1. NASA’s Dangerous Mathematics: Black Holes and Dividing by Zero

2. Black holes are where God divided by zero.
- Steven Wright

3. Part 1
Dividing by Zero

4. Whole Number Operations
Draw a picture that helps to demonstrate the meanings of (and answers to) each of the following:
– 3 7 * 3 21 / 3

8. Building off of Addition:
Subtraction
Multiplication
Division

9. Division as “Repeated Subtraction”
We subtracted a total of 7 “3’s” to go from 21 to 0, so
21 ÷ 3 = 7

10. Dealing with Zero
Use concepts and methods from the previous problems to find the answers to the following:

11. Division and Zero
The Question: Starting at 0, how many times do we need to subtract 8 to get to 0?
The Question: Starting at 8, how many times do we need to subtract 0 to get to 0? 8 -0
We’re already there. We’ve already accomplished the goal.
We subtracted 0 “8’s”, so
0 ÷ 8 = 0
The goal is impossible.

12. Dividing by zero is problematic.
Bottom Line:
Dividing by zero is problematic.

13. Part 2
Black Holes

14. To make a black hole, first we need a star …
Blow up your balloon, but not too large – it needs to stay somewhat spherical. Tie off the end of the balloon.
Cover the inflated balloon with about 4 sheets of aluminum foil.
Now, you have a star!

15. Finding Density
Use a tape measure to measure the approximate circumference of your model star.
Then, assuming that your star has a mass of approximately 30 grams, calculate the density of your star.

16. A few formulas that might help …
C = 2πr D = M÷V V = (4/3) π r3

17. SUPERNOVA! Now, it’s time for your star to “go supernova” …
Squeeze your star or use a sharp instrument to pop the balloon.
After the balloon has popped, squeeze the shell of the collapsed star, shaping it into a spherical ball.
Make the “supernova remnant” as small of a sphere as you can.
Then, measure the circumference of the “supernova remnant” and calculate the remnant’s density.

18. Sometimes, a supernova results in a black hole …
What would it take for our model star to become a REAL black hole? Before we can answer that, we need to know what a black hole is.

19. Definition: a place where the escape velocity is faster
What is a Black Hole? ?
Definition: a place where the
escape velocity is faster
than the speed of light.

20. Definition: the velocity at which something must travel
Escape Velocity ?
Definition: the velocity at
which something must travel
away from an object
such that the gravity
of the object
cannot stop it.

21. Escape Velocity (2) Escape velocity depends on the gravity
of the object

22. Escape Velocity (3) ^ Low mass Medium mass High mass Very high mass
for a given radius ^
Low
mass
Medium
mass
High
mass
Very
high
mass

23. Escape Velocity (4)
radius decreases
increases

24. Gravity Formula (modified)
Force = G∙M ÷ (r2) What happens to the gravitational force as M increases? What happens to the force as r decreases?

25. What is a Black Hole? Nothing can escape.
It’s an object of high enough mass
And/or small enough radius
such that
the escape velocity is
faster than the speed of light.
Nothing can escape.

26. To Make a Black Hole R = 2GM/c2 G = 6.67 x 10-8 and c = 3 x 1010
The radius of the “event horizon” of a black hole with a given mass can be found using the following formula:
R = 2GM/c2
And the following constants:
G = 6.67 x and c = 3 x 1010
Knowing that, we can calculate the radius of the event horizon of a black hole with a mass of 30 grams!

27. Our Model Black Hole
For a typical balloon and foil assembly of 30 grams to become a black hole, the radius would have to be … 4 x cm CHALLENGE: How would you explain that size to a student?

28. And this is where it starts to get strange …
Once the object gets so dense that not even the speed of light is fast enough to escape (if it gets close to the object, that is), then the familiar laws of the universe can no longer handle it. The object collapses to an extremely heavy POINT or “SINGULARITY,” ripping a hole in spacetime itself.

29. So when we talk about the “size” of a black hole:
We are really talking about its MASS.
And when we talk about
the “radius” of a black hole:
We are talking about the farthest distance from the singularity that light cannot escape from.

30. The Key is ENERGY
Obviously, it would take a lot of energy to turn our model star into a black hole (more energy than we have at our disposal)
The same is true throughout the universe. Black holes result from extremely violent, energetic events
Super Super Novae
Merging Neutron Stars
We cannot make a black hole out of our balloon and foil “star”!

31. Just for fun, though …
If we COULD turn our aluminum foil/balloon ball into a black hole, what would happen?

32. Part 3
Put it all together …

33. Black holes are where God divided by zero.
- Steven Wright
In your own words, explain what Steven Wright might have meant when he said this.

34. Let’s go back to the density formula:
D = M ÷ V

35. Reflection
What math/science concepts and skills have we used in these activities?
What other math/science connections can you make to these concepts?
What other (not necessarily math/science) skills and concepts are related to these activities?
How might you use these activities in your classroom?