Download Convert Between Base 10 and Base x Systems (iii)

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Mathlete8: Convert Between Base 10 and Base x Systems (iii)
Name: _____________________________
5/2/2014
Date: ____________________
Convert Between Base 10 and Base x Systems (iii):
Writing Fractions in a Base x System
EXPANDED FORM OF A NON-INTEGER NUMBER
All numbers in the base 10 system can be expressed by an expanded form. For
example, 0.386210 (read 0.3862 base 10)
0.3862 = 3x10-1 + 8x10-2 + 6x10-3 + 2x10-4
In other words, the place values in the base 10 system are in the power of 10.
So, if you want to express the number using a different number x as the base, the
“place values” are based on power of x.
Example 1. Find the value of 0.10112 (read 0.1011 base 2) in the base 10 system.
0.10112 = 1x2-1 + 0x2-2 + 1x2-3+ 1x2-4
= (11/16)10
Ans. (11/16)10
Example 2. Convert (11/16)10 into base 2.
Make sure that the denominator is in the power of 2. Since 16 = 24, that means that your
answer will contain 4 decimal places. Then we use the “bus” method to analyze the
numerator like before:
2
11
2
5
2
2
R1
R1
2 1
0
R0
R1
Ç Read the remainders from bottom up,
Ans. 10112
Example 3. Convert 202.023 to base 7, round off to two decimal places if necessary.
Convert the integer part:
2023 = 2010 = 267
Convert the fraction part:
0.023 = 0(3-1) + 2(3-2) = 2/9 = 0.2222210
To go from base 10 to base 7, we'll multiply this base 10 fraction portion by 7, take the
fractional part and do it again. We'll do this three times, twice for the number of significant
figures, and once to see if we need to round.
7(0.22222...) = 1.55555...
7(0.55555...) = 3.88888...
7(0.88888...) = 6.22222...
Reading the integer parts forwards, the base 7 fraction
is: 0.136. Since 6 is above the midway point, we'll round
up to 0.14.
Therefore, the answer is approximately 26.147.
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Base 10 to Base A (iii)
IQ Abacus Math & Language School
Mathlete8: Convert Between Base 10 and Base x Systems (iii)
Name: _____________________________
Score: (
Date: ____________________
) / 10
Answer the questions below. Round off to 3 decimal places if necessary.
1. Convert 4/27 to base 3.
6. Convert 0.375 to base 2 and base 5.
2. Convert 143/729 to base 9.
7. Convert 0.1345 to base 10.
3. Convert 159/343 to base 7.
8. Convert 2.7038 to base 6.
4. Convert 322/343 to base 7.
9. Convert 0.2349 to base 2.
5. Convert 0.101112 to base 10.
10. Convert 3.256 to base 8.
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Base 10 to Base A (iii)