Quick, what number is this?
If you thought:
That is clearly one-hundred-and-one…
… then that would be entirely reasonable!
However, those three symbols:
… 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 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 has a value, according to it’s position or place.
In base 10, those values are:
| Value expressed as a power | |||
|---|---|---|---|
| Value expressed in standard form |
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 we are filling in the digits in the chart like so:
| Value expressed as a power | |||
|---|---|---|---|
| Value expressed in standard form | |||
… and in expanded form, we know that:
You’ve probably not thought of 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 and mean one-hundred-and-one – that is, we are using base 10 – we should write it like this instead:
Base 2
In base 2, has a value of five.
To indicate that we are expressing a value in base 2, we write it like this:
So how does have a value of five?
It’s all about the base of the power assigned to each place:
| Value expressed as a power | |||
|---|---|---|---|
| Value expressed in standard form | |||
In expanded form:
So, this is how we know that in base 2 has a value of five in base 10.
Expressed using symbols, that is:
Another example
It is true that:
How?
| Value expressed as a power | ||||
|---|---|---|---|---|
| Value expressed in standard form | ||||
In expanded form:
Exercises
Try doing the following conversions in your graph paper notebook: