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Operations on Real Numbers (positive and negative),[object Object],Real numbers include numbers on both sides of the number line.  We need to be able to perform arithmetic operations ( +, -, x, ÷) on all real numbers, whether they are positive or negative. ,[object Object]
Positive and Negative Numbers,[object Object],    Remember that on the number line, values increase in positive value when we move right, and numbers increase in negative value when we move left.,[object Object],  -3  -2 -1  0   1   2   3  ,[object Object],If we think of positive and negative numbers in terms of money, we can consider positive value as assets and negative value as debt.  When we accumulate more assets we have more money, but accumulation of debt sends us further into the negative direction.,[object Object]
Add a Positive Amount,[object Object],     So, if we are adding a positive amount to a number, we move to the right along the number line. ,[object Object],-5  -4   -3  -2   -1     0     1     2    3     4     5,[object Object],3 + 1.5 = 4.5,[object Object],We start at the point representing 3 and move 1.5 units to the right,[object Object],-5  -4   -3  -2   -1     0     1     2    3     4     5  ,[object Object],-5 + 2 = -3     ,[object Object],We start at the point -5 and move 2 units to the right.  We are still in debt, but moving in the right direction.,[object Object]
Add a Negative Amount,[object Object],     If we add a negative amount to a number, we move that number of units to the left.,[object Object],-6   -5  -4   -3  -2   -1    0     1     2    3     4     5     6,[object Object],5.5 + (-4) = 1.5,[object Object],We start at the point 5.5 and move 4 units to the left,[object Object],-6  -5  -4   -3  -2   -1    0     1     2    3     4     5     6 ,[object Object],-2 + (-4) = -6   ,[object Object],We start at the point -2 and move 4 units to the left,[object Object]
Rules for Addition of Real Numbers,[object Object],We can make some general rules so we don’t have to draw a number line for every problem:,[object Object],When adding numbers with like signs, we add their absolute values and use the sign of the original addends.,[object Object],			2.5 + 3 = |2.5| + |3| = 5.5,[object Object],		-1 + (-4) = - (|-1| + |-4|) = - (1 + 4) =  -5,[object Object]
Rules for Addition of Real Numbers,[object Object],When adding numbers with different signs, find the difference of their absolute values and use the sign of the addend with the greater absolute value. ,[object Object],	     -5 + 6 = + (|6| - |-5|) = + (6 – 5) = 1,[object Object],    37 + (-42) = - (|-42|- |37|) = - (42 – 37) = -5,[object Object]
Subtraction,[object Object],    Subtraction is the opposite of addition, so we can restate any subtraction problem as adding the inverse (additive inverse) of the number.  This is helpful when we are working with negative and positive numbers. ,[object Object],	5 – 2 = 5 + (-2) = 3,[object Object],	-1.25 – 2.5 = -1.25 + (-2.5) = -3.75,[object Object],	-1 – (-1.3) = -1 + 1.3 = 0.3,[object Object]
Multiplication ,[object Object],We know that when we multiply two positive numbers we get a positive result.,[object Object],				2 • 4 = 8,[object Object],But what if one or both of the numbers are negative?,[object Object],Think in terms of money…. If $2 is deducted from your bank account 4 times, you are down $8. ,[object Object],(-2) • 4 = -8 ,[object Object],However, if that deduction were reversed 4 times, then you would be ahead $8.,[object Object],(-2) • (-4) = 8,[object Object]
Division,[object Object],Here is an example of division involving negative numbers ,[object Object],Let’s say you have a debt of $30, represented by -30 ,[object Object],Now, two good friends come along and offer to help pay the debt and split it 3 ways,[object Object],      (-30) ÷3 = -10          You are still in debt, but only $10 now.  ,[object Object],Now, if the debt is reversed as well as divided, the problem looks like this:,[object Object],      (-30) ÷ (-3) = 10       The result is a credit to each  rather than debt.,[object Object]
Generalization,[object Object],We can make some general rules for multiplication and division of real numbers. ,[object Object],If the signs are the same (both positive or both negative) we get a positive result. ,[object Object],		2 • 2 = 4   		(-2) • (-2) = 4  ,[object Object], 		4 ÷ 2 = 2   		(-4) ÷ (-2) = 2,[object Object],If the signs are different, then the result is negative. ,[object Object],		2 • (-2) = -4		(-2) • 2 = -4,[object Object],	4 ÷ (-2) = -2		(-4) ÷ 2 = -2,[object Object]
Multiple factors,[object Object],What if we have more than two factors ?,[object Object],(-2)  • (-3) • (    ) • (-5) • (6) ,[object Object],Multiplying the first two negative numbers results in a positive 6.,[object Object],6,[object Object],Now we have 2 positives that also result in a positive value.,[object Object],The next two result in negative,[object Object],--,[object Object],Final answer is negative,[object Object],-45,[object Object],Note that if we have an odd number of negatives, the result will be negative and if we have an even number ( or zero) negatives, the result will be positive,[object Object]
Impact on Exponents,[object Object],     What happens when we raise a negative number to some power?,[object Object],            = (-2) • (-2) • (-2) • (-2) • (-2) = -32    ,[object Object],we have an odd number of negative factors,[object Object],           = (-2) • (-2) • (-2) • (-2) • (-2) • (-2) = 64  ,[object Object],we have an even number of negative factors.,[object Object],     To generalize:  When raising a negative number to a power, if the exponent is odd, the result will be negative; if the exponent is even, the result will be positive.,[object Object]
Practice,[object Object],    Now, let’s combine what we know about Order of Operations with what we know about operations on positive and negative numbers.,[object Object],First we perform operations within Parentheses,[object Object],Next, resolve the Exponents,[object Object],Multiply and Divide from left to right,[object Object],Add and Subtract from left to right,[object Object],-,[object Object],-(64) – 4(-27) ÷ 9,[object Object],-64 + 108  ÷  9,[object Object],-64 +  12 ,[object Object],-52,[object Object]

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