#C1.3

Abstraction and Binary Numbers

Quick, what number is this?

101

If you thought:

That is clearly one-hundred-and-one...

... then that would be entirely reasonable!

However, those three symbols:

101

... can also represent the number five.

Why is this?

The answer lays in what number system is assumed.

Number systems

A number system is an example of an abstraction.

Most people would look at $101$ and see one-hundred-and-one.

That is because, way back in elementary school, they learned the concept of place value, and the base 10 number system was used.

Base 10

Each digit in $101$ has a value, according to it's position or place.

In base 10, those values are:

Value expressed as a power	$10^{2}$	$10^{1}$	$10^{0}$
Value expressed in standard form	$100$	$10$	$1$

The value of each place is a power with a base of 10.

The exponent of the power increases as you move from right to left.

So when we are reading $101$ we are filling in the digits in the chart like so:

Value expressed as a power	$10^{2}$	$10^{1}$	$10^{0}$
Value expressed in standard form	$100$	$10$	$1$
	$1$	$0$	$1$

... and in expanded form, we know that:

\begin{aligned} = 100 \times 1 + 10 \times 0 + 1 \times 1 \\ = 100 + 0 + 1 \\ = 101 \end{aligned}

You've probably not thought of $101$ in quite that detailed a manner in a long time.

Going forward in this course, when we are writing numeric values, we must be careful to annotate the number system.

We do this by appending a subscript. When we write $101$ and mean one-hundred-and-one – that is, we are using base 10 – we should write it like this instead:

101_{10}

Base 2

In base 2, $101$ has a value of five.

To indicate that we are expressing a value in base 2, we write it like this:

101_{2}

So how does $101_{2}$ have a value of five?

It's all about the base of the power assigned to each place:

Value expressed as a power	$2^{2}$	$2^{1}$	$2^{0}$
Value expressed in standard form	$4$	$2$	$1$
	$1$	$0$	$1$

In expanded form:

\begin{aligned} = 4 \times 1 + 2 \times 0 + 1 \times 1 \\ = 4 + 0 + 1 \\ = 5 \end{aligned}

So, this is how we know that $101$ in base 2 has a value of five in base 10.

Expressed using symbols, that is: $101_{2} = 5_{10}$

Another example

It is true that:

1001_{2} = 9_{10}

How?

Value expressed as a power	$2^{3}$	$2^{2}$	$2^{1}$	$2^{0}$
Value expressed in standard form	$8$	$4$	$2$	$1$
	$1$	$0$	$0$	$1$

In expanded form:

\begin{aligned} = 8 \times 1 + 4 \times 0 + 2 \times 0 + 1 \times 1 \\ = 8 + 0 + 0 + 1 \\ = 9 \end{aligned}

Exercises

Try doing the following conversions in your graph paper notebook:

$1011_{2} =_{10}$
$0011_{2} =_{10}$
$1111_{2} =_{10}$
$14_{10} =_{2}$
$13_{10} =_{2}$
$12_{10} =_{2}$