Binary Guide Part 1

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Introduction
The numbering system we use on a daily basis is base 10 or the decimal system. It turns out that
while the numbering system we use is a natural choice for humans, when computers are concerned,
a binary numbering system is more suitable.

Base 10 or Decimal Numbers
Let’s revisit how we were taught to read and write decimal numbers. This will help us understand
the structure of binary numbers.
Digits: There are 10 digits

0 1 2 3 4 5 6 7 8 9

Place Value: The position of each digit determines what that value is. Recall that from right
to left the first 5 place values are: ones, tens, hundreds, thousands, ten thousands.
ten thousands
10,000

thousands
1,000

hundreds
100

tens
10

ones
1

Base 10: Each place value can be represented in terms of 10 by using exponents. 10 is the base
and each place value is a power of 10 The table below shows the place values represented in
terms of 10.
104
10,000

103
1,000

102 101
100 10

100
1

Expanded Form: Consider the decimal number 123. Recall that the digit and place value
determine what that number is. We can add the quantities in each place value to get the number
123.
100s place

10s place

1s place

1

2

3

(1 × 100)
100

(2 × 10)
20

(3 × 1)
3

123 = (1 × 100) + (2 × 10) + (3 × 1)
123 = 100 + 20 + 3

Quick Practice: Write the following numbers in their expanded form like the example above.
432 =

76 =

8632 =

154 =

90 =

5120 =

801 =

43 =

2922 =

Base 2 or Binary Numbers
A binary number system follows the same structure, but instead of powers of 10, place values are
powers of 2 as shown below:
23
8

Binary

22
4

21
2

Digits: There are 2 digits

20
1

Decimal

103
1,000

102 101
100 10

100
1

0 and 1

Expanded Form: Consider the binary number 1110. Recall that the digit and place value
determine what that number is. We can add the quantities in each place value to rewrite 1110.
8

4

2

1

1

1

1

0

(1 × 8) (1 × 4) (1 × 2) (0 × 1)
8
4
2
0

1110 = (1 × 8) + (1 × 4) + (1 × 2) + (0 × 1)
1110 = 8 + 4 + 2 + 0
1110 = 14

Shortcut: Since there are only two digits, 0 and 1, you can simply add every place value with a
digit of 1.

8

4

2

1

1

1

1

1

8 + 4 + 2 + 1 = 15

8

4

2

1

1

0

0

1

8+1=9

Quick Practice: Convert the following binary numbers to decimal numbers as shown above. You
23 22 21 20
may use a table with binary place values.
8 4 2 1

0010=

0110=

0011=

1001=

1111=

0100=

0001=

0011=

0110=

1000=

0111=

1101=
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