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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”.

• 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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# Pre-Algebra 4 - Whole Numbers, Integers, and the Number Line

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Chun Sang Suen posted on 2018/08/10
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