What are binary numbers and what they mean?
The binary number system is a positional number system with base . It uses only two digits:
01For example, the binary number 101_2 is written in base .
To avoid confusion, we usually write a small index showing the base of the number. For example:
A binary number is usually written in the form:
Here:
The value of this number is:
So each digit tells us whether the corresponding power of two is present in the number.
Let us write in binary.
We decompose it into powers of two:
So the binary representation is:
This shows an important idea: the binary representation of a number is just a sum of powers of two.
There is a simple algorithm for converting a decimal number into binary:
Convert to binary:
Now read the remainders from bottom to top:
So:
To convert from the binary system to the decimal system, we can use the formula . For example:
11 / 2 = 5 (remainder 1)
5 / 2 = 2 (remainder 1)
2 / 2 = 1 (remainder 0)
1 / 2 = 0 (remainder 1)
1011
#include <bits/stdc++.h>
using namespace std;
int main() {
int x = 11;
string s = "";
while (x > 0) {
if (x % 2 == 0) {
s += '0';
} else {
s += '1';
}
x /= 2;
}
reverse(s.begin(), s.end());
}
1011 => 8+2+1 => 11
#include <bits/stdc++.h>
using namespace std;
int main() {
string s = "1011";
int x = 0;
for (int i = s.size() - 1, p = 1; i >= 0; i--) {
if (s[i] == '1') {
x += p;
}
p *= 2;
}
}