Base-2

Base-2 is a number system based on two numbers, 0 and 1. It’s also called the Binary number system if you’re studying computer science.

Binary is the basic fundamental if you start learning about computers. How is a computer identical to this binary number? Computers are man-made compute machines, and machines only recognize two conditions: On and Off.

These binary numbers represent those two states.

Why is it called Base-2 #

Because it is based on 2 numbers, so is Octal which is also called Base-8, Hexadecimal is also called Base-16. even the Decimal number that you use everyday is also called Base-10.

Can we create our own number system? *The answer is yes, we can.

Conversion to Binary (Base-2) #

There’s no need to explain Decimal because it’s the system you use every day. Conversion to Binary involves the modulo operation, which is finding the remainder of the quotient to determine whether each unit gets \(0\) or \(1\).

For example, we will find the binary number of 1:

$$ 1_{10} = 1_2 $$

Since 1 is a binary number

$$ 3_{10} = n_2 $$

We only have 2 numbers, 0 and 1, so how do we represent 3 in binary?

The formula goes like this:

$$ 3\div 2 = 1 $$

If divided, the result is \(1\), this is called Quotient. Save the quotient for later use.

We don’t need the quotient to get the binary, we need the Remainder. For that we use modulo.

$$ 3 \space{mod}\space 2 = 1 $$

We get the first binary: \(1\). But you have to repeat this until the last quotient is zero.

How do you do that? The previous quotient must be divided by \(2\) continuously until it runs out.

$$ 3 \space{mod}\space 2 = 1 \qquad{quot.} = 1\\ 1 \space{mod}\space 2 = 1 \qquad{quot.} = 0 $$

The binary of \(3_{10}\) is \(11_2\)

Don’t forget to record the remainder based on the smallest to largest quotient (from bottom to top).

More Examples #

$$ 15_{10} = n_2 $$

$$ 15 \space{mod}\space 2 = 1 \qquad{quot.} = 7\\ 7 \space{mod}\space 2 = 1 \qquad{quot.} = 3\\ 3 \space{mod}\space 2 = 1 \qquad{quot.} = 1\\ 1 \space{mod}\space 2 = 1 \qquad{quot.} = 0\\ $$

$$ 15_{10} = 1111_2 $$

Another one:

$$ 22_{10} = n_2 $$

$$ 22 \space{mod}\space 2 = 0 \qquad{quot.} = 11\\ 11 \space{mod}\space 2 = 1 \qquad{quot.} = 5\\ 5 \space{mod}\space 2 = 1 \qquad{quot.} = 2\\ 2 \space{mod}\space 2 = 0 \qquad{quot.} = 1\\ 1 \space{mod}\space 2 = 1 \qquad{quot.} = 0\\ $$

$$ 22_{10} = 10110_2 $$

Can we create ourselves a number system? #

Yes.