# O.3.7 — Converting between binary and decimal

In order to understand the bit manipulation operators, it is first necessary to understand how integers are represented in binary. We talked a little bit about this in section 2.4 -- Integers, and will expand upon it here.

Consider a normal decimal number, such as 5623. We intuitively understand that these digits mean (5 * 1000) + (6 * 100) + (2 * 10) + (3 * 1). Because there are 10 decimal numbers, the value of each digit increases by a factor of 10.

Binary numbers work the same way, except because there are only 2 binary digits (0 and 1), the value of each digit increases by a factor of 2. Just like commas are often used to make a large decimal number easy to read (e.g. 1,427,435), we often write binary numbers in groups of 4 bits to make them easier to read (e.g. 1101 0101).

As a reminder, in binary, we count from 0 to 15 like this:

Decimal Value Binary Value
0 0
1 1
2 10
3 11
4 100
5 101
6 110
7 111
8 1000
9 1001
10 1010
11 1011
12 1100
13 1101
14 1110
15 1111

Converting binary to decimal

In the following examples, we assume that we’re dealing with unsigned integers.

Consider the 8 bit (1 byte) binary number 0101 1110. Binary 0101 1110 means (0 * 128) + (1 * 64) + (0 * 32) + (1 * 16) + (1 * 8) + (1 * 4) + (1 * 2) + (0 * 1). If we sum up all of these parts, we get the decimal number 64 + 16 + 8 + 4 + 2 = 94.

Here is the same process in table format. We multiply each binary digit by its digit value (determined by its position). Summing up all these values gives us the total.

Converting 0101 1110 to decimal:

 Binary digit 0 1 0 1 1 1 1 0 * Digit value 128 64 32 16 8 4 2 1 = Total (94) 0 64 0 16 8 4 2 0

Let’s convert 1001 0111 to decimal:

 Binary digit 1 0 0 1 0 1 1 1 * Digit value 128 64 32 16 8 4 2 1 = Total (151) 128 0 0 16 0 4 2 1

1001 0111 binary = 151 in decimal.

This can easily be extended to 16 or 32 bit binary numbers simply by adding more columns. Note that it’s easiest to start on the right end, and work your way left, multiplying the digit value by 2 as you go.

Method 1 for converting decimal to binary

Converting from decimal to binary is a little more tricky, but still pretty straightforward. There are two good methods to do this.

The first method involves continually dividing by 2, and writing down the remainders. The binary number is constructed at the end from the remainders, from the bottom up.

Converting 148 from decimal to binary (using r to denote a remainder):

148 / 2 = 74 r0
74 / 2 = 37 r0
37 / 2 = 18 r1
18 / 2 = 9 r0
9 / 2 = 4 r1
4 / 2 = 2 r0
2 / 2 = 1 r0
1 / 2 = 0 r1

Writing all of the remainders from the bottom up: 1001 0100

148 decimal = 1001 0100 binary.

You can verify this answer by converting the binary back to decimal:

(1 * 128) + (0 * 64) + (0 * 32) + (1 * 16) + (0 * 8) + (1 * 4) + (0 * 2) + (0 * 1) = 148

Method 2 for converting decimal to binary

The second method involves working backwards to figure out what each of the bits must be. This method can be easier with small binary numbers.

Consider the decimal number 148 again. What’s the largest power of 2 that’s smaller than 148? 128, so we’ll start there.

Is 148 >= 128? Yes, so the 128 bit must be 1. 148 - 128 = 20, which means we need to find bits worth 20 more.
Is 20 >= 64? No, so the 64 bit must be 0.
Is 20 >= 32? No, so the 32 bit must be 0.
Is 20 >= 16? Yes, so the 16 bit must be 1. 20 - 16 = 4, which means we need to find bits worth 4 more.

Is 4 >= 8? No, so the 8 bit must be 0.
Is 4 >= 4? Yes, so the 4 bit must be 1. 4 - 4 = 0, which means all the rest of the bits must be 0.

148 = (1 * 128) + (0 * 64) + (0 * 32) + (1 * 16) + (0 * 8) + (1 * 4) + (0 * 2) + (0 * 1) = 1001 0100

In table format:

 Binary number 1 0 0 1 0 1 0 0 * Digit value 128 64 32 16 8 4 2 1 = Total (148) 128 0 0 16 0 4 0 0

Another example

Let’s convert 117 to binary using method 1:

117 / 2 = 58 r1
58 / 2 = 29 r0
29 / 2 = 14 r1
14 / 2 = 7 r0
7 / 2 = 3 r1
3 / 2 = 1 r1
1 / 2 = 0 r1

Constructing the number from the remainders from the bottom up, 117 = 111 0101 binary

And using method 2:

The largest power of 2 less than 117 is 64.

Is 117 >= 64? Yes, so the 64 bit must be 1. 117 - 64 = 53.
Is 53 >= 32? Yes, so the 32 bit must be 1. 53 - 32 = 21.
Is 21 >= 16? Yes, so the 16 bit must be 1. 21 - 16 = 5.

Is 5 >= 8? No, so the 8 bit must be 0.
Is 5 >= 4? Yes, so the 4 bit must be 1. 5 - 4 = 1.
Is 1 >= 2? No, so the 2 bit must be 0.
Is 1 >= 1? Yes, so the 1 bit must be 1.

117 decimal = 111 0101 binary.

In some cases (we’ll see one in just a moment), it’s useful to be able to add two binary numbers. Adding binary numbers is surprisingly easy (maybe even easier than adding decimal numbers), although it may seem odd at first because you’re not used to it.

Consider two small binary numbers:
0110 (6 in decimal) +
0111 (7 in decimal)

Let’s add these. First, line them up, as we have above. Then, starting from the right and working left, we add each column of digits, just like we do in a decimal number. However, because a binary digit can only be a 0 or a 1, there are only 4 possibilities:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 0, carry a 1 over to the next column

Let’s do the first column:

```0110 (6 in decimal) +
0111 (7 in decimal)
----
1
```

0 + 1 = 1. Easy.

Second column:

``` 1
0110 (6 in decimal) +
0111 (7 in decimal)
----
01
```

1 + 1 = 0, with a carried one into the next column

Third column:

```11
0110 (6 in decimal) +
0111 (7 in decimal)
----
101
```

This one is a little trickier. Normally, 1 + 1 = 0, with a carried one into the next column. However, we already have a 1 carried from the previous column, so we need to add 1. Thus, we end up with a 1 in this column, with a 1 carried over to the next column

Last column:

```11
0110 (6 in decimal) +
0111 (7 in decimal)
----
1101
```

0 + 0 = 0, but there’s a carried 1, so we add 1. 1101 = 13 in decimal.

Now, how do we add 1 to any given binary number (such as 1011 0011)? The same as above, only the bottom number is binary 1.

```       1  (carry column)
1011 0011 (original binary number)
0000 0001 (1 in binary)
---------
1011 0100
```

Signed numbers and two’s complement

In the above examples, we’ve dealt solely with unsigned integers. In this section, we’ll take a look at how signed numbers (which can be negative) are dealt with.

Signed integers are typically stored using a method known as two’s complement. In two’s complement, the leftmost (most significant) bit is used as the sign bit. A 0 sign bit means the number is positive, and a 1 sign bit means the number is negative.

Positive signed numbers are stored just like positive unsigned numbers (with the sign bit set to 0).

Negative signed numbers are stored as the inverse of the positive number, plus 1.

Converting integers to binary two’s complement

For example, here’s how we convert -5 to binary two’s complement:

First we figure out the binary representation for 5: 0000 0101
Then we invert all of the bits: 1111 1010
Then we add 1: 1111 1011

Converting -76 to binary:

Positive 76 in binary: 0100 1100
Invert all the bits: 1011 0011

Why do we add 1? Consider the number 0. If a negative value was simply represented as the inverse of the positive number, 0 would have two representations: 0000 0000 (positive zero) and 1111 1111 (negative zero). By adding 1, 1111 1111 intentionally overflows and becomes 0000 0000. This prevents 0 from having two representations, and simplifies some of the internal logic needed to do arithmetic with negative numbers.

Converting binary two’s complement to integers

To convert a two’s complement binary number back into decimal, first look at the sign bit.

If the sign bit is 0, just convert the number as shown for unsigned numbers above.

If the sign bit is 1, then we invert the bits, add 1, then convert to decimal, then make that decimal number negative (because the sign bit was originally negative).

For example, to convert 1001 1110 from two’s complement into a decimal number:
Given: 1001 1110
Invert the bits: 0110 0001
Convert to decimal: (0 * 128) + (1 * 64) + (1 * 32) + (0 * 16) + (0 * 8) + (0 * 4) + (1 * 2) + (0 * 1) = 64 + 32 + 2 = 98
Since the original sign bit was negative, the final value is -98.

If adding in binary is difficult for you, you can convert to decimal first, and then add 1.

Why types matter

Consider the binary value 1011 0100. What value does this represent? You’d probably say 180, and if this were standard unsigned binary number, you’d be right.

However, if this value was stored using two’s complement, it would be -76.

And if the value were encoded some other way, it could be something else entirely.

So how does C++ know whether to print a variable containing binary 1011 0100 as 180 or -76?

Way back in section 2.1 -- Basic addressing and variable declaration, we said, “When you assign a value to a data type, the compiler and CPU takes care of the details of encoding your value into the appropriate sequence of bits for that data type. When you ask for your value back, your number is “reconstituted” from the sequence of bits in memory.”

So the answer is: it uses the type of the variable to convert the underlying binary representation back into the expected form. So if the variable type was an unsigned integer, it would know that 1011 0100 was standard binary, and should be printed as 180. If the variable was a signed integer, it would know that 1011 0100 was encoded using two’s complement (assuming that’s what it was using), and should be printed as -76.

What about converting floating point numbers from/to binary?

How floating point numbers get converted from/to binary is quite a bit more complicated, and not something you’re likely to ever need to know. However, if you’re curious, see this site, which does a good job of explaining the topic in detail.

Quiz

1) Convert 0100 1101 to decimal.
2) Convert 93 to an 8-bit unsigned binary number.
3) Convert -93 to an 8-bit signed binary number (using two’s complement).
4) Convert 1010 0010 to an unsigned decimal number.
5) Convert 1010 0010 to a signed decimal number (assume two’s complement).

6) Write a program that asks the user to input a number between 0 and 255. Print this number as an 8-bit binary number (of the form #### ####). Don’t use any bitwise operators.

Hint: Use method 2. Assume the largest power of 2 is 128.
Hint: Write a function to test whether your input number is greater than some power of 2. If so, print ‘1’ and return your number minus the power of 2.

### 413 comments to O.3.7 — Converting between binary and decimal

• kanishkaditya shukla

hey nascardriver,

A problem arose when I was trying to implement your suggestion of moving 'int y' into main function(in my previous code given below).

What happened:-
if I move 'int y' into main function and send it to 'binconverter()' as argument then I can't change the
value of 'y' using the statement in 'binconverter()' function [ y=y-pow(2,(power-1)); ],
it simply won't work as 'y' is sent as an argument and assigned to  a perimeter in
'binconverter()' the perimeter runs out of scope for main function as the function ends.
and the changed value of 'y' is discarded as the command return to main function.hence,for the next statement
in main function( which needed the changed value ), 'y' uses the same value as perimeter again and again as perimeter
and hence, results in incorrect answer.

please let me know if I am wrong(and whats wrong), also if I am right then please suggest an alternative
method of fixing this.
sorry for annoying you.

• nascardriver

You're right. Print in `binconverter` and `return y`, have a look at the solution if you don't know how to do it. You'll later learn about references, they'll allow functions to make persistent changes to arguments.

• kanishkaditya shukla

Thanks got it now.

Tried it and it worked perfectly.
Thanks

• kanishkaditya shukla

hey alex and nascardriver,I hope you guys are having a good day.

NOTE:-i have used "using namespace std" in the program irrespective of the best practice (sorry for that) in order to keep the project short(however it didn't seem to work to be honest :-) ).

int input(int power);
char binconverter(int power);

INPUT.CPP

#include <iostream>
# include <cmath>
using namespace std;

int input(int power)
{int x;
cout << "please enter the no. to be converted into binary\n";
cin>> x;
if(x>=0)
{return x;}
else{return ((pow(2,power))+x);};
}

BIN CONVERTER.CPP

#include <iostream>
# include <cmath>
using namespace std;

int y{input(8)};

char binconverter(int power)
{

if(y<=pow(2,power)&&y>=pow(2,(power-1)))
{
y=y-pow(2,(power-1));
return '1';
}
else {
return '0';
};
}

MAIN.CPP

#include <iostream>

using namespace std;

int main()
{
int startpower{8};
/*cout <<"please enter how many bits do you want\n";
cin>> startpower;*/
char a{ binconverter(startpower)};
--startpower;
char b{ binconverter(startpower)};
--startpower;
char c{ binconverter(startpower)};
--startpower;
char d{ binconverter(startpower)};
--startpower;
char e{ binconverter(startpower)};
--startpower;
char f{binconverter(startpower)};
--startpower;
char g{ binconverter(startpower)};
--startpower;
char h{ binconverter(startpower)};
cout<<a<<b<<c<<d<<" "<<e<<f<<g<<h;
return 0;
}

hey nascardriver can you help me to figure out how can i take the user input for the number of bit and create as many as variables needed in order to assign them the value and print it.

FOR EXAMPLE:- if I want to convert a decimal number into a 16 bit binary number then in order to do this (in my program) I will have to create 16 variables(a,b,c....)and the program will become long,is there any shorter way of doing this so that the program automatically creates variables when needed.

• nascardriver

- Inconsistent formatting. Use your editor's auto-formatting feature.
- Use `std::pow` instead of `pow`.
- Don't call functions from outside of functions. (`input`).
- The last character your program prints should be a line feed to prevent mushed lines.

You're looking for dynamically allocated arrays. They're covered later.

• kanishkaditya shukla

1. what do you mean by "Don't call functions from outside of functions. (`input`)"?
whats the problem in doing so?,

(As I am working quite hard to become perfect in these topics.)

It would be very helpful if you explain these question using example (where required) instead of using some scientific word(ex:-mushed lines) as I don't know most of those yet.

• nascardriver

"mushed" is informal, it's when you take something and make it small, eg. mashed potatoes. The next line printed in the console will be appended to the last line you printed.

> whats the problem in doing so?
You don't know the order global variables get initialized in. It could happen that `input` tries to access something that doesn't exist yet.

> what should i do to remove this error from above code?
Move `y` into `main` and pass it to `binconverter`.

> The last character your program prints should be a line feed

• kanishkaditya shukla

• timbo
• Matthew E