Negative Numbers

Professor Dave Explains
17 Aug 201705:27
EducationalLearning
32 Likes 10 Comments

TLDRThe script explains the concept of negative numbers, using the example of owing more apples than you have. Negative numbers are positioned left of 0 on the number line, descending to negative infinity. More negative numbers represent greater debt, just as smaller bank balances indicate greater need. Rules for operating with negative numbers are covered - adding a negative number subtracts, subtracting a negative number adds. Multiplication and division with negatives may yield positive or negative results based on the sign of each factor. These principles are tied to number lines and bank accounts to provide an intuitive understanding.

Takeaways
  • 😀 When subtracting a number greater than what you have, you end up with a negative number representing how many you are short
  • 😮 Negative numbers extend the number line to negative infinity on the left side
  • 😃 Negative numbers to the left on the number line have lower values than those to the right
  • 🤔 Understanding negative numbers helps explain overdraft bank accounts
  • 😊 Adding a negative number is the same as subtracting the positive version of that number
  • 🧐 Subtracting a negative number cancels out the minus signs and acts like adding
  • 😲 Multiplying or dividing by a negative number results in a negative answer
  • 🤯 Multiplying two negative numbers cancels out the minus signs and results in a positive
  • 😃 Dividing two negative numbers works because anything divided by itself is 1
  • 🧐 Tricks help remember arithmetic identities and appreciate mathematics deeply
Q & A

  • What happens when someone wants to buy more apples from you than you actually have?

    -You could say you have a negative number of apples. For example, if you have 0 apples and someone wants 15, you could owe them 15 apples, meaning you have -15 apples.

  • How does a number line help understand negative numbers?

    -A number line continues negatively to the left of 0. This shows the increasing magnitude of negative numbers, with numbers further left being smaller (more negative).

  • How are negative numbers similar to an overdrawn bank account balance?

    -Having a negative bank account balance means owing the bank money, just like having a negative number of apples means owing apples. More negative balance = owing more money.

  • What is the effect of adding a negative number?

    -Adding a negative number is the same as subtracting the positive version of that number. For example, 5 + -2 = 5 - 2 = 3.

  • What is the effect of subtracting a negative number?

    -Subtracting a negative number is the same as adding the positive version of that number. For example, 5 - -2 = 5 + 2 = 7.

  • Why don't the minus signs simply cancel out when you subtract a negative number?

    -The minus signs don't actually cancel out - subtracting a negative number is effectively the same as adding a positive number. It's like the negative amount is being taken away.

  • What is a common source of errors when working with negative numbers?

    -Incorrectly subtracting negative numbers is a very common mistake. Students often make errors by not realizing it is equivalent to adding.

  • What happens with multiplication and division of negative numbers?

    -If one number is negative, the result will be negative. If both numbers are negative, the minus signs cancel out and the result is positive.

  • How can you easily remember rules for operations with negative numbers?

    -Viewing each negative number as -1 times the positive number can help. The -1 factors then cancel out when you multiply or divide two negative numbers.

  • Why go beyond simply memorizing rules for negative number arithmetic?

    -Memorization alone doesn't lead to a deep understanding. Making connections through tricks like factoring out -1 helps appreciate the logic behind the rules.

Outlines
00:00
😀 Introducing Negative Numbers

Paragraph 1 introduces the concept of negative numbers using the analogy of owing more apples than you have. It explains how a negative number represents having less than zero of something. The number line is used to visualize negative numbers extending left from zero to negative infinity.

😊 Comparing & Operating with Negative Numbers

Paragraph 2 explains that negative numbers to the right on the number line are greater than those to the left. This means -2 is greater than -5, unlike with positive numbers. Basic operations like addition, subtraction, multiplication and division are shown using negative numbers. Subtracting a negative number is like adding a positive number.

Mindmap
Keywords
💡subtraction
Subtraction is a basic mathematical operation for determining how much is left when some quantity is taken away. It is key to understanding negative numbers, as the professor illustrates with the example of having 10 apples but then selling 15 - requiring you to acquire 5 more just to get to 0.
💡number line
A number line is used to visually represent and compare numbers, with higher numbers farther to the right. It is crucial for grasping negative numbers, as they extend left from 0 down to negative infinity, indicating numbers less than none.
💡negative
A negative number refers to some quantity less than zero, illustrated in the transcript with dollars owed to a bank (negative balance) or needing additional apples just reach 0. Negative numbers are to the left of 0 on the number line.
💡positive
Positive numbers refer to quantities greater than 0. They reside to the right of 0 on the number line, increasing up to positive infinity.
💡addition
Adding negative numbers is equivalent to subtracting their positive counterparts. For example, 5 + -2 equals 5 - 2 = 3, as visually confirmed on the number line.
💡subtraction
Subtracting negative numbers is akin to adding their positive versions, since subtracting a debt increases your quantity. For instance, 5 - -2 equals 5 + 2 = 7 per the number line.
💡multiplication
Multiplying negative numbers results in a negative product if one factor is negative. This aligns with repeatedly adding a negative quantity.
💡division
Dividing by a negative denominator yields a negative quotient, again aligning with the logic of negative numbers.
💡arithmetic identities
Arithmetic identities refer to mathematical rules for transforming operations in valid ways, often using properties like negatives or division by 1. These identities simplify working with numbers.
💡comprehend
To comprehend something means to understand it fully. The professor checks for comprehension of negative number arithmetic by the end of the video explanation.
Highlights

We learned about simple subtraction, whereby if you have ten apples, and someone buys three, of them from you, you have seven left.

But what happens if someone wants more apples than you have? Let's say someone wants fifteen apples.

You could promise them fifteen apples, saying you’ll get five more tomorrow, so in a sense, you could say that after the transaction you have negative five apples.

This is because once you get more, you immediately owe five of them, so it’s like you will need five apples just to get to zero apples.

But on the other side of zero, if we continue subtracting, we start with negative one and we can go all the way to negative infinity.

This means that negative two is greater than negative five, which can be counterintuitive, because five is greater than two.

If you overdraft, your balance will be in the negative, and if you owe the bank ten dollars, you have more money than if you owe twenty dollars, because you will need less money to get back to zero.

Adding a negative number is the same as subtracting the positive version of that number.

Subtracting a negative number is the same as adding the positive version of that number, because the minus signs cancel each other out.

If your bank account is overdrawn by ten dollars, and the bank were to forgive or subtract that overdraft, you’d suddenly be at zero, which is ten more than negative ten.

Incorrectly subtracting negative numbers is a very common source of error for math students, so make sure you don’t make these kinds of mistakes.

With multiplication and division, if one of the numbers is negative, the answer will be negative.

However, if both numbers are negative, again the minus signs cancel each other out and the result will be positive.

Negative one times fifteen divided by negative one times three is clearly positive five, because the negative ones cancel out, since anything over itself is one.

Little tricks like this make it easier to remember all kinds of arithmetic identities, and help us go beyond simple memorization, which by itself doesn’t always lead to a deep appreciation of mathematics.

Transcripts

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