# use integers and rational numbers

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Grade 8 Algebra 1 Lesson Thu. 9 Sept. 2009 Today’s Big ideas How do you compare numbers, and how do you calculate with negative numbers? Lesson 2.1 (plus a bit)

Use Integers and Rational Numbers
Vocabulary Natural (or Counting) numbers 1, 2, 3, . . . Whole numbers 0, 1, 2, 3, . . . Integers . . . , -2, -1, 0, 1, 2, . . . Rational numbers – any number that can be written as the fraction a/b, where a and b are any integers as long as b ≠ 0. This means that a rational number must be either a repeating or a terminating decimal. Irrational numbers – numbers that cannot be written as the ratio of two integers. This means that irrational numbers must be non-repeating, non-terminating decimals. Real numbers - the combined set of rational and irrational numbers.
Real Numbers

Rational Numbers         ‐3/4         25/7                           0.333…  Integers       . . . , ‐3, ‐2, ‐1  Whole Numbers      0  Natural Numbers  1, 2, 3, . . .  Irrational Numbers

0.23223222322223…

The Number Line

1 3 Numbers to the right are greater than numbers to their left e.g. −2 > −3 and - 5 > −5 2 4
Opposites – two numbers that are the same distance from zero on the number line, but on opposite sides of it e.g. 3 and -3; -x and x € Note: opposites are actually additive inverses of each other. a number + its additive inverse = the additive identity (0) e.g. Zero is called the additive identity because when you add it to any number, it doesn’t change that number’s value (identity). This same concept holds for multiplication. The multiplicative identity is 1 because if you multiply any number by 1, that number’s value (identity) remains unchanged. Also, a number x its multiplicative inverse = the multiplicative identity (1) e.g. Absolute value – the absolute value of n, (note that a distance is never negative). E.g. , is the distance n is from zero on the number line

− 7 = 7 and

7 =7

Conditional Statement – has a hypothesis and a conclusion

€ e.g. if a is a positive number, then a = a
hypothesis

conditional statement

conclusion € If even one example (called a counter-example) satisfies the hypothesis, but the conclusion is false, then the conditional statement is false. Lesson 2.2

If the numbers have the same sign, add their absolute values and keep that sign. If the numbers have different signs, find the difference in their absolute values and use the sign of the one with the larger absolute value. (c.f. Martians and Terrans) Examples
Same signs Different signs 6 + 4 = 10 − 6 + 4 = −2 − 6 + −4 = −10 6 + −4 = 2

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More Properties Addition is commutative i.e. a + b = b + a Addition is associative i.e. a + (b + c) = (a + b) + c
€ e.g. 5 + -3 = -3 + 5 e.g. - 4 + (7 + 3) = (-4 + 7) + 3

Lesson 2.3

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Subtracting Real Numbers

Change the problem to adding the opposite. i.e. a − b = a + −b Examples
6 −11 = 6 + −11 = −5 −6 −11 = −6 + −11 = −17 6 − (−11) = 6 + 11 = 17 −11− (−6) = −11+ 6 = −5

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Note that subtraction is neither commutative nor associative.
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Homework • Sheets 1.1, 1.2 and 1.3 – odd numbers only • Kaleidoscope due next period (Monday)

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