AboutJosh Expertise When I work through problems, I emphasize principles and key ideas which I believe are worth noting. I will try to answer questions in the following areas, but not at the advanced level. Algebra. Sequences & Series. Trigonometry. Functions & Graphs. Coordinate Geometry. Quadratic Polynomials. Exponentials & Logarithms. Basic Calculus. Probability, Permutations and Combinations. Mathematical Induction. Complex numbers. Physics problems.
Experience
Experience: I have worked as a teaching assistant in college. My hope is that more people will share knowledge without boundary, give help without seeking recognition or monetary rewards.
Supplementary Website: See a selection of past questions in my maths repository under "Question Archive"
Education Credentials: Bachelor degree in Engineering Science."Everyone struggles with something."
Question Hello Josh.
My math teacher states that n*0 = 0, n=integer.
So :
2*3=6 which means 6/3=2
15*3=45 which means 45/3=15
10*4=40 which means 40/4=10
Now :
2*0=0 which means 0/0=2 ?????
Isn't it right that result of n*0 is undefined?
Thanks.
Answer Hello Joe,
This is a good question because zero plays an important role in mathematics.
It is true that any number multiplied by zero is zero. This is an essential requirement for constructing a vector field. Without going into the specifics, we can think of a "vector field" as just a space where numbers live and they are governed by sensible rules which enable us to ADD, MULTIPLY and COMBINE numbers.
The ground rules or arithmetic as we know them, things like addition and subtraction (cancellation) that we use in algebra would not work if this property "n*0=0" is not satisfied.
n*0 = 0 is a valid statement for any real number.
However, its INVERSE technically does not exist, if we divide both sides by zero, the right hand side of the equation becomes "0/0", we say that this entity is UNDEFINED. To get around this problem, known as "singularity", we forbid division by zero in any circumstance.
In other words, starting with n*a=b, we can divide both sides by "a" to obtain n=b/a only if "a" is non-zero.
The next part of my response briefly considers what useful purpose this serves. Why would anyone contemplate having a zero in the denominator?
Two areas immediately come to mind. One is in the area of CALCULUS, another is in an area known as COMPLEX ANALYSIS. Of course, it is not possible for us to even scratch the surface of these advance topics which are normally studied in senior years at high school, all the way to university. For now, we will have to settle with the following notions just to get a flavor.
In a nutshell, CALCULUS is about the study of the rate of change of a quantity. For instance, if we consider the physics equation for speed (S), it is given by the distance travelled (D) divided by the time interval (t). So, we have an equation like S=D/t. If the speed is not constant, for example, an object is accelerating, then the instantaneous speed is given approximately by dD/dt, which represents the distance travelled over a small time span. To get more accurate, we have to shrink this window down to a tiny time segment. This idea is a repetitive theme in the study of calculus. We are often interested in exploring system behavior in the limit as things (e.g., the denominator "t" in this example) tend toward zero. This is very similar to your consideration, sort of like "dividing something by zero", the difference lies in not letting "t" become zero, but only close to zero.
A second area (and there are many) where zeros etc. are useful comes from engineering, it is concerned with the design and analysis of systems and their stability. Typically, something of interest (e.g., an acoustic filter) is modelled generally as "b/a". However, the terms "b" and "a" here can be any expression (e.g., a function like a=x+2 or something more complicated) not just a number. This sort of consideration is quite commonly encountered in many areas of science. One interesting result called Cauchy's theorem may be used to find the answer of integrals which are otherwise difficult to compute. [One would have to study complex numbers and much more to make sense of this]
Summary:
Anyway, here are some of the main points of our discussion.
1. n*0 = 0 (the existence of zero) is a result with practical significance for algebra. "n*0" is not undefined, it is just zero. But "n/0" is undefined.
2. Given n*a=b, we can divide both sides by "a" only if "a" is non-zero. Otherwise, the term "b/0" is undefined (it goes to infinity).
3. Considering an expression of the form "b/a", more generally, "b" and "a" can themselves be some mathematical expression, not only numbers. In the context of mathematical modeling, much interest in the analysis of "b/a" as the term "a" tends toward zero comes from being able to better understand the behavior of a system.
