0.4.2 The Binary Number System

Binary Uses Only Two Digits

The binary number system is a base-2 number system.

It uses only two digits:

0
1

That may seem limiting at first, but position allows binary to represent many different values.

Just as decimal reuses ten digits across different place values, binary reuses 0 and 1 across powers of 2.

Binary Place Values Grow by Powers of 2

Starting from the rightmost position, binary place values are:

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

These are powers of 2:

2^0 = 1
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16

Each move one position to the left doubles the place value.

Read a Binary Number by Its Active Places

Consider:

1011

Line the digits up with their binary place values:

Add the active place values:

8 + 2 + 1 = 11

So:

1011₂ = 11₁₀

The small base labels are sometimes used to make the representation explicit:

• subscript 2 means binary;
• subscript 10 means decimal.

A Zero Means That Place Is Not Included

Consider:

1001

The place values are:

8 4 2 1

The binary digits are:

1 0 0 1

Only the 8 and 1 places are active.

Therefore:

8 + 1 = 9

A 0 does not remove the position.

It means that place contributes zero to the current value.

Counting in Binary

Decimal counting begins:

0
1
2
3
4
5

Binary counting begins:

0
1
10
11
100
101
110
111
1000

Notice what happens after 1.

There is no binary digit 2, so the next position is used.

This is similar to decimal moving from 9 to 10.

The base determines when another position becomes necessary.

Binary and Decimal Can Represent the Same Quantity

The quantity six can be represented as decimal:

6

or binary:

110

Binary 110 has:

• one 4;
• one 2;
• zero 1s.

So:

4 + 2 = 6

The symbols are different.

The represented quantity is the same.

Why Binary Matters in Digital Systems

Digital hardware works naturally with two-state distinctions.

At a simplified level, computing systems can represent information using two distinguishable states that are modeled as:

0
1

Those binary digits can be combined to represent:

• numbers;
• text;
• colors;
• instructions;
• images;
• sound;
• and many other forms of digital information.

Binary is therefore not limited to “binary numbers.”

It is a fundamental representation used throughout digital systems.

Keep the Place-Value Pattern

When reading binary, do not treat a sequence such as:

101101

as a strange decimal-looking number.

Read it by place value.

From right to left:

1, 2, 4, 8, 16, 32

A 1 means the place contributes its value.

A 0 means it does not.

That is the core reading strategy for small binary values.