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, , , ,

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?

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,

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?

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)

• 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)

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?

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)

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?

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

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

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!