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

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

Part 1 Dividing by Zero

Whole Number Operations Draw a picture that helps to demonstrate the meanings of (and answers to) each of the following: 12 + 3 15 – 3 7 * 3 21 / 3

Building off of Addition: Subtraction Multiplication Division

Division as “Repeated Subtraction” 21 -3 18 15 12 9 6 3 0 We subtracted a total of 7 “3’s” to go from 21 to 0, so 21 ÷ 3 = 7

Dealing with Zero Use concepts and methods from the previous problems to find the answers to the following: 0 8 8 0

Division and Zero 0 8 The Question: Starting at 0, how many times do we need to subtract 8 to get to 0? 8 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.

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

Part 2 Black Holes

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!

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.

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

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.

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.

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.

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.

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

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

Escape Velocity (4) radius decreases increases

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

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.

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 10-8 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!

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 10-27 cm CHALLENGE: How would you explain that size to a student?

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.

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.

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”!

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

Part 3 Put it all together …

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.

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

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?