Understanding Negative Numbers The adding subtraction multiplying and
Understanding Negative Numbers • The adding, subtraction, multiplying and dividing by negative numbers is a difficult idea for many students • This presentation attempts to solve that.
A new invention • The idea of negative numbers is a ‘new’ idea, less than 2, 000 years old. • There are several ways to think of negative numbers
Number lines • The number line is one way to think of negative numbers =4 -3 -10 2+ 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 2 plus another 2 is 4 (or 2 + 2 = 4) 2 minus 5 is NEGATIVE 3 (or 2 - 5 = -3) 10
10 5 4 3 2 1 0 -1 -2 -3 -4 -5 -10 Vertical Number line • But a number line can be vertical if you want You can think of these numbers relative to the ground. Or maybe like a well The black positive numbers are an amount above the ground, the red negative numbers are below the ground. One might bury a hamster at – 3’ from the surface of the ground ‘
Money and Negative Numbers • We all understand money! • We can think of a negative amount of money as a ‘bill’, or something owing. So even after you get a positive amount of money the debt will cancel it out. I have $10 +10 Where is that $10 you owe me? ? + + -10 = 10 and negative 10 is 0
A Negative is an Opposite • A negative number is an ‘opposite’ number of a positive number • When a number and its ‘opposite’ are added together we get zero Sometimes an ‘opposite’ is also called “an additive inverse” The opposite of 5 is – 5 8 is – 8 5 plus – 5 is 0 8+ (-8) = 0 The opposite of 3 is -3 The opposite of – 7 is +7 -7 + 7 = 0
Adding a Negative Number is 10 Subtracting 5 4 3 2 1 0 -1 -2 -3 -4 -5 -10 • Adding a negative is no different than subtracting! • If I have $10 and I have to add in the fact I have a debt of $10, I really have 0 dollars. 10 plus (-10) is zero OR 10 + (-10) = 0 • If I have $20 dollars, but I have to add in the fact that I owe 5 dollars, then I really have I hope the $15. hamster was dead 20+ (-5) = 15 before it was buried! • If I bury my hamster at – 2 feet from the surface, and the frost pushes it up 1 foot, my hamster is at – 1 foot from the surface. -2 + 1 = -1
10 Practice Adding Positive and Negative Numbers 5 4 3 2 1 0 -1 -2 -3 -4 -5 2 -2 3 -2 -3 -2 10 0 -1 5 4 3 2 1 0 -1 -2 -3 -4 -5 3 – 1= 1 – 3= 4 + (-1)= -1 + (-2)= -4 + 2= -1+ 1= 0 + (-1)= -10 You try these, click the left mouse button for the answers
Another way to look at Adding and Subtracting • Instead of number lines and rules, sometimes it helps to think of markers – Sort of like the way we thing of money + + + 3 + 3=6 + + + 6 - 2=4 Six take away two
Adding and subtracting with markers • What if we made red markers Opposites? That is negative. And each red marker (negative) cancelled a green marker (Positive)? + + + - 3 + (-2) = 1 Which is the same as 3– 2=1 - -3 + (-2) = -5 Which is the same as -3 – 2 = -5 See why we say that adding a negative is the same as subtracting
Subtracting a Negative number • Subtracting a negative number is hard to imagine – You can imagine a number line and going to the left (or down) a negative amount, which really means you are going to the right (or up) – But the markers are best - - - – 5 take away – 2 Or: – 5 – (– 2) – 5 –(– 2)= – 3 The same as – 5 +2 = – 3
See that again! - - - – 5 take away – 2 Or: – 5 – (– 2) – 5 –(– 2)= – 3 The same as – 5 +2 = – 3 So taking away a negative is the same as adding
More to Follow • More to follow! • Come back for more
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