Addition and Subtraction of Small Numbers

Professor Dave Explains
3 Aug 201708:39

Summary

TLDR教授戴夫解释了加法和减法的基本概念。他指出数学源自于人类早期的语言和符号,用于描述周围的世界。数学起初很简单,随着时间推移变得复杂。加法表示两个数字的和,而减法则表示两个数字的差。通过举例说明,他介绍了这些运算的基本性质,如加法的交换律和结合律,并强调了理解数学符号背后实际意义的重要性,帮助学习者更好地掌握数学的实用性。

Takeaways

  • 📚 数学起源于简单的计数和符号,用于交流思想和描述环境。
  • 🚀 数学的发展是必要的,并且至今仍在继续,尽管现代数学的前沿领域对许多人来说可能很抽象。
  • 🔢 人类最早发展的数学类型是算术,它涉及计数和代表计数的符号。
  • 👶 计数和算术的需要随着文明的形成和贸易的开始而产生,用于跟踪库存和定价商品。
  • 👋 我们基于十进制的计数系统源于我们习惯于用手指计数,尽管也有基于二十或六十的系统。
  • 🍎 加法是最基本的算术运算,代表两个数字的组合,形成一个单一的数字或和。
  • 🍏 减法是加法的逆运算,它不是找到两个数字的和,而是找到它们的差。
  • 🔄 加法具有交换性,即加数的顺序不影响结果。
  • 🔄 加法还具有结合性,即连续两次加法的执行顺序不影响最终结果。
  • 🚫 减法不具有交换性和结合性,减数的顺序和组合方式会影响结果。
  • 🌐 数学符号,无论多么复杂,都具有具体的含义,它们根植于物理世界。
  • 💡 理解数学结构代表的实际意义,可以帮助我们认识到它们的实用性和力量。

Q & A

  • 数学的起源是什么?

    -数学起源于我们的语言,是一套帮助我们沟通思想和描述周围环境的模型和符号。

  • 为什么数学在今天变得如此复杂?

    -数学在过去几百年里变得极其复杂,这是因为所有现有的、有时困难的数学都必须被学习,以便人们能够对数学领域做出贡献。

  • 人类最早发展的数学类型是什么?

    -人类最早发展的数学类型无疑是算术。

  • 为什么我们使用十进制系统?

    -我们使用十进制系统是因为我们习惯用十个手指计数,尽管十这个数字是完全任意的。

  • 加法是什么?

    -加法是最基础的算术运算,它代表两个数字的组合,形成一个单一的数字或和。

  • 如何用数学符号表示加法?

    -我们会把数字2,然后是加号或加号符号,接着是数字3,后面是等号,最后是数字5。

  • 减法是什么?

    -减法是加法的逆运算,它不是找到两个数字的和,而是找到它们的差。

  • 如何用数学符号表示减法?

    -我们可以写一个方程,上面是数字5,然后是减号符号,接着是一个1,然后是等号,最后是数字4。

  • 加法有哪些数学属性?

    -加法是交换律的,即加数的顺序不重要。加法也是结合律的,即连续两次加法的执行顺序不重要。

  • 减法有哪些数学属性?

    -减法不是交换律的,减数的顺序很重要。减法也不是结合律的,连续两次减法的执行顺序会影响结果。

  • 数学符号是如何与现实世界联系的?

    -无论数学符号看起来多么抽象,它们都与现实世界有联系,即使这种联系不如加号那样直接。

  • 这个系列的目标是什么?

    -这个系列的目标是使所有其他数学运算像加法和减法一样易于理解,一旦你深刻理解了数学结构的代表意义,它们就不再显得任意和令人恼火,而是变得强大,因为它们的实用性变得明显。

Outlines

00:00

📚 数学的基础:加法和减法

Professor Dave 在本段中介绍了数学的起源和发展,强调数学最初是作为语言的一部分,通过模型和符号帮助我们沟通和描述周围环境。他提到,尽管数学在几百年的时间里变得极其复杂,但基础数学,如算术,是文明发展的必要工具。算术包括计数、表示数字的符号,以及用于处理数字的方法。Dave 教授解释了十进制系统的选择,尽管它是任意的,但它在我们的逻辑中根深蒂固。接着,他介绍了加法和减法的基本操作,包括它们的定义、数学表示以及如何通过这些操作解决实际问题。

05:01

🔢 数字的性质和运算规则

在第二段中,Professor Dave 讨论了加法和减法的一些基本性质。他指出加法具有交换律和结合律,即加数的顺序和组合方式不影响结果。然而,减法不具有这些性质,因为被减数和减数的顺序对结果有影响。Dave 教授还提到,尽管数学符号可能看起来抽象,但它们都有实际的物理世界根源。他的目标是使其他数学运算像加法和减法一样易于理解。最后,他强调了理解数学构造的重要性,因为理解它们代表的实际意义可以使数学看起来更有用而不是令人困惑。

Mindmap

Keywords

💡数学

数学是研究数量、结构、空间和变化等概念的一门学科。在视频中,数学被描述为起源于人类对计数和符号的需求,用于沟通思想和描述环境。数学的发展与人类文明的进步紧密相关,它从简单的计数和算术开始,逐渐演变成今天我们所知道的复杂抽象的数学体系。

💡算术

算术是数学的一个分支,主要研究数字及其基本运算,如加法、减法、乘法和除法。视频中提到,算术是人类最早发展的数学形式,它帮助人们计数和解决日常生活中的实际问题,如计算部落人数、交易商品等。

💡加法

加法是最基本的算术运算之一,它表示将两个或多个数合并成一个总和。视频中以得到两个苹果和三个苹果为例,通过加法运算得出总共有五个苹果。加法不仅是一种数学运算,也是我们日常生活中经常用到的一种计算方法。

💡减法

减法是算术中的另一种基本运算,它用于找出两个数之间的差值。视频中通过吃掉一个苹果的例子,解释了减法的概念,即从总数中去掉一部分,得到剩余的数量。减法在日常生活中的应用非常广泛,如计算剩余物品的数量。

💡数轴

数轴是一种数学工具,用于表示和比较数字的大小和顺序。视频中提到,通过数轴可以可视化减法运算,例如十四减去十一等于三,因为三是从十一到十四的距离。数轴帮助我们直观地理解数字之间的关系。

💡交换律

交换律是数学中的一个基本概念,指的是在某些运算中,操作数的顺序可以互换而不影响结果。视频中特别指出,加法具有交换律,即无论先加哪个数,结果都是相同的,例如二加三等于五,三加二也等于五。

💡结合律

结合律是数学中的另一个重要概念,它指的是在进行连续的某种运算时,无论怎样组合这些数,最终的结果都是相同的。视频中提到,加法具有结合律,无论是先加前两个数还是后两个数,最终的和都是相同的。

💡非交换律

非交换律指的是在某些运算中,操作数的顺序会影响最终的结果。视频中解释说,减法不具有交换律,即从一个较大的数中减去一个较小的数,与从小的数中减去大的数,结果是不同的,例如三减二不等于二减三。

💡非结合律

非结合律意味着在连续进行某种运算时,不同的组合方式会导致不同的结果。视频中通过五减三再减二与三减二再减五的例子,说明了减法不具有结合律,不同的计算顺序会产生不同的结果。

💡数学符号

数学符号是数学语言的一部分,用于表示数学概念和运算。视频中提到了加号(+)、减号(-)和等号(=)等符号,它们帮助我们构建数学表达式和方程,如二加三等于五,五减一等于四等。

💡数学抽象

数学抽象是指将数学概念从具体的物理世界中抽离出来,形成更为普遍和理论化的数学结构。视频中提到,尽管现代数学的前沿领域可能非常抽象,难以理解,但它们仍然与现实世界有着联系,并且具有实际应用价值。

Highlights

数学并非一开始就是一系列看似随意的运算,旨在迷惑和挫败学生。

数学的理解在过去几百年变得极其复杂。

数学起源于语言的一部分,是帮助我们沟通思想和描述环境的模型和符号。

数学创新因需求而产生,并且仍然如此。

今天的数学前沿位于抽象领域,很少有人能理解。

算术是人类最早发展的数学类型。

计数后不久,人类需要符号来代表计数数字,以及操作这些数字的方法。

文明形成后,我们需要能够跟踪库存,适当地定价物品。

许多计数系统基于十进制,尽管也有基于二十或六十的系统。

我们今天使用的数字系统基于十,这个数字是完全任意的。

加法是最基本的算术运算,代表两个数字的组合。

减法是加法的逆运算,它不是找到两个数字的和,而是它们的差。

加法是交换律的,即加数的顺序不重要。

减法不是交换律的,被减数和减数的顺序很重要。

加法也是结合律的,即连续两次加法的执行顺序不重要。

减法不是结合律的,连续减法的顺序会影响结果。

无论我们选择数学的哪个方向,无论方程看起来多么复杂,我们必须记住这些符号都有具体的含义。

本系列的目标之一是使所有其他数学运算像加法和减法一样易于理解。

Transcripts

00:00

Hey it’s Professor Dave; let’s talk about addition and subtraction.

00:10

As we have just come to understand, math never started out as a bunch of seemingly arbitrary

00:15

operations meant to confuse and frustrate students.

00:20

It is simply that our understanding of math has become extremely sophisticated over the

00:25

past few hundred years, and the fact that all of this existing and sometimes difficult

00:31

math must be learned before anyone can contribute to the field is what is responsible for the

00:37

way that so many people despise this subject.

00:42

But let’s remember that math began very simply as part of our language, a set of models

00:47

and symbols that helped us to communicate ideas and describe our surroundings.

00:55

Mathematical innovations arose by necessity, and they still do, it’s simply that the

01:00

frontier of today’s math lies in an abstract place that very few can understand.

01:06

By the end of this series, maybe we can all get there, but for now, let’s start at the

01:11

very beginning.

01:13

What was the first kind of math that was developed by the human race?

01:18

That would undoubtedly be arithmetic.

01:22

Shortly after humans were able to count, we needed symbols to represent those counting

01:27

numbers, as well as methods to manipulate those numbers in ways that represent real-life

01:33

concepts.

01:35

How many people are in the tribe?

01:38

Two kids were just born, how many are there now?

01:41

How many years has the tribe’s wisest elder been alive?

01:46

Once civilizations formed and we began to trade goods with one another, we needed to

01:51

be able to keep track of inventory, price items appropriately, and so forth.

01:58

How many apples are in the basket?

02:01

How many do you want, and how many are now left?

02:04

Being that we like to count on our fingers, and we have ten of them, many counting systems

02:09

were based on a system of ten.

02:12

There were others based on twenty, or even sixty, but the one we use today is based on

02:18

ten, so rather than diving into the anthropology of arithmetic, let’s keep our study focused

02:24

on what can be readily applied.

02:27

Presently, our conceptualization of the number ten as the basis for our numerical system

02:33

is so ingrained in our collective logic that we sometimes forget that this number is completely

02:39

arbitrary.

02:41

If we had only eight fingers, things would be totally different.

02:45

But ten is what we went with, and it works just fine.

02:49

Getting back to the apples, the basic operations we will learn first are addition and subtraction.

02:57

Addition is the most basic arithmetic operation, and it represents the combination of two numbers

03:03

to become a single number, or a sum.

03:07

If you get two apples from one vendor, and then three from another, how many apples did

03:11

you get?

03:13

Of course we can easily count the resulting pile and see that there are five.

03:18

But how do we represent this mathematically?

03:21

Using the symbols that are common of today, we would put the number two, then the addition

03:28

or plus symbol, and then the number three, followed by an equals sign and then the number

03:35

five.

03:36

This is an equation, which is a statement of equality.

03:41

The expression on the left is numerically equivalent with the expression on the right.

03:47

This particular equation reads, “two plus three equals five”, with the word plus essentially

03:53

meaning “and”, and the equals sign meaning “is”.

03:57

Two and three is five, so five is the sum of this additive operation.

04:05

The next operation that became necessary was subtraction, which is the inverse, or opposite

04:10

of addition, in that it doesn’t find the sum of two numbers, it finds their difference.

04:16

You bring all five of your apples home and you eat one of them.

04:21

If one apple has been subtracted, how many are left?

04:24

Again, it is easy to count and see that there are four left.

04:28

But this result can also be calculated, which is much different from counting.

04:33

We can write another equation with a five, then the minus symbol, followed by a one,

04:39

then the equals sign, and the number four.

04:42

This reads, “five minus one equals four”, which essentially means five less one is four,

04:49

so four is the difference between one and five.

04:52

On a number line, this is the distance between the two numbers, and this is an excellent

04:57

way to visualize subtraction.

05:00

Fourteen minus eleven is three, because three is the difference between the two numbers.

05:06

It takes three to get from eleven to fourteen.

05:11

Now that we have become familiar with the symbolic representation of these simple operations,

05:17

we should discuss some applicable properties of numbers.

05:22

Addition is commutative, in that the order in which numbers are added does not matter.

05:29

Two plus three equals five, and three plus two also equals five.

05:35

Subtraction is not commutative.

05:37

It does indeed matter which number is being subtracted from the other.

05:41

Three minus two is not the same as two minus three.

05:45

Addition is also associative.

05:48

This means that if performing two successive additions, the order in which they are performed

05:53

does not matter.

05:55

Two plus three plus four will be nine no matter which numbers we add first.

06:00

We can add the first two to get five, and then add that to four, or we can add the latter

06:05

two to get seven, and then add that to two.

06:08

The result is the same.

06:12

Subtraction is not associative.

06:14

If we write down five minus three minus two, we could do five minus three first.

06:19

That gives us two, and subtracting the other two, we get zero.

06:23

If instead we do the three minus two first, we get one, and five minus one is four.

06:29

So we can see that subtraction is not associative.

06:33

We will learn all about the order of operations later, as well as other ways in which these

06:38

kinds of properties become less obvious yet very important.

06:44

No matter how far we choose to go with math, and no matter how complicated equations appear

06:49

to be, with symbols like square roots and logarithms and integrals, we must always remember

06:56

that these symbols mean something concrete.

07:00

They are rooted in the physical world, even if in a way that is more difficult to immediately

07:05

conceptualize than the plus symbol.

07:08

One goal of this series will be to make all the other mathematical operations as intelligible

07:14

and relatable as addition and subtraction.

07:19

Once you intimately understand what mathematical constructs represent, they no longer seem

07:25

arbitrary and infuriating, but instead powerful, as their utility becomes apparent.

07:32

So let’s move forward and learn some more arithmetic, but first, let’s check comprehension.

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