Topic 12: Directed Numbers



Topic 12: Directed Numbers12.1 What is an Integer?An integer is any positive or negative whole number and 0 such as 1, -2, 10, 7, 216, -155. Integers have a direction, positive or negative, therefore they are also known as directed numbers.Positive (+) Integers are all around us, we use them every day and all the time.She is 40 years old; This bottle contains 500ml of water; He is 156cm tall; There are 365 days in a year, unless it is a leap year in which case there will be 366 days and February will have 29 days instead of 28.A positive number is a number above 0. We may denote a positive number by a plus sign in front of the number, however it is common to put no sign at all: 5, 12, 254Negative (?) Integers ?? ……..we need to think about this one!Bank Accounts – Below €0Temperature – Below 0?C Elevators – Below ground level Depth – Below sea level Golf – Under parA negative number is a number that goes below 0. We denote a negative number by a minus sign: -2, -5, -1912.1.1 The Number LineWe can illustrate integers on a number line. It helps us understand positive and negative numbers better!-46672565405SMALLER00SMALLER3219450869950BIGGER00BIGGER12.2 Adding and Subtracting IntegersThe number line helps us to add and subtract any two integers. The 1st number is your starting point The sign of the 2nd number tells you the direction (positive or negative) The 2nd number tells you how many steps in that direction you must takeLet’s say you have 3+4, now we know that the answer to this sum is 7 because we are used to adding positive integers. Put your finger on the first number ‘3’ and move ‘4’ steps in the positive direction. Where do you land? 7!+ 3+4=7What about -6 + 3 ? Put your finger on -6 and move 3 steps in the positive direction. Where do you land? -3!-6+3=-3What about 3-7? Put your finger on 3 and move 7 steps in the negative direction. Where do you land? -4!3-7=-4What about -7-3? Put your finger on -7 and move 3 steps in the negative direction.Where do you land? -10!-7-3=-10In SummaryBoth negative? Answer will be negative -3-4=- 7 (keep walking in the negative direction)Both positive? Answer will be positive +3+4=+7 (keep walking in the positive direction)Different signs? If the bigger number has a negative sign the answer will be negative, if the bigger number has a positive sign the answer will be positive, always find the difference between the numbers and put the sign: -3 + 4 = 1 -4 + 3 = -1Such questions may also be reasoned out by considering debt which is money that you owe. Consider the negative numbers as money that you owe and the positive numbers as money that you have in your pocket:-6+3=-3 I am 6 euros in debt (-6) and I have 3 euro in my pocket (+3), I can pay 3 euro but I will still be 3 euro in debt (-3)3-7=-4 I have 3 euros in my pocket (+3) and I am 7 euro (-7) in debt, I can pay 3 euro but I will still be 4 euro in debt (-4)-7-3=-10 I am 7 euro in debt (-7) and another 3 euro in debt (-3), so in all I am 10 euro in debt (-10)12.2.1 When the Signs are Touching4705350600710Don’t not go (--)means GO (+)00Don’t not go (--)means GO (+)Sometimes, we might have two signs next to each other. In such a case these two signs must become one sign and then we work out the sum as in section 12.2 above.2562225333375002552065952500Same signs become a positive: 4 - - 4 = 4 + 4 = 8 4 ++ 4 = 4 + 4 = 84712970251460Don’t go (-+)means don’t (-)0Don’t go (-+)means don’t (-)28575003365500028575001460500Different signs become a negative:4+ - 4 = 4 - 4 = 0 4 -+ 4 = 4 - 4 = 0 Examples:a) 4-- 3=4+3=7b) -6-+5=-6-5=-11c) 4+- 3=4-3=1 d) 3+ - 9=3-9=-6e) 2+ - 9-1=2-9-1=-7-1=-812.3 Multiplying and Dividing IntegersWhat does 2 x 4 mean? (we, know that the answer is 8)2 for 4 times 2 + 2 + 2 + 2 = 8or 4 for 2 times 4 + 4 = 8 Here we are remembering that multiplication is repeated additionSo what does -2 x 4 mean?-2 for 4 times - 2 - 2 - 2 - 2 = -8-2 × 4 = -8Therefore, we can say that multiplying a negative number by a positive number and vice versa will give a negative answer21805906350 - × + = -+ × - = -0 - × + = -+ × - = -So, using this new information we are now able to multiply:Positive x Positive = PositivePositive x Negative = NegativeNegative x Positive = Negative39014402740660+3 300+3 338862002329180+3 300+3 33863340808990+3 00+3 3855720412750+3 300+3 338862001959610+3 300+3 338709601578610+3 300+3 338557201205230+3 300+3 336576002569210003627120220091000360045018649950035814001541145003537585122301000351091587630000350139054165500So, we just need to figure out what answer a Negative x Negative will give. Let us consider the ‘negative’ 3 times table: 3 ×-3=-9 2 × -3=-63 1 × -3=-30 ×-3=0-1 × -3=3-2 × -3=6-3 ×-3=9 -4 × -3=122162175762000 + × + =+ - × - =+0 + × + =+ - × - =+In this pattern we are adding 3 each time, so we can see that multiplying two negative numbers will give us a positive answer. There is a more complicated explanation for why this is so, but for the moment we shall allow this pattern to suffice.In summary therefore we can say:Multiplying numbers having SAME SIGNS gives a POSITIVE ANSWERMultiplying numbers having DIFFERENT SIGNS gives a NEGATIVE ANSWERThese same rules apply to dividing directed numbers as division is repeated subtraction:-4 ÷ 2 means divide -4 between 2 people then each person will get -2. If two people are 4 euro in debt (-4) and they share the debt equally they each need to give 2 euro (-2).In conclusion, to multiply or divide directed numbers, you must multiply or divide the two numbers normally to get the answer and then you must decide what sign to put in front of the answer - remembering that same signs give a positive answer whilst different signs give a negative answer.Examples:a) -8 × -2 = 16 -?×?-?=?+ b) -8 × 2 = -16 (-?×?+?=?-)c) -24 ÷ -6 = 4 (-?÷?-?=?+) d) 24 ÷ -6 = -4 (?+?÷?-?=?-)To conclude, here is a summary on operations with negative numbers:Topic 13: Introducing Algebra172974031623000Remember when in junior school you used to fill in the empty boxes?1729740143065500In algebra, the same thing happens but we use letters instead of boxes to hold a place for numbers that we do not know yet.Algebra is when we use a letter to represent (stand for) a numberBut why use a letter when we can use a number?Unfortunately, we can’t always put a number or value to an amount, so that’s when we use letters. There are 12 eggs in this box! There are x smarties If I take a smartie, there will be x-1 smarties-18097566738500Consider this container of paperclips, we can’t tell how many paper clips there are in all, so we use a letter to represent this number. We can however, use this letter to say things about the paper clips:Let’s say there are n paper clipsIf I take 2 paper clips out, there will now be: n-2 paper clipsIf I add 10 paper clips to the n we started with there will now be: n+10 paper clipsIf I add another container with the same amount of paper clips, there will be: n+n = 2 × n = 2n paper clipsn-2, n+10 and 2n are called expressions. Expressions are a combination of numbers (1,2,3..), letters (a,b,c…) and operators (+,-, ×..) WITHOUT an EQUALS sign. Expressions represent a number.Important Points on ExpressionsWhen a letter and a number are next to each other, there is a ‘hidden’ × (times) n?+?n?+?n?+?n?=?4?×?n? which we write as 4nIn expressions it is common practice to write the number in front of the letter not the other way round: 4n NOT n4 13.1 Collecting like termsA term is part of an expressionLike terms contain the same letterYou can simplify an expression by collecting like terms:y + y + y= 3yb + b + b +b +b= 5ba + a + a +b +b= 3a+2bg+g-g-5r=g-5rNote that, when there is no number in front of a letter as in the last example, it means that there is a hidden ‘1’ g = 1g When the expressions get longer, simplifying by collecting like terms gets trickier. I would suggest to circle, underline or mark the like terms making sure to include the signs. Collect each set separately, then put the answers to each set together to formulate the simplified expression.1447800369570002790825379095003752850631190330517532194500180975033147000Example 1: Simplify the expression 5a + 3b + 3 – 2a + b - 5 5a + 3b + 3 – 2a + b - 5left118300500-9525056388000-1524003067050023907759525a terms 5a – 2a = 3ab terms 3b+ b = 4bnumber terms 3- 5 = -2Ans: 3a + 4b – 2399097554610032385005556251590675545465367665133655100286702531750000200025032702500124777528384500Example 2: Simplify the expression 4h + 3-h+3f-9h-5+h-1 4h + 3 -h +3f -9h -5 + h- 1 19050116840000-7620029210000h terms 4h-h-9h+h = -5h f terms just 3f→nothing to collect!number terms 3- 5-1 = -3Ans: -5h+3f – 313.2 SubstitutionPicture a football match. We have players on the pitch and players on the bench. At any time during the game the coach may choose to substitute a player. What does this mean? It means that a player on the bench will take the place of a player on the pitch.Substitution in Maths is exactly the same, only instead of players on the pitch we have letters and instead of players on the bench we have numbers. In substitution we replace letters by given numbers to find the value of an expression.Example 1: Evaluate the expression 2y + 7 when y=4The letter y goes out and is replaced by the number 4. Now, remember that when a number and a letter are next to each other there is a hidden times.2y + 72(4) + 72×4 + 78 + 7Ans: 15Example 2: Evaluate the expression 3a-2b when a=10 and b=4The letter a goes out and is replaced by the number 10 and the letter b goes out and is replaced by the number 4. Now, remember that when a number and a letter are next to each other there is a hidden times.3a-2b310-2(4)3×10-2×4 30- 8Ans: 22Example 3: Evaluate the expression pq-4 when p=7 and q=2When a letter and a letter are next to each other there is a hidden times too.pq-4(7)(2)-47×2-4 14-4Ans: 1013.3 EquationsAn equation is like a balanceAn equals sign tells us that one side balances the other side exactlyTo find the value of x, we must get it on its own on one side of the balance (equals)To get x on its own we must add or remove the same amount from both sides to keep the scale in balanceIn the example above, to get x on its own we must remove 3 from the left hand side (LHS). To keep the scale in balance, we must also remove 3 from the right hand side (RHS). We are then left with x on the LHS and 2 on the RHS, which implies that x=2. We have solved an equation!This year we will be solving simple equations which we can categorize into three types. Let us look at how to solve each type in turn using the balancing method.Equation Type 1A + 2 = 7A + 2 - 2 = 7 – 2A = 5Equation Type 23x = 123x÷3 = 12÷3x=4Equation Type 3 = Type 1 + Type 22B + 1 = 5(Type 1) 2B + 1-1 = 5-12B = 4(Type 2)2B÷2 = 4÷2B = 2More examplesType 1Type 2Type 3m + 4 = 11m + 4 – 4 = 11 – 4 m = 72b = 62b ÷ 2 = 6 ÷ 2 b = 32e + 3 = 72e + 3 – 3 = 7 – 3 2e = 4 2e ÷ 2 = 4 ÷ 2 e = 2p – 6 = 10p – 6 + 6 = 10 + 6 p = 163h = 93h ÷ 3 = 9 ÷ 3 h = 34c – 5 = 114c – 5 + 5 = 11 + 5 4c = 16 4c ÷ 4 = 16 ÷ 4 c = 4To solve an equation, you always want to get the letter(unknown value) on its own. To do this you must get rid of the numbers attached to the letter by doing the reverse operation (- if +, + if -, ÷ if x). This reverse operation must be done to both sides, to keep the equation in balance.The best thing about solving equations is that you can always check whether your answer is correct.How?By using substitutionWhen we get our answer we can substitute it into our starting equation and check whether the equation balances. p – 6 = 10 p = 16 16 – 6 = 10 3h = 9 h = 33 x 3 = 9 4c – 5 = 11 c = 44 x 4 – 5 = 1116 – 5 = 11 ................
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