9.6.5: largest number with a specified number of digits. (a) what is the decimal representation of (10000)7? (b) what is the largest number that can be represented with four digits in base 7? (give the base-7 representation of the number as well as its decimal representation.) (c) what is the relationship between the values of the two numbers in the previous two questions?
The Correct Answer and Explanation is:
(a) Decimal Representation of ( (10000)_7 )
To convert the base-7 number ( (10000)_7 ) into decimal, we can expand it using powers of 7. The base-7 number ( 10000_7 ) can be represented as:
[
10000_7 = 1 \times 7^4 + 0 \times 7^3 + 0 \times 7^2 + 0 \times 7^1 + 0 \times 7^0
]
Simplifying:
[
1 \times 7^4 = 1 \times 2401 = 2401
]
[
0 \times 7^3 = 0
]
[
0 \times 7^2 = 0
]
[
0 \times 7^1 = 0
]
[
0 \times 7^0 = 0
]
So the decimal representation of ( 10000_7 ) is:
[
10000_7 = 2401_{10}
]
(b) Largest Number Representable with Four Digits in Base-7
In base-7, the largest digit is 6 (since base-7 uses digits from 0 to 6). Therefore, the largest number that can be represented with four digits in base-7 is ( 6666_7 ).
To convert ( 6666_7 ) into decimal, we expand it in powers of 7:
[
6666_7 = 6 \times 7^3 + 6 \times 7^2 + 6 \times 7^1 + 6 \times 7^0
]
Calculating the powers of 7:
[
7^3 = 343, \quad 7^2 = 49, \quad 7^1 = 7, \quad 7^0 = 1
]
Now calculate the decimal value:
[
6 \times 343 = 2058
]
[
6 \times 49 = 294
]
[
6 \times 7 = 42
]
[
6 \times 1 = 6
]
Adding them together:
[
6666_7 = 2058 + 294 + 42 + 6 = 2400_{10}
]
Thus, the largest number that can be represented with four digits in base-7 is ( 6666_7 ), which equals ( 2400_{10} ).
(c) Relationship Between the Two Values
In the previous two questions, we determined the decimal equivalents of two base-7 numbers:
- ( 10000_7 = 2401_{10} )
- ( 6666_7 = 2400_{10} )
The relationship between the two decimal values is that ( 10000_7 ) (or ( 2401_{10} )) is greater than ( 6666_7 ) (or ( 2400_{10} )) by exactly 1.
This shows that the largest number representable with four digits in base-7 (which is ( 6666_7 )) is only 1 less than ( 10000_7 ), the next number in base-7 that requires five digits. This highlights a key property of number systems: as the number of digits increases, the magnitude of the number grows exponentially based on the base, and the smallest number that increases the total number of digits (e.g., moving from four to five digits in base-7) is always one more than the largest number representable with the previous number of digits.