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Master of Arts in Teaching (MAT)

0! = 1

The fact that 0! = 1 is a useful convention in probability. The number of ways to arrange
N distinct objects is given by N! and it is a convention to say that there is only one way to
arrange zero objects . 1! equals 1 is also a useful convention in probability, it makes sense to say
that there is only one way to arrange 1 object. Using something called the Gamma Function I
could find an approximation for 3. I do not believe that there is a
number such that x! = 0 because after learning how to find (1.5!) no matter how small the
fraction or decimal is it will not become zero. This is similar to the idea of limits in Calculus.

The Difference Between No Slope and a Zero Slope

Students sometimes have difficulty understanding the meaning of the number zero in the
context of slope, they may have a hard time determining if the slope of the line is zero or if the
line is undefined (i.e. has no slope). I explain to the students the difference between zero slope
and no slope by talking about a “skier dude” in Colorado. In this example I draw skier dude
skiing on a horizontal line and note that skier dude would say (in reference to the slope) "this is a
total zero dude." Thus a horizontal line has a slope of zero. Then I draw a vertical line on the
board with skier dude next to the top of the vertical line and ask what skier dude would say about
this ski slope. By noting that skier dude would say "oh, no" (because there would be no way of
surviving the slope), I make the connection that a vertical line has no slope. During this lesson I
also tie together the previous knowledge of being able to have a zero in the numerator but not in
the denominator with the slope formula of rise over run. This seems to help the students
understand slope a little better.

The Angle of Zero Degrees

It can be difficult for some geometry instructors to understand why there is such a thing
as a zero-degree angle. It is possible that the difficulty lies in their understanding of the
definition of an angle; that an angle is formed when two rays share a common endpoint . In a
zero-degree angle, according to this definition, one ray is directly on top of (coincidental with)
another ray; so some may consider that to just be one ray, and therefore not an angle at all. Other
instructors may believe an angle is defined as the space between two rays (with a common
endpoint), and since there is no space between the rays in a zero-degree angle, there is no angle
formed. These same difficulties may arise when studying a 360-degree angle. Similarly, it can
also be difficult for some to understand an angle of 180 degrees, since two rays which share a
vertex and go in opposite directions form a straight line and no longer appear to be two rays.
The fact that a protractor has a measure of zero on it provides evidence that there is such a thing
as a zero-degree angle. A practical example of one can be considered in the firing of a projectile
from one building to the top of another building with the same exact elevation. Discounting any
effects from wind or gravity, the angle at which the projectile is fired would be equal to zero
degrees.

Zero as a Number

The natural numbers are the numbers that are first learned when a child is growing up.
They consist of the infinite set {1, 2, 3, ….}. I tell my students that they are the numbers that
come naturally to them, like when they were learning to count on “Sesame Street”. The natural
numbers are closed under addition. This means that when you add two natural numbers you get
another natural number. The same rule applies when you are multiplying two natural numbers.
We can extend the natural numbers to the set of whole numbers by including zero. The set of
whole numbers is also closed under addition and multiplication. Whole numbers also include the
zero-multiplication property; specifically when you multiply a natural number by zero we get
zero for an answer. To explain this first of all it has to be understood that, when applied to whole
numbers, multiplication is repeated addition. For example 4 * 3 is the same as 4 + 4 + 4. When
you multiply 5 * 0 you can change it to 0 * 5 by using the commutative property of
multiplication. The expression 0 * 5 expands to 0 + 0 + 0 + 0 + 0 which has a sum of zero.
Since these steps will always work when you replace 5 with any whole number, we see that any
whole number multiplied by zero will give you zero for an answer.

A general proof, assuming the additive property of zero, follows:
 

Given, by the addition property of zero
Multiplication Property of Equality
Distributive Property of Equality
Transitive Property of Equality (steps 2 and 3)
Subtraction Property of Equality
Subtraction
Addition property of zero

Thus the addition property of zero determines the multiplicative property of zero.

"Proof " that 0 = 1

If a person does not understand all of the rules about using the number zero some very
strange statements in Algebra can be “proved”. For example here is a proof that 0 = 1which
results from a lack of understanding of the properties about the number zero:

Let x = 0 [Given]
Then x ( x -1) = 0 [Multiplication property of 0]
Therefore x – 1 = 0 [Divide both sides by x; Cancellation property]
And x = 1 [Adding 1 to each side of the equation ]
Therefore, 0 = 1.  

This “proof” violates a very important property: that division by zero is undefined. Notice that
to arrive at step 3 from step 2, you have to divide by x, but since x = 0, this is equivalent to
dividing
by zero. Thus the argument is invalid.

Given that a and b are integers such that a = b, prove 0 = 2.

1.Given

2.Reflexive Property of Equality

3. Multiplication Property of Equality

4. Distributive Property

5. Addition Property of Equality

6. Associative Property of Equality
   
7. a(a - b) = a(b + 2) -b(b + 2) 7. Distributive Property
   
8. a(a - b) = (a - b) (b + 2) 8. Distributive Property
   
9. a = b + 2 9. Division Property of Equality
   
10. b = b + 2 10. Transitive Property (steps 1, 9)
   
11. Therefore 0 = 2 11. Subtraction Property of Equality

In this "proof" the error is dividing by (a - b) between steps 8 and 9. The reason this is incorrect
is since a = b, which was the given information, then a - b = 0. Thus the argument is incorrect.

The Zero Metaphor


There are many metaphors in society which use the word zero. One example is the "zerobalanced
budget". This is a philosophy in business or in government where the amount of
money taken in should be the same as the amount of money being spent. This is a concept that is
very hard for groups like local and national governments to achieve; thus the big deficit we have
here in the United States.

Another metaphor using the number zero is "Ground Zero". This means that an observer
is at distance zero from a blast site, a phrase which was coined during the first atom bomb test in
1946. It would be interesting to find out why the Lincoln Journal Star calls their Friday
entertainment section of the newspaper "Ground Zero" since there is no obvious connection
between entertainment and the explosion of an atomic bomb.

In the context of measuring temperature, scientists use "absolute zero" to identify the
temperature at which molecules stop moving. Although a temperature of absolute zero has never
been achieved, temperatures very close to it have been reached in laboratories.

A website called "The School of Wisdom", indicates the geometrical representation for
zero-dimensional object is a point. It is only a representation because physically it is impossible
to draw a point with no length or width.

According to a train of thought called Tao, “The Zero Dimension is the point, the
infinitely small place holder. It exists not in space, but in time only. It is the moment between
past and future, the subject, zero.” An article describing the Tao philosophy also stated that the
Zero Dimension is the home of the natural numbers and the subject point, zero, is pure
awareness.

Conclusion

The number zero is a very powerful tool in mathematics which has many different
applications and rules. I have learned many truly amazing ways to think about and work with
zero from writing this paper. I never realized how many different rules there were for the
number zero in mathematics. This helped me realize why I have to be very careful in my
teaching when the number zero is involved in the lesson. The number zero is a very special
number which is much more important and powerful than what its name is associated with in our
society.

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