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  • Hello. I'm Professor Von Schmohawk and welcome to Why U.

  • For tens of thousands of years, people invented different ways of counting things.

  • Things such as gazelles

  • or coconuts

  • or people

  • or days.

  • In mathematics, the counting numbers are callednatural numbers”.

  • Natural numbers start with one.

  • There is no limit to the largest natural number.

  • Natural numbers do not include the number zero.

  • When people started counting things

  • it probably seemed pointless to invent a number for "no things".

  • Why would you say "The number of bananas we have is zero."

  • when you could just say

  • "Yes, we have no bananas!"

  • However, once positional notation was invented

  • a symbol to represent zero was needed as a place holder for columns containing no digits.

  • For instance, the number 2009 represents

  • two thousands

  • plus zero hundreds

  • plus zero tens

  • plus nine ones.

  • Without the zero symbol, this number could get quite confusing.

  • At some point, people started including zero along with the natural numbers.

  • The natural numbers plus zero became known as thewhole numbers”.

  • Zero is a number with a unique property.

  • When you add zero to any number the value of that number is unchanged.

  • In mathematics, anidentity elementis a number that

  • leaves the value of something unchanged when a particular mathematical operation is performed.

  • So zero is known as theadditive identity”.

  • One is also a number with a unique identity property.

  • When any number is multiplied by one its value is unchanged

  • so one is known as themultiplicative identity”.

  • The existence of a number which is an additive identity

  • and a number which is a multiplicative identity

  • is an important property for a number system.

  • Up until now, we have thought of numbers as quantities.

  • But what if we visualize numbers as distances?

  • If we think of numbers as representing distances from some point

  • then we can arrange the numbers on a line like the numbers on a ruler.

  • The point from which the distances are measured is called theorigin”.

  • It makes sense to place the number zero at the origin

  • since it represents zero distance from that point.

  • We must now choose some distance for the number one.

  • This distance is called theunit distance”.

  • Every whole number then corresponds to a multiple of that unit distance.

  • This way of representing numbers is called a “number line”.

  • Since there are an infinite number of whole numbers

  • we place an arrow on the right end of the number line

  • to show that it goes on forever in that direction.

  • The natural numbers and the whole numbers

  • both can be represented as points on this number line.

  • Addition can be thought of as adding distances on the number line.

  • For example, adding one unit distance

  • to one unit distance

  • gives us a distance of two units.

  • Adding a distance of three units

  • to a distance of four units

  • gives a distance of seven units.

  • Likewise, if we subtract a distance of four units from a distance of seven units

  • we get a distance of three units.

  • When any two whole numbers are added

  • we always get another whole number.

  • Therefore, we say that whole numbers areclosedunder the operation of addition.

  • A group being closed under some operation

  • means that the operation will always create a result

  • which is also a member of that same group.

  • But are the whole numbers closed under subtraction?

  • If you subtract a larger whole number from a smaller whole number

  • there is no whole number which can represent the result.

  • This is because we would need a negative number to represent the result

  • and whole numbers do not include negative numbers.

  • Therefore the whole numbers are not closed under subtraction.

  • However, if we expand our collection of numbers to include negative numbers

  • then we can always find a number to represent the result

  • of any addition or subtraction operation.

  • These whole numbers which can be positive, negative, or zero are calledintegers”.

  • No matter how we add or subtract integers

  • the result can always be represented by some integer.

  • Therefore integers are closed under both addition and subtraction.

  • You may be wondering what a negative number actually means.

  • As recently as the 18th century

  • negative numbers were not accepted as legitimate numbers by many mathematicians.

  • It was thought that only positive numbers represented things in the real world.

  • However, the idea that negative numbers

  • don't actually represent anything in the real world is debatable

  • as anyone who has ever overdrawn their bank account can tell you.

  • Someone who owes more money than they have

  • could be thought of as having less than zero money

  • or having a negative net worth.

  • Death Valley is below sea level

  • so the altitude of Death Valley could be thought of as a negative altitude.

  • A vacuum cleaner creates an air pressure which is less than atmospheric pressure

  • so it can be thought of as creating a negative pressure.

  • Integers can be represented on a number line just like natural numbers and whole numbers

  • but now the number line must go off to infinity in both directions.

  • With a positive or negative sign

  • a number can be thought of as representing not only a distance, but also a direction.

  • Just as a positive number can be thought of

  • as representing a distance to the right of the origin

  • a negative number can be thought of as representing a distance to the left.

  • Adding a positive integer means moving that number of units to the right.

  • For example, if we add a positive integer to the number two

  • we start at two on the number line

  • and then move that number of units to the right.

  • Adding a negative integer means moving that number of units to the left.

  • In fact, adding a negative number is exactly the same thing as subtracting a positive number.

  • So we can think of subtraction as just the addition of a negative number.

  • For example, the problem

  • two

  • plus three

  • minus six

  • minus two

  • plus four

  • are instructions to start at two on the number line

  • then move to the right three units

  • then move to the left six units

  • then left another two units

  • and finally to the right four units.

  • At the end of the journey you will be at the one position.

  • So far we have seen that adding a positive integer

  • means moving that number of units to the right.

  • Subtracting a positive integer

  • means moving that number of units, but in the opposite direction.

  • We have also seen that adding a negative integer

  • means moving that number of units to the left.

  • So subtracting a negative integer must mean to move that number of units to the right.

  • Subtracting a negative number is the same as adding a positive number.

  • The distance from a number to the origin

  • is called itsmagnitudeorabsolute value”.

  • For instance, the numbers positive three and negative three

  • have opposite signs but the same magnitude

  • since they are located the same distance from the origin.

  • If you take any number, positive or negative

  • and add a number of the same magnitude but the opposite sign

  • the result will be zero.

  • This number of equal magnitude and opposite sign is called the number's “additive inverse”.

  • Any number plus its additive inverse is zero.

  • For example, the additive inverse of positive three is negative three

  • and the additive inverse of negative three is positive three.

  • With the invention of integers, we now have a much more powerful number system.

  • Since the integers are closed under addition and subtraction

  • we can represent the result of adding or subtracting any numbers in our system.

  • However, as we shall soon see

  • there are still some operations which cannot be represented using only integers.

Hello. I'm Professor Von Schmohawk and welcome to Why U.

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B1 中級 美國腔

代數前4 - 整數、整數和數位線 (Pre-Algebra 4 - Whole Numbers, Integers, and the Number Line)

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    Chun Sang Suen 發佈於 2021 年 01 月 14 日
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