that if I write

float q = (1.0/3.0);

that this problem doesn't occur, but I'm just wondering why I can't use (1/3) since q is defined as a float. This page says it's just a convention to have the decimal point.

• lharriger

Think it through as follows:

float x = 1 reads "put INT 1 into FLOAT x." This changes its type from int to float. The same is true for float y = 3.
Thus float z = x/y divides two floats and returns a float.

However for float q = (1/3), this is a two part statement.
The first part (1/3) reads "divide INT 1 by INT 3". Since this is division of two integers, this means it must return an integer (the floor), which in this case is 0.
The second part is then q = 0, which reads "put INT 0 into float q."

An important thing to keep in mind is that division on a float is different than division on an integer. The literal 1 is read as an integer, however, the literal 1.0 is read as a float/double. This is why q = (1.0/3.0) is different than q = (1/3).

Hope this helped.

• duyquyen1710

thank you!

• Alex

Yup, 1 / 3 performs integer division (which gives an answer of 0, as the fractional component is dropped) , whereas 1.0 / 3.0 performs floating point division (which gives an answer of 0.333333...)

Hi there! Congratulation, very good explanation!!! Just what I was looking for.

Thank you.

• Stacey Brewer

When I run the following code, the values seem really wrong when output. What is going wrong here?

• sanjiv

your C ++ compiler has a tendency to roundoff 8th precision onwards.
For any value lesser then 8;
It will display 1 lesser than called for.

• Noha

Didn't understand how 0.1 is represented in binary by 0.00011001100110011…

• jeremy

In decimal, .1 is tenths, .01 is hundredths, .001 is thousandths and so on. Likewise, in binary, .1 is halves, .01 is quarters, .001 is eights, and so on.
0.000110011... would be equal to 1/16 + 1/32 + 1/256 + 1/512 + ...

• Kostas81

Ok, so in binary, we have to approximate the 1 of decimal 0.1 with an infinite sum. What I still don't understand is why we don't use all the "weights", that is, all the powers of 2, but only 1/16, 1/32, 1/256, 1/512 and so on, that is, the 4th position (2^4 = 16), the 5th, the 8th, the 9th, and so on. In other words, why don't we have 0.011111111........ which is equal to 1/2 + 1/4 + 1/8 + 1/16 + ...? It also approaches 1! (I am referring of course to the decimal part of 0.1, that is, the 1.)

• johnbobz

That's exactly why you can't use every power of 1/2. The infinite sum would add up to 1, which is ten times the number we require. In order for the sum to add up to 0.1, you would need to add Sum[(1/2)^4n + (1/2)^(4n+1)], taking n from 1 to infinity. You can try it yourself if you want.

• Kostas81

Yes, you are right, this sum indeed converges to 0.1, whereas the sum I used converges to 1.0. Thank you for the clear and concise explanation!

• Kostas81

There is also a good explanation in Wikipedia (yes, sometimes - not often though - Wikipedia has good articles):

"Fractions in binary

Fractions in binary only terminate if the denominator has 2 as the only prime factor. As a result, 1/10 does not have a finite binary representation, and this causes 10 × 0.1 not to be precisely equal to 1 in floating point arithmetic. As an example, to interpret the binary expression for 1/3 = .010101..., this means: 1/3 = 0 × 2^(-1) + 1 × 2^(-2) + 0 × 2^(-3) + 1 × 2^(-4) + ... = 0.3125 + ... An exact value cannot be found with a sum of a finite number of inverse powers of two, and zeros and ones alternate forever."

Follows a table of the conversion (fractional approximations) for fractions from decimal to binary. For the ones who are interested:

http://en.wikipedia.org/wiki/Binary_numeral_system

• Vivek Hiremath

How do I convert a Float say

x = 1234.567890123456789

to
y = 1234.5678901234 (small float ..10 decimal places only)

Something similar to setPrecision, to use NOT for display/Printing, but to use as a value for calculations / pass it on to a Database etc ?

• I'm not sure what the best way to do this is. For small numbers, you can multiply by 10^x, cast to an integer to drop the remaining decimals, then divide by 10^x. However, if your number is too large you'll overflow the int when you do the casting so I won't say this is foolproof.

• Chris

I set the precision level to 4, and added cout for the 2 values, fValue1 + fValue2.

I got fValue1 IS actually rounded off to 1.345 and fValue2 IS actually 1.123, expecting now the get the result of 2.468, but still reports 'fTotal is not 2.468'

Why is that?

Chris

• Rounding error. The numbers printed on your screen by cout are rounded in this case, so you're not seeing the full representation. However, when you do the comparison, it does so with the actual numbers, not the rounded ones, which can lead to rounding issues.

• iseneirik

how can i make the value a user inputs into a float?

then when i run the program from main() and i put in 2 values like eg. x = 10 y = 3 then the answer is 3 instead of 3.333333

• You are already storing the user input values as a float. The problem is that your function is returning an integer, so it's truncating the result of x/y. Change your function to return a float and you will be good.

• Sam

Hi Alex, really great guide. Enjoying the challenges its throwing up so far. With regards to functions returning integer's, how do you change a function to actually return a float? My little "add" program is working but it cuts of the decimal? so if I have x = 4.5 + y = 5.5 it will give me the answer of 9! im probably missing something ridiculously obvious but if you could help that would be great!

• Alex

Just change the return type of the function from int to float (or even better, double).

However, if x and y are ints, and you try to assign fractional values to them (e.g. 4.5 or 5.5) the fractions will get lost. So you may need to make those floats/doubles as well.

• this is very help full site

• So with all the rounding errors and precision problems, how do programmers deal with operations that need to display something that would end up with a precision or rounding error? Or am I just over-thinking things?

• Most of the time it's simply not necessary to display a number to the number of significant digits where precision/rounding errors creep in. Generally with floating point numbers, programs will truncate the display to 2-5 decimals.

• Lyle

For applications which need highly precise and extreme ranges, developers rely on libraries developed specifically to handle big and high precision numbers. Phrases describing them include:   arbitrary-precision arithmetic,  bignum arithmetic, multiple precision arithmetic, or sometimes infinite-precision arithmetic.   That's how they get past issues of rounding and precision errors because they are written to maintain accuracy using a variety of techniques.  Beyond this tutorial.

• Coming form a Java background, I wonder if anyone can advise me a C++ library with a similar function as Java's BigDecimal.
Preferably one that works on Linux with gcc(so not the decimal type from Visual C++)

• Masss

Is there something wrong with my code?

The compiler brings up a problem with setprecision()..

Thanks

• You need to include iomanip.h to use setprecision() that way.

• Jefferson

What do the f's after some of the float and double values mean?

• By default, if you type a floating point value into C++ it's typed as a double. Consequently, if you do something like this:

You're assigning a double to a float, which loses precision, and the compiler will probably complain.

Putting an "f" after the value means that you intend that value to be a float, not a double. Then when you do this:

You're assigning a float value to a float variable, which makes more sense.

• jsieckman

I missed something. How is it that 4.53 is a double and not a float?

• rameye

The 4.53 is a literal constant of type `double` by default. When you add the f suffix to it like 4.53f it then becomes a literal constant of type `float`

• cpplx

it might be good if you add this explanation in the lesson instead of just a comment in the code. I was wondering myself about the usage of the f suffix due overlooking the comment.

• Alex

Good idea. Done.

• Jim

Alex,
It seems kind of dumb to inatialize a variable as a float, and then specify that you want a float to be output, by adding the f suffix. {example 9.87654321f }
Except for the fact that all floats are defaulted as double in C++.  I've ran your example with all the numbers with the f suffix and without and get the same output.  My PC has i7 processor.  Why use float at all, since both use four bytes?