Binary Representation

Section 11.2 Binary Representation

Before we move on to presenting numbers with arbitrary base

b

where

b

is a natural number greater than

1,

we consider one more special case. One of the most common bases other than base

10

is base

2 .

While base

10

numbers are written with the ten symbols

0,

1,

2,

3,

4,

5,

6,

7,

8,

and

9

numbers in base

2

are written using the two symbols

0 and 1.

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Base

2

representation is particular of interest because digital devices (such as computers) work with the two states on and off, which are represented by

0

and

1 .

To distinguish numbers in base

2

representation from numbers in base

10

representation we add a subscript

2

to the number in base

2

representation. For example, we write

1001_{2} = 9

🔗

where we interpret

1001_{2}

as a number in base

2

representation and

9

as a number in base

10

representation. Numbers in base

2

representation are also called binary numbers.

🔗

Subsection Base $2$ Expansion

🔗

The values of the places of base

10

numbers are the powers of

10 .

As before we write these from right to left because (by convention) the least significant digit of numbers in decimal representation is on the right. For an

n

digit decimal (that is, bas

10

) number we have the place values:

🔗

10^{n - 1},

10^{n - 2},

\dots,

10^{4} = 10 000,

10^{3} = 1000,

10^{2} = 100,

10^{1} = 10,

10^{0} = 1 .

🔗

Similarly the place values of base

2

numbers are the powers of two. For an

n

digit base

2

(or binary) number we have the place values:

🔗

2^{n - 1},

2^{n - 2},

\dots,

2^{4} = 16,

2^{3} = 8,

2^{2} = 4,

2^{1} = 2,

2^{0} = 1 .

🔗

For a binary number

a = (r_{n - 1} \dots r_{2} r_{1} r_{0})_{2}

🔗

where for

r_{i} \in {0, 1}

we have

i \in {0, \dots, n - 1}

the binary (base

2

) expansion

a = r_{n - 1} \cdot 2^{n - 1} + r_{n - 2} \cdot 2^{n - 2} + \dots + r_{1} \cdot 2 + r_{0} .

🔗

This immediately yields a method for converting base

2

numbers to base

10 .

🔗

In the video in Figure 11.5 we recap the material covered above and present examples.

🔗

Figure 11.5. Binary Numbers by Matt Farmer and Stephen Steward

🔗

For the remainder of the section we continue considering the conversion of base

2

to base

10

and also count in base

2 .

🔗

Example 11.6. Conversion from binary to decimal.

🔗

We convert the binary number

100101_{2}

to base

10 .

We have

100101_{2} = 1 \cdot 2^{5} + 0 \cdot 2^{4} + 0 \cdot 2^{3} + 1 \cdot 2^{2} + 0 \cdot 2^{1} + 1 \cdot 2^{0} = 32 + 4 + 1 = 37 .

🔗

We have found that the decimal representation of the base

2

number

100101_{2}

37 .

🔗

In Figure 11.7 we give more examples of numbers in base

2,

their base

2

expansion, and the number in decimal representation.

$n$ in	base $2$ digits of $n$				base $2$ expansion of $n$	$n$ in
base $2$	$2^{3}$	$2^{2}$	$2^{1}$	$2^{0}$		base 10
$0_{2}$				$0$	$0 \cdot 1$	$0$
$1_{2}$				$1$	$1 \cdot 1$	$1$
$10_{2}$			$1$	$0$	$1 \cdot 2 + 0 \cdot 1$	$2$
$11_{2}$			$1$	$1$	$1 \cdot 2 + 1 \cdot 1$	$3$
$100_{2}$		$1$	$0$	$0$	$1 \cdot 2^{2} + 0 \cdot 2 + 0 \cdot 1$	$4$
$101_{2}$		$1$	$0$	$1$	$1 \cdot 2^{2} + 0 \cdot 2 + 1 \cdot 1$	$5$
$110_{2}$		$1$	$1$	$0$	$1 \cdot 2^{2} + 1 \cdot 2 + 0 \cdot 1$	$6$
$111_{2}$		$1$	$1$	$1$	$1 \cdot 2^{2} + 1 \cdot 2 + 1 \cdot 1$	$7$
$1000_{2}$	$1$	$0$	$0$	$0$	$1 \cdot 2^{3} + 0 \cdot 2^{2} + 0 \cdot 2 + 0 \cdot 0$	$8$
$1001_{2}$	$1$	$0$	$0$	$1$	$1 \cdot 2^{3} + 0 \cdot 2^{2} + 0 \cdot 2 + 1 \cdot 1$	$9$
$1010_{2}$	$1$	$0$	$1$	$0$	$1 \cdot 2^{3} + 0 \cdot 2^{2} + 1 \cdot 2 + 0 \cdot 1$	$10$

🔗

Figure 11.7. Binary (base

2

) numbers, their base

2

digits, their base

2

expansion, and in base

10 .

The two symbols used in binary numbers are

0

and

1 .

Recall that

2^{0} = 1

and that

2^{1} = 2 .

🔗

So to convert a number in base

2

representation

write down the base $2$ expansion, which consists of the digits of the base $2$ representation converted to decimal and the place values, which are the powers of $2$
evaluate this expression to obtain the base $10$ representation.

🔗

Try yourself.

Checkpoint 11.8. Convert from binary to decimal.

Give the expanded base

2

form of

111011_{2} .

\cdot 2^{6} +

\cdot 2^{5} +

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2 +

\cdot 1

Give

111011_{2}

in decimal representation.

🔗

Subsection Counting in Base $2$

🔗

For a better understanding of binary (or base

2

) numbers, we consider counting in that representation.

🔗

When we start counting using only the two symbols 0 and 1. As in the case of decimal numbers we start with zero.

🔗

0_{2},

🔗

Still with one digit we can also write the number one:

🔗

1_{2}

🔗

So with one digit we were able to count zero and one. As in the case of decimal numbers we add one more digit and obtain:

🔗

10_{2},

11_{2}

🔗

With two digits we have counted to three. As we cannot go further with those two digits, we continue with:

🔗

100_{2},

101_{2},

110_{2},

111_{2}

🔗

With three digits we have counted to seven. Adding one more digit we continue with:

🔗

1000_{2},

1001_{2},

1010_{2},

1011_{2},

1100_{2},

1101_{2},

1110_{2},

1111_{2}

🔗

With 4 digits we have counted from zero to fifteen. By now the pattern is clear and we can keep counting like this indefinitely, adding one more digit when we have exhausted all combinations with the current number of digits.

🔗

Considering the numbers above we see that

10_{2}

is two,

100_{2}

is four, and

1000_{2}

is eight.

🔗

In Checkpoint 11.9 apply the pattern described above to count in base

2 .

Checkpoint 11.9. Count in base $2$ .

Count in base 2

In the first column enter the numbers in base

2 .

Recall that the characters used to represent base

2

numbers are:

In the other columns enter the values for the digits of the base

2

expansions.

For your convenience the last number in each row is the corresponding decimal number.

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 0

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 1

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 2

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 3

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 4

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 5

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 6

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 7

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 8

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 9

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 10

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 11

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 12

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 13

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 14

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 15

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 16

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 17

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 18

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 19

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 20

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 21

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 22

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 23

_{2} =

\cdot 2^{4} +

\cdot 2^{3} +

\cdot 2^{2} +

\cdot 2^{1} +

\cdot 2^{0} = 24

described in detail following the image — 🔗
Figure 11.10. *Su Doku* by Randall Munroe (`https://xkcd.com/74`).

Prev Top Next

Section 11.2 Binary Representation

Subsection Base 2 Expansion

Example 11.6. Conversion from binary to decimal.

Checkpoint 11.8. Convert from binary to decimal.

Subsection Counting in Base 2

Checkpoint 11.9. Count in base 2.

Subsection Base $2$ Expansion

Subsection Counting in Base $2$

Checkpoint 11.9. Count in base $2$ .