0.4.4 Reading Small Binary Numbers

Start From the Right

The easiest way to read a small binary number is to write the place values from right to left.

The first positions are:

1, 2, 4, 8, 16, 32, 64, 128

Each position is worth twice the position to its right.

For a four-bit value:

8 4 2 1

For an eight-bit value:

128 64 32 16 8 4 2 1

Example: 0101

Write the binary digits under the place values:

Place value	8	4	2	1
Binary digit	0	1	0	1

Use only the places containing 1.

4 + 1 = 5

Therefore:

0101₂ = 5₁₀

The leading zero does not change the numeric value.

It can still be useful when a fixed number of bit positions is being shown.

Example: 1010

Place value	8	4	2	1
Binary digit	1	0	1	0

The active places are:

8 + 2

So:

1010₂ = 10₁₀

Example: 1111

Every place is active:

8 + 4 + 2 + 1 = 15

Therefore:

1111₂ = 15₁₀

Four binary digits can represent sixteen different patterns:

through:

When interpreted as unsigned numbers, those patterns represent decimal values 0 through 15.

Example: 00110110

Use the 8-bit place values:

Place value	128	64	32	16	8	4	2	1
Binary digit	0	0	1	1	0	1	1	0

The active places are:

32 + 16 + 4 + 2

Add them:

So:

00110110₂ = 54₁₀

Convert Decimal to Binary by Finding Powers of 2

You can also move in the other direction.

Suppose the decimal value is:

Find powers of 2 that add to 13.

The largest useful value is 8.

Remaining:

13 - 8 = 5

Use 4.

Remaining:

5 - 4 = 1

The 2 place is not needed.

Use 1.

So the places are:

8 4 2 1
1 1 0 1

Therefore:

13₁₀ = 1101₂

Place Values Prevent Guessing

When binary is unfamiliar, it is easy to look at:

and accidentally read it like decimal “one thousand one hundred one.”

Do not read binary as a decimal-looking string.

Use place values:

8 + 4 + 0 + 1 = 13

That process is slower at first, but it builds the correct mental model.

Repeated Patterns Become Familiar

With practice, small values become recognizable.

Decimal	Binary
0	0000
1	0001
2	0010
3	0011
4	0100
5	0101
6	0110
7	0111
8	1000
9	1001
10	1010
11	1011
12	1100
13	1101
14	1110
15	1111

You do not need to memorize the entire table immediately.

Notice the pattern.

Each new value follows binary counting, and each column corresponds to one power of 2.

The Four-Bit Range Leads Directly to Hexadecimal

Four bits can represent sixteen different patterns.

That matches the number of single-digit symbols used by hexadecimal.

This is why binary and hexadecimal fit together so neatly.

The next Learning Activities introduce hexadecimal and then connect one hexadecimal digit directly to a group of four binary bits.