Chapter 2 Rational Numbers notesjohnstonsd36.weebly.com/uploads/2/1/3/3/21338878/chapter... ·...
Transcript of Chapter 2 Rational Numbers notesjohnstonsd36.weebly.com/uploads/2/1/3/3/21338878/chapter... ·...
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2.1 Comparing and Ordering Rational Numbers
Rational Numbers: formed when one integer is divided by another integer
where
Rational numbers can be positive, negative, or zero and include integers, fractions, mixed numbers and many decimals.
For example, , , ,
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Ordering Rational Numbers
Rational numbers can be compared by placing them on a number line. Larger rational numbers are to the right, smaller rational numbers are to the left.
Example: Place -4, 3, -0.5 and 1/2 on the number line. Order the numbers from smallest to largest.
Comparing Fractions
• Fractions can be compared by finding a common denominator.
• Once you have the same denominator, then the fraction with the largest numerator is the largest rational number.
Example #1: Which is larger: 2/3 or 3/4?
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Example #2: Which is larger: -1/2 or +1/4?
Example #3: Which is larger: -1/2 or -1/4?
Comparing Decimals
• When comparing decimal numbers, first look at the portion of the numbers to the left of the decimal point (the whole numbers).
• If the whole numbers are different, then the decimal with the larger whole number is larger!
Example #1: Which is larger: 42.15 or 32.23?
• If the whole numbers are the same, compare the numbers to the right of the decimal point.
• To be ‘fair’, they have to have the same number of decimal places. If one number has less decimal places,
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add zeros to right of the number until all numbers have the same number of decimal places.
• The decimals can then be compared to find out which one is larger.
Example #2: Which is larger: 42.153 or 42.16?
Example #3: Which is larger: -3.254 or -3.23?
Comparing Decimals to Fractions:
• If you are asked to compare a decimal to a fraction, you can convert the fraction to a decimal first, and then compare your numbers.
Example #1: Which is larger: 0.9 or 7/8?
Example #2: Which is larger: 2.7 or 2 2/3?
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2.2 Problem Solving with Rational #’s in Decimal Form
Rules for operations positive and negative rational numbers are the same as for positive and negative integers.
• To add rational numbers of with the same sign, add the absolute values. The sum has the same sign.
+2.2 + 3.2 (-3.62) + (-7.21)
• To add rational numbers with different signs, subtract
the smaller absolute value from the larger absolute value. The sum has the same sign of the number with the larger absolute value.
+7.83 + (-2.21) (-9.1) + (+6.3)
• Subtracting a rational number is equivalent to adding
its opposite (-6.82) – (+2.51) (-2.3) – (-3.7)
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• The product/quotient of 2 rational numbers having the same sign is positive. The product/quotient of 2 rational numbers having different signs is negative.
(-6.2) x (-3.1) (-8.4) ÷ (+2.1)
ORDER OF OPERATIONS
- just like with integers, the order of operations for rational numbers follows PE(MD)(AS):
- perform operations inside parentheses () first - divide and multiply in order from left to right - add and subtract in order from left to right
Solve:
6.2 ÷ 3.1 + 6.2 x (-3.0) 2.5 + 5 x (3.32-6.22)
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PROBLEM:
A hot air balloon climbs at 0.8 m/s for 10 s. It then descends at 0.6 m/s for 6 s.
a) what is the overall change in altitude? b) What is the average rate of change in altitude?
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2.3 Problem Solving with Rational #s in Fractional Form
• Rational numbers expressed as proper or improper fractions can be added, subtracted, multiplied, and divided the same way as positive fractions.
• The sign rules for integer operations also apply to rational numbers expressed as fractions.
a) b)
c) d)
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Problem: Tim had $50. When he went to the mall, he spent 1/5 of the money on food, another 1/2 on clothes, and 1/10 on bus fare. How much money does he have left at the end of the day?
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2.4 Determining Square Roots of Rational Numbers
If the side length represents a number, then the area of the square models the square of that number.
If the area of the square represents a number, then the side length of the square models the square root of that number.
Study the table below of perfect squares.
Thus because ____________ The perfect square of 8 is ____ because _____________
Perfect Squares
Factors Square Root
1 4 2x2 9 3x3 16 4x4 25 5x5 36 6x6 . .
100 10x10
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A perfect square is the product of ___ equal __________ The square root of a _________ square can be determined exactly. The square root of an _________ square can only be approximated (the answer on your calculator is rounded) Determine if the following numbers are perfect squares: a) 3.61 b) 1/4
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c) 0.73 HINT: To be a perfect square, a number has to be an even number of decimals, and the square root has half the number of decimals as its square!