Binary numbers (fundamental) question

[tomcenjerrym]

I have 2 questions:

FIRST
I read from John H. Mathews’ Numerical Methods using MATLAB, 3E that the number
1563 can be expanded in form of 10-base expansion as:

(1 × 10$$^{3}$$) − (5 × 10$$^{2}$$) + (6 × 10$$^{1}$$) + (3 × 10$$^{0}$$) …......... (1)

Which results 1000 − 500 + 60 + 3 equal to 563.

But the number 1563 should be expanded in form of 10-base expansion as:

1563 = 1000 + 500 + 60 + 3

Which means:

(1 × 10$$^{3}$$) + (5 × 10$$^{2}$$) + (6 × 10$$^{1}$$) + (3 × 10$$^{0}$$) …......... (2)

Hence, (1) ≠ (2) or 1563 ≠ 563.

So, can anyone explain me about these?

SECOND
If the number 1563 is expressible in expanded form of 10-base expansion as:

1563 = 1000 + 500 + 60 + 3
=(1 × 10$$^{3}$$) − (5 × 10$$^{2}$$) + (6 × 10$$^{1}$$) + (3 × 10$$^{0}$$)

how can I express the number 1563 in expanded form of 2-base expansion?

Please advance

 

[ganstaman]

I would have assumed the "-(5x10^2)" to be a typo and your (2) equation to be correct.

If you understand why the base 10 expansion is correct, then you should be able to expand it into base 2 if I simply gave you the following list:

2^0 = 1
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
2^9 = 512
2^10 = 1024

 

[tacman]

FIRST:

The reason why the two forms of expansion (1) and (2) are different is because they are using different powers of 10. In the first form (1), the powers of 10 are decreasing from left to right, while in the second form (2), the powers of 10 are increasing from left to right.

In the first form (1), the number 1563 is being expanded as 1563 = 1*10^3 - 5*10^2 + 6*10^1 + 3*10^0. This means that the first digit (1) is multiplied by 10^3, the second digit (5) is multiplied by 10^2, and so on. This is the traditional way of expanding numbers in base 10.

In the second form (2), the number 1563 is being expanded as 1563 = 1*10^3 + 5*10^2 + 6*10^1 + 3*10^0. This means that the first digit (1) is multiplied by 10^3, the second digit (5) is multiplied by 10^2, and so on. This is not the traditional way of expanding numbers in base 10, but it still results in the correct answer of 1563.

To better understand this, let's look at a simpler example of the number 123. In the traditional way of expanding numbers in base 10, we would write 123 = 1*10^2 + 2*10^1 + 3*10^0. However, we could also write it as 123 = 1*10^2 - 8*10^1 + 11*10^0. Both forms are different, but they still result in the same number 123.

In summary, both forms (1) and (2) are correct ways of expanding the number 1563 in base 10, but they are just using different powers of 10.

SECOND:

To express the number 1563 in expanded form of 2-base expansion, we need to use powers of 2 instead of powers of 10. In base 2, the digits can only be 0 or 1, so we would write 1563 as:

1563 = 1*2^10 + 1*2^9 + 1*2^