Positive And Negative Numbers.

football is a game of games and losses let's suppose that in two consecutive plays a team moves forward for a gain of five yards then is thrown back for a loss of three yards directions on these two plays we can call a gain of five yards positive 5 yards and a loss of three yards negative 3 yards positive and negative numbers like these are called signed numbers and they can be used to indicate direction or quality to avoid confusing these signs of quality with addition or subtraction signs signed numbers are sometimes enclosed in parentheses now let's go back to the football game for a moment and think about the counting numbers you used in arithmetic we can picture numbers such as these as corresponding to points on a line to illustrate the use of this number line we can do some very simple arithmetic operations to show 2 + 4 we count off two units then four more 1 2 3 4 the answer is 6 now a simple subtraction illustration 5 - 2 start at five and count back two units 1 2 and there's the answer three but suppose the problem is 2 - 5 for instance if a team is penalized 5 yards after a gain of two yards how would you show this on the number line you'd start at two and count back five but you can see that this is not possible with just the numbers of arithmetic how can it be done by enlarging the system of numbers to do this numbers are added to the number line that are less than zero counting to the left these numbers to the left of zero are called negative while the numbers to the right are now called positive this enlarged system is the system of positive and negative numbers movement to the right is positive and numbers become larger movement to the left is negative and numbers become smaller remember these are not signs of operation as the plus and minus signs used in arithmetic these positive and negative signs are signs of direction or quality the value of a number without regard to its sign is called its absolute value and may be written this way between vertical bars the absolute value of a number shows its distance from zero but not which side of zero the number is on let's consider the rules for using positive and negative numbers by reading a thermometer we can illustrate the rules for adding signed numbers you can think of a thermometer as a vertical number line for temperatures above zero we use positive numbers for temperatures below Z we use negative numbers let's suppose that the column moves up from 0 to 20° now it moves up 10 more de we find the overall change by adding the absolute value 20 to the absolute value 10 the answer is 30 the total distance moved 30 is an absolute value written between vertical bars because both movements were in the positive direction the result is 30 above zero so the sum of postive 20 and pos1 is POS 30 in this second example the column moves in the negative Direction down 10° from zero and then 10 more degrees in the same direction you can see that the total movement is 20° a -2 a -10 plus a -10 equals a -20 now you can see one rule for adding signed numbers in these two examples we added the absolute values of the numbers and kept the same sign in the answer so the sum of two positive numbers is a positive number the sum of two negative numbers is a negative number now how do we add numbers with opposite signs indicating opposite movement on the number line if the column moves 20° up from zero then 10 de down the total movement is 30° but we know that the net change from 0 is only 10° in the positive direction a positive 10 let's take another example if the column moves 20° down from zero then up 10° 10° in the negative Direction represents the net change from zero a negative 10 so to add a positive and a negative number you find the difference between their absolute values keeping the sign of the number with the larger absolute value to understand how we subtract signed numbers let's recall what subtraction in arithmetic really means we can think of this problem as meaning that from five objects two are to be taken away leaving three we can also think of this problem in another way asking what is 5 - 2 is the same as asking what number added to two equals 5 in other words 5 is equal to 2+ three the answer was found by thinking of subtraction in terms of addition this relation between addition and subtraction is especially useful when using signed numbers let's work this same subtraction problem using the number line to find the difference between pos5 and pos2 locate the numbers on theine line then count the number of units from pos2 to POS 5 the difference is three units in the positive direction a positive three now let's find the difference between two numbers in a different way what number when added to pos5 gives the answer pos3 a -2 two units in the negative Direction so subtracting a positive two produce the same result as adding a negative -2 which leads to this simple rule for subtracting signed numbers change the sign of the subtend and then proceed as in addition there's another way to think of this Rule positive2 and -2 have the same absolute value but different signs such numbers are called additive inverses or opposites in this case positive2 is the additive inverse of -2 and vice versa to subtract positive2 we added its opposite -2 so we can state the rule for subtraction in another way to subtract assigned number you add its opposite now let's look at this simple problem in subtraction in a metic this would mean from four objects seven are to be taken this is physically impossible but as you've seen a greater number can be subtracted from a smaller one by using signed numbers you change the sign of the subtend and add so to positive4 you add seven units in the negative Direction the answer is -3 so we've seen the number line used to demon demonstrate the rules for adding and subtracting signed numbers now let's see what the rules are for multiplying signed numbers we'll let a railroad represent a number line we'll let the zero point on the number line be represented by a city we'll think about three elements in solving the problems that follow speed time and distance we'll call distance east of the city positive and distances west of the city negative we'll think of a train speed East as positive we'll think of a train speed West as negative the third element is time we'll call the time positive after a train leaves the city going in either direction the light area on the right half of the clock will represent positive time we'll call the time negative before a train arrives at the city coming from either direction the light area on the left half of the clock will represent negative time the first problem will deal with just positive numbers a train going east has been averaging 30 mph since it left the city 2 hours ago what is its position the 30 and two are both positive because speed is to the East and time is hours after departure if the train has been traveling East at 30 mph for 2 hours it is 60 Mi east of the city distances east of the city are positive so positive 30 time pos2 equal POS 60 a positive number multiplied by a positive number equals a positive number now let's deal with two negative numbers suppose a train moving West will average 30 mph until it arrives at the city in 2 hours what is its present position the 30 is negative and the two is negative this is because the speed is in a negative direction to the west and the time is negative before arrival at the city so the train is here 2 hours before arriving 60 m east of the city the answer is positive 60 again this illustrates a second rule the product of two negative numbers is a positive number now let's multiply two numbers with unlike signs a train going west has been averaging 30 mph since it left the city 2 hours ago because movement is West the 30 is negative the two is positive because it represents time after leaving the city now what is the Train's present position in 2 hours the train will be 60 M west of the city -60 is the answer the product of a negative and a positive number is a negative number now let's change the problem to reverse the signs of the 30 and 2 a train is traveling East and will average 30 mph for 2 hours until it reaches the city because the speed is Eastward 30 is positive but two is negative this is because the two represents hours before arrival now what is the Train's position at this time 2 hours before arrival it is 60 m rest of the city ative 60 a positive number times a negative number equals a negative number so now we can State the general rules for multiplying two signed numbers when two numbers with like signs are multiplied the product is positive when two numbers with unlike signs are multiplied the product is negative the rules for division are the same to show this we make use of the fact that multiplication and division are inverse operations for example 60 / 2 = 30 because 30 * 2 = 60 you saw that when the 30 and 2 were positive the answer was positive 60 changing the problem to division you can see that the quotient of two positive numbers is positive because the product of a positive and negative number is negative the quotient of two negative numbers is positive because the product of two negatives is a positive the quotient of a positive and negative number is negative changing this fourth example to a division problem again illustrates that the quotient of two numbers with unlike signs is negative now let's review the rules for working with signed numbers when adding two numbers if they have like Signs add the absolute values keeping the same sign if the numbers have unlike signs keep the sign of the number with a greater absolute value and find the difference between the absolute values to subtract signed numbers mentally change the sign of the subah hand and proceed as in addition the rules for multiplication are these when multiplying two numbers with like Signs the product is positive when multiplying two numbers with unlike signs the product is negative the rules for division are the same as those for multiplication the quotient of two numbers with like Signs is [Music] positive and the quotient of two numbers with unlike signs is [Applause] negative your understanding of positive and negative numbers and the rules that govern their use is basic to your progress in mathematics [Music]