Binary Numbers

We’re used to counting in a base-10 numbering system, called the decimal system. In the decimal system each place is 10 times larger than the next place and there are 10 number symbols (0 through 9).

Consider the number: 231.

\begin{align} 10^2 \end{align}

\begin{align} 10^1 \end{align}

\begin{align} 10^0 \end{align}

100’s place

10’s place

1’s place

2

3

1

Binary is a base-2 counting system, where there are just two number symbols, 0 and 1. Binary is the simplest possible counting system. It works for today’s computers because 0 represents “off” and 1 represnts “on”, the two states of a CMOS gate.

Here’s the number 231 in binary.

\begin{align} 2^7 \end{align}

\begin{align} 2^6 \end{align}

\begin{align} 2^5 \end{align}

\begin{align} 2^4 \end{align}

\begin{align} 2^3 \end{align}

\begin{align} 2^2 \end{align}

\begin{align} 2^1 \end{align}

\begin{align} 2^0 \end{align}

128’s place

64’s place

32’s place

16’s place

8’s place

4’s place

2’s place

1’s place

1

1

1

0

0

1

1

1

As you can see it takes more digits to represent a number in binary than it does in decimal. The word “bit” is short for “binary digit”.

Converting from Decimal to Binary

Converting from decimal to binary is easy once you know the trick. You do it with a two nstep algorithm. Follow these steps.

  1. Is your number even or odd? If it’s odd write a 1. If it’s even write a 0.

  2. Divide your number by 2, ignore any fractional part.

  3. Repeat until you get to zero!

Here’s how to convert 231.

Step

Number

Even or Odd?

Bits so far

1

231

Odd

1

2

115

Odd

11

3

57

Odd

111

4

28

Even

0111

5

14

Even

00111

6

7

Odd

100111

7

3

Odd

1100111

8

1

Odd

11100111

9

0

Stop!

11100111

Converting from Binary to Decimal

Converting from binary to decimal is also easy. Write your binary number and add together all of the place values where your number has a one. Ignore the zeros.

128

64

32

16

8

4

2

1

1

1

1

0

0

1

1

1

128

64

32

4

2

1

128 + 64 + 32 + 4 + 2 + 1 = 231

Exercise

Pick a few numbers and convert them. Use my program to check your work.

[ ]:
number = int(input("Please enter an integer: "))
binary = ''

while number > 0:
    if number % 2 == 0:
        binary = '0' + binary
        print (f'{number} is even. ({binary})')
    else:
        binary = '1' + binary
        print (f'{number} is odd. ({binary})')

    number = int(number / 2)

print ("All done!")
print ("The binary is:", binary)

Hexadecimal

Binary is hard to work with because it takes a lot of bits to write most numbers. Decimal is hard to work with because it’s cumbersome to convert between binary and decimal. So what’s a nerd to do?

The hexadecimal counting system is a base-16 counting system. It has 16 number symbols. Hexadecimal borrows the letters A through F to represent number values.

Symbol

Value

0-9

Same as decimal

a

10

b

11

c

12

d

13

e

14

f

15

The great thing about hexadecimal is that you can convert four bits to a decimal digit easily. Here’s how.

Bits

Hex

0000

0

0001

1

0010

2

0011

3

0100

4

0101

5

0110

6

0111

7

1000

8

1001

9

1010

a

1011

b

1100

c

1101

d

1110

e

1111

f

Hexadecimal numbers are often written with a “0x” at the beginning. That makes it harder to confuse them with decimal numbers. So when you see “10” think ten and when you see “0x10” think sixteen. When you convert binary to hexadecimal split your binary number into groups of four bits.

Here’s how to convert 231:

1110

0111

e

7

So…

230 = 11100111 (binary) = 0xe7