Rational Numbers
A rational number is any number that can be expressed in the fractional form
a and b are both integers and the denominator is not equal to zero.
a
, where
b
When a rational number is expressed as a decimal, the number will appear as a
terminating decimal, which means the decimal stops, or as a repeating decimal, which
means the decimal goes on forever, repeating the same values over and over.
(While rational numbers can be expressed as decimals or fractions, this lesson will focus
on decimals while Lesson 5 will explore fractions in more detail.)
Computing with Decimal Numbers
example 1
Compute:
When adding (or subtracting) rational numbers expressed as decimals, always line up
the decimal points vertically, then add each column from right to left. If any numbers
carry over (if a column adds up to more than 9) add the tens digit of that column to
the next column (to the left). Make sure the decimal place in the answer lines up with
the columns of numbers being added:
The answer is 108.72.
example 2
Compute:
When multiplying decimal numbers, as opposed to when adding (or subtracting), do
not line up the decimal points. Line up the numbers on the right:
Next, multiply the bottom digit the farthest to the right by each digit in the top
number, moving from right to left. If any product is greater than nine, write down the
digit in the ones place and carry over the digit in the tens place. The carried-over digit
will be added to the next product found:
Before multiplying the next digit in the bottom number (the 2 in this case) by each
digit in the top number, first place a zero in the first column (to the right), then
multiply the same way as before:
Now add the two rows of numbers you just created and move the decimal point to the
left the same number of places as the total number of digits that are to the right of the
decimal in each of the numbers you just multiplied together. For example, 43.68 has
two digits to the right of the decimal place, and 2.5 has one digit to the right of the
decimal. So there are a total of three digits to the right of the decimal in those
numbers:
Moving the decimal point three places to the left gives us an answer of 109.2.
example 3
Compute:
When dividing with decimals (without a calculator), the number you are dividing by
(called the divisor) needs to be a whole number. In this problem, the divisor is 1.8,
which not a whole number. So, move the decimal point to the right until it is a whole
number. [You can do this as long as you move the decimal point of the number you
are dividing into (called the dividend) the same number of spaces to the right.]
Now divide, placing the decimal point in the quotient (the answer) directly above the
decimal point in the dividend:
The answer is 2.3.
example 4
The difference between two temperature readings was 7 degrees. Which of the
following could be the two temperature readings?
A.
B.
C.
D.
−7º and 1º
−4º and 3º
−1º and 7º
−5º and 12º
A difference refers to an answer found by subtraction. To see which answer has two
numbers with a difference of 7, subtract each smaller number from the larger number
in each answer.
What this question is really testing is to see if you understand how to subtract a
negative number. When you subtract a negative number, it’s the same as adding that
same positive number:
Applying this to each answer:
A) 1 – (−7) = 1 + 7 = 8
B) 3 – (−4) = 3 + 4 = 7
C) 7 – (−1) = 7 + 1 = 8
D) 12 – (−5) = 12 + 5 = 17
The answer is B.
Comparing Decimal Numbers
example 5
Erin determined the masses of some samples for her science project. The mass of each
sample is listed below.
Sample
Mass (grams)
1
17
2
16.7
3
17.6
4
16.67
Which of the following correctly lists the samples in order from the least mass to the
greatest mass?
A.
B.
C.
D.
1, 2, 3, 4
2, 3, 4, 1
2, 4, 1, 3
4, 2, 1, 3
Numbers can be ordered from smallest to largest by comparing the place value of
each number. Going from left to right, check the tens places, ones place, tenths place,
and so on, and see which digits are larger.
In this problem, all four numbers have a “1” in the tens place, so they can’t be
compared using the tens digit. Two of the numbers have a “7” in the ones place, while
the other two numbers have a “6” in the ones place, so these first two numbers are
bigger. Now we just need to figure out, among the two larger numbers (17 and 17.6),
which one is larger than the other, and among the smaller numbers (16.7 and 16.67),
which one is larger.
The next place value is the tenths place, so compare those digits. If there is no digit
there, a “0” can be used for that place value. Since 6 is larger than 0, 17.6 is larger
than 17 (or 17.0). And since 7 is larger than 6, 16.7 is larger than 16.67.
Listing all four numbers in order, from smallest to largest:
The answer is D.
Irrational Numbers
An irrational number is any number that cannot be expressed as a rational number.
There are two kinds of irrational numbers: irrational numbers known by other symbols,
such as π (pi), or square roots that can’t be simplified into rational numbers.
π is the ratio of the circumference of a circle to its diameter, and it equals
3.14159265358979323846…. (π is usually rounded off to 3.14.)
Pi is a number that, when expressed in decimal form, never ends, and never repeats. This
is true for all irrational numbers, as any irrational number cannot be expressed as a
terminating decimal or repeating decimal. (Note: Square roots are rational numbers if
they are the square root of a rational number that’s been squared.)
but
example 6
Which of the following is an irrational number?
A.
B.
4
3
24
C. 81
D. −4.07
a
(where a and b are both
b
integers and the denominator is not equal to zero). Any number expressed as a
decimal is a rational number, which also includes integers. Square roots (and �) are
typically the only irrational numbers you’ll ever see.
A rational number can be expressed in the fractional form
Knowing this definition for a rational number, it’s easy to eliminate answers A and D.
The remaining answers are both square roots, but only one can be irrational if there is
only one answer to this question.
The key to figuring out which square root is an irrational number is being able to
recognize a perfect square, which is the square of an integer. (More on this in
Lesson 3.) Since 81 is equal to 9 squared (or 9 × 9), 81 is equal to 9. Therefore,
81 is actually a rational number.
The answer is B.
Name ______________________________
Compute (without using a calculator):
1) 1.06 + 2.7 =
2) 14 + 3.14 + 0.113 =
3) 9.8 – 6.23 =
4) 4.05 – 7.1 =
5) 6.22 × 3.7 =
6) 20.7 × 1.36 =
7) 8.88 ÷ 2.4 =
8) 11.25 ÷ 0.75 =
Re-write the given numbers in order from smallest to largest:
9) 15.45, 14.54, 15.55, 14.94
10) 39, 38.9, 39.8, 38
11) 1,000,100; 1,000,001;
1,001,000; 1,000,010
12) 14.89, 14.9, 14.889, 14.8
Label each number as rational or irrational:
13) 0
6
15)
17)
14) 3.14
1
4
19) 2.34
16)
9
5
18) �
20)
2
7