Using Integer Exponents

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來自 Duke University 的課程

Data Science Math Skills

1278 個評分

Duke University

Data Science Math Skills

1278 個評分

Data science courses contain math—no avoiding that! This course is designed to teach learners the basic math you will need in order to be successful in almost any data science math course and was created for learners who have basic math skills but may not have taken algebra or pre-calculus. Data Science Math Skills introduces the core math that data science is built upon, with no extra complexity, introducing unfamiliar ideas and math symbols one-at-a-time. Learners who complete this course will master the vocabulary, notation, concepts, and algebra rules that all data scientists must know before moving on to more advanced material. Topics include: ~Set theory, including Venn diagrams ~Properties of the real number line ~Interval notation and algebra with inequalities ~Uses for summation and Sigma notation ~Math on the Cartesian (x,y) plane, slope and distance formulas ~Graphing and describing functions and their inverses on the x-y plane, ~The concept of instantaneous rate of change and tangent lines to a curve ~Exponents, logarithms, and the natural log function. ~Probability theory, including Bayes’ theorem. While this course is intended as a general introduction to the math skills needed for data science, it can be considered a prerequisite for learners interested in the course, "Mastering Data Analysis in Excel," which is part of the Excel to MySQL Data Science Specialization. Learners who master Data Science Math Skills will be fully prepared for success with the more advanced math concepts introduced in "Mastering Data Analysis in Excel." Good luck and we hope you enjoy the course!

從本節課中

Measuring Rates of Change

This module begins a very gentle introduction to the calculus concept of the derivative. The first lesson, "This is About the Derivative Stuff," will give basic definitions, work a few examples, and show you how to apply these concepts to the real-world problem of optimization. We then turn to exponents and logarithms, and explain the rules and notation for these math tools. Finally we learn about the rate of change of continuous growth, and the special constant known as “e” that captures this concept in a single number—near 2.718.

Using Integer Exponents7:48

Simplification Rules for Algebra using Exponents11:08

How Logarithms and Exponents are Related12:36

The Change of Base Formula4:23

The Rate of Growth of Continuous Processes11:09

與講師見面

Daniel Egger
Executive in Residence and Director, Center for Quantitative Modeling
Pratt School of Engineering, Duke University
Paul Bendich
Assistant research professor of Mathematics; Associate Director for Curricular Engagement at the Information Initiative at Duke
Mathematics

Hello, I'm Daniel Egger.

Welcome to the Data Science Math Skills Section on Exponents and Logarithms.

We'll be beginning now with the basic integer exponents.

And we'll start with positive integer exponents.

Positive Integer Exponents are numbers used for counting, like 1,

2, 3, and so on.

When we have a number like 9, which is equal to 3 times 3.

Three is a factor of nine, and it repeats, it occurs twice.

In the number 27, the factor 3 occurs 3 times.

And of course in the number 81, the factor 3 occurs 4 times.

We can use positive integer exponents, which are numbers placed in the upper

right corner of a factor to count how many times the same factor repeats in a number.

So 9 equals 3 times 3 equals 3 To the 2nd,

where 2 is the exponent.

27 equals 3 times 3 equals 3 to

the 3rd, where 3 is the exponent.

And 81 equals 3 times 3 times 3,

equals 3 to the 4, where 4 is the exponent and so on.

So we read this as 3 raised to the 4th power or simply 3 to the 4th for short.

So let's look at another example.

The number 248,832 is just equal to 12, times 12,

times 12, times 12, times 12.

And you can see the great convenience in being able

to write this as 12 raised to the 5th power,

where 5 is the exponent and this number is 12 to the 5th.

Note that for historical reasons, numbers raise to the second

power have special names as well as a regular name.

So this is 4 to the 2nd but it's also 4 squared.

And this is 4 to the 3rd,

but it's also 4 cubed.

Next we're going to talk about 0 as an exponent.

These problems are easy, why do I say that?

Because by the definition of exponents any number

except 0 raised to the 0 with power equals 1.

So for example, 3 to the 0 = 1, 2 to the 0 = 1,

2pi to the 0 = 1, 1 / x cubed to the 0 = 1 so

long as x does not equal 0, and so on.

Even for mathematicians zero to the zero a little too weird and

so that is simply undefined.

Next we'll look at negative integer exponents.

2 raised to the exponent

-1 is read 2 to the -1.

And it is equal 1 over 2 to the 1 or 1/2.

2 to the minus 2 is equal to 1 over 2 to the 2, or one quarter.

And 2 to the minus 3 is equal to 1 over 2 to the 3, or one eighth.

So you should see the pattern here.

Raising a number to a negative exponent is the same as dividing it

by the same integer if that integer were positive.

Now let's consider dividing by a negative exponent.

By the exact same logic, 1 over 2 to

the -1, simply = 2 to the 1 or 2.

1/2 to the -2 simply = 2

to the 2nd or 2 squared or 4.

1/2 to the -3 simply = 2 to the 3rd or 2 cubed or 8.

We express the general rule for

negative exponents using letters to stand for just about any number.

And the general rule looks like this,

x to the minus n equals 1 over x to the n and

1 over x to the minus n equals x to the n.

One of the most common uses of exponents is for

something that we call scientific notation.

Scientific notation is a way to write

numbers that otherwise would have a very large number of zeros in them.

What we do is we take the part of the number that has significant digits,

here 5,972 and we put one digit to the left of the decimal place.

So we have 5.972 and

then we multiply by 10 raise to the appropriate exponent.

Which in this case is 24.

And 10 to the 24 means that you are going to have,

you're going to move the decimal place 24 spaces to the right.

Get it?

So let's try another one.

If we have a number less than one,

we are going to be moving to decimal place to the left.

And we are going to have a negative exponent.

So in this case, we have the mass of an electron, and

this would be 9.109 times10 to the -31 because

we're moving the decimal place 31 places to the left.

Key thing to remember is that we only

need to keep these significant non-zero digits and

we always have one digit, to the left of the decimal place.

One digit to the left.

And that's all you need to know for scientific notation.

And that concludes our first video on exponents.

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