Teachable Moments
Prepared by: Sa’diyya Hendrickson
Why is 0.9 = 1?
I. The Conceptual Dilemma
Recall that in a base 10 number system, 0.9 = 0.999 . . . =
9
10
+
9
100
+
9
1000
+ ...
The expanded form is an infinite sum! Is it possible for an infinite sum to equal a finite quantity?
While it may be difficult to believe, the answer is yes!
Consider a square with side lengths equal to 1. What is its area?
Area = length ⇥ width
=1·1
= 1 unit2
Now, let’s consider separating the area of the square into fractional parts, as shown below:
From the diagram, we can see that:
1=
1
2
+
1
4
+
1
8
+
1
16
+
1
32
+ ···
This is another example wherein we can express the number 1 as an infinite sum!
II. Proof: 0.9 = 1
Let x = 0.9 = 0.99. Then, 10x = 9.9. From this, we have:
)
Therefore,
Making Math Possible
10x =
9.9
x =
0.9
9x =
9.0
9x
=
9
9
9
x = 1,
1 of 1
as required.
⇤
c Sa’diyya Hendrickson