Math 208, Section 2.2 Solutions: Numbers and the Decimal System
1. Write the following decimal numbers as powers of 10:
a. 0.000001
I know that .1 = 10-1, and 0.000001 is five decimal places further to the left, which
means that I subtract 5 from -1 to get the exponent, so I know that 0.000001 = 1 x 10-6.
b. 10,000,000
I know that 10 = 1 x 101 and 10,000,000 is six decimal places further to the right, which
means that I add 6 to 1 to get the exponent, so I know that 10,000,000 = 1 x 107.
c. The number whose decimal representation is a 1 followed by 50 zeros
Using the same reasoning as in (b), the number is 49 decimal places further to the right
than 10, which means that I add 49 to 1 to get the exponent, so I know that the decimal
representation is 1 x 1050.
d. The number whose decimal representation is a decimal point followed by five
zeros, followed by a 1
Using the same reasoning as in (a), the number is five decimal places further to the left
than .1 = 10-1, so I subtract 5 from -1 to get the exponent. Thus the decimal
representation of the number is 1 x 10-6.
e. The number whose decimal representation is a decimal point followed by 50 zeros,
followed by a 1
Using the same reasoning as in (a), the number is 50 decimal places further to the left
than .1 = 10-1, so I subtract 50 from -1 to get the exponent. Thus the decimal
representation of the number is 1 x 10-51.
2. Write the following decimal numbers in expanded form:
a. Skip
b. 20,030.04
20,030.04 = 2*10,000 + 0*1,000 + 0*100 + 3*10 + 0*1 + 0*.1 + 4 *.01
= 2*104 + 0*103 + 0*102 + 3*101 + 0*100 + 0*10-1 + 4*10-2
I wrote 2 times 10,000 because 2 is in the ten thousands place. There are zeros in the
thousands place and the hundreds place, so I wrote 0 times 1,000 plus 0 times 100. I
wrote 3 times ten because there is a 3 in the tens place. There are zeros in the ones place
and the tenths place, so I wrote 0 times 1 and 0 times .1. I wrote 4 times .01 because
there is a 4 in the hundredths place. Then, I added them all together to get the expanded
form. I continued and wrote each product using exponents.
c. 4.006080
4.006080 = 4*1 + 6*.001 + 8*.00001 + 0*.000001
= 4*101 + 6*10-3 + 8*10-5 + 0*10-6
I wrote 4 times 1 because 4 is in the ones place. There are zeros in the tenths place and
the hundredths place, so didn’t write anything for them, but I wrote 6 times .001
because there is a 6 in the thousandths place. There is a zero in the ten thousandths
place, so I didn’t write anything for that, but I wrote 8 times .00001 because there is a 8
in the hundred thousandths place. I had to write 0 times .000001 because otherwise we
wouldn’t know that there is a 0 in the millionths place. Then, I added them all together
to get the expanded form. I continued and wrote each product using exponents.
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3. Write the following numbers in ordinary decimal notation:
a. Two-hundred thousand, forty-six
200,046
I wrote the 2 in the hundred thousands place because the problems said “two-hundred
thousand”. There are no ten thousands mentioned and no thousands mentioned, so the
ten thousands place and the thousands place get zeros. There are also no hundreds
mentioned, so the hundreds place gets a zero. Since the problem says “forty-six,” I put a
4 in the tens place and a 6 in the ones place.
b. Two-hundred forty-six thousand
246,000
The problem says two-hundred forty-six thousand, which means that 246 goes in the
thousands place. But, each place can only hold one digit, so I know that 6 goes in the
thousands place, the 4 goes in the ten thousands place, and the 2 goes in the hundred
thousands place. There are no hundreds, tens or ones mentioned, so those places get
zeros.
c. Two-hundred forty-six thousandths
0.246
There are no whole numbers mentioned, so I put a 0 in the ones place. Then, there are
246 thousandths, which means that there are 2 tenths, 4 hundredths, and 6 thousands.
So, I wrote 246 after the decimal.
d. Two-hundred four thousand, six
204,006
There are 204 thousands, which means that there are 2 hundred thousands, 0 ten
thousands, and 4 thousands. So, I wrote a 2 in the hundred thousands place, a 0 in the
ten thousands place, and a 4 in the thousands place. There is no mention of hundreds or
tens, so those places get a zero. Then the problem said six, so I wrote a 6 in the ones
place.
e. Two-hundred four thousand and six tenths
204,000.6
There are 204 thousands, which means that there are 2 hundred thousands, 0 ten
thousands, and 4 thousands. So, I wrote a 2 in the hundred thousands place, a 0 in the
ten thousands place, and a 4 in the thousands place. There is no mention of hundreds,
tens, or ones so those places get a zero. Then the problem said six tenths, so I wrote a 6
in the tenths place.
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