Math - Number Bases - Challenge

Ok I understand how to convert the number into binary but how do you convert it back? for 91 it was equal to 1011011 and then you said to do 1+2+8+16+64 where exactly do these numbers come from? is it multiplying the last by 2? and then just skipping the zeros? Thank you!

Beginning from the right to left: number1 · 2^position(0)+number2 · 2^position(1)+…

So, beginning ALWAYS from the right and beginning ALWAYS counting from position ZERO, the first one is a 1, position 0, so it would be 1·2^0, the second one, is also a 1, position 1th, so 1·2^1, the third one is a zero, 2th position, so 0·2^2… and so on:

1·2^0+1·2^1+0·2^2+1·2^3+1·2^4+0·2^5+1·2^6 = 1·1+1·2+0·4+1·8+1·16+0·32+1·64=1+2+8+16+64

I don’t know if I have explained well enough … :sweat_smile:

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Ah I understand now so it will be raised to the position it is in starting right to left at position 0. Thank you!

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I went with 173 decimal and got 10101101 for the binary representation and AD for the hex representation.

To solve the hex, I began with convoluted arithmetic but about halfway through remembered something from my comp science classes decades ago. You can break the binary representation into chunks of four bits and then translate each bit to it’s hex representation. For example, 11110001 is F1 in hex. The first chunk of 4 bits, 1111, equals 15 decimal, or F hex. The second chunk, 0001, is 1 decimal, or 1 hex.

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Great work @archbishopFPP.
Being able to break the binary string into bytes is one of the reasons hex is used a fair bit in computer science. They play very well together and hex is generally more human readable - especially when the binary string start getting longer.

For instance, if you had; 10110101100111011110
You could quickly break this down to 1011 (B) || 0101 (5) || 1001 (9) || 1101 (D) || 1110 (E) -> B59DE

Converting to decimal is still a bit of pain in either case but this trick for binary to hex conversion can be incredibly useful.

Here is my answer to the challenge:

Bin:

Hex:
image

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For 173:

Binary: 10101101
Hexadecimal: AD

For 34:
Binary: 100010
Hexadecimal: 22

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For 173 :

bin
173/2 = 86 r 1
86/2 = 43 r 0
43/2 = 21 r 1
21 / 2 = 10 r 1
10 / 2 = 5 r 0
5 / 2 = 2 r 1
2/2 = 1 r 0
1/2 = 0 r 1

10101101

hexa
173/16 = 10 r 13
10/16 = 0 r 10

AD

For 42 :

42 / 2 = 21 r 0
21/ 2 = 10 r 1
10 / 2 = 5 r 0
5/2 = 2 r 1
2/2 = 1 r 0
1/2 = 0 r 1

101010

42 / 16 = 2 r 10
2 / 16 = 0 r 2
2A

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For 173 I got

AD in hex --> ( 16 * 10 ) + 13
10101101 in binary —> 1+ 4+ 8+32 + 128

Then tried 289

121 in hex --> 16^2 +2(16^1) + 1(16^0) = 256 + 32 + 1
100100001 --> 1 + 32 + 256

The results I got for 173:

Hex: AD
Bin: 10101101

Also made a post here how I verified my results with code:

Number 723
In bin: 1011010011
In hex: 2D3

Challenge 173:

In bin:

173 / 2 = 86 r 1
56 / 2 = 43 r 0
43 / 2 = 21 r 1
21 / 2 = 10 r 1
10 / 2 = 5 r 0
5 / 2 = 2 r 1
2 / 2 = 1 r 0
1 / 2 = 0 r 1

10101101

In hex:

173 / 16 = 10 r 13
10 / 16 = 0 r 10

10 = A
13 = D

AD

New number:
number is 2046

In bin:

2046 / 2 = 1023 r 0
1023 / 2 = 511 r 1
511 / 2 = 255 r 1
255 / 2 = 127 r 1
127 / 2 = 63 r 1
63 / 2 = 31 r 1
31 / 2 = 15 r 1
15 / 2 = 7 r 1
7 / 2 = 3 r 1
3 / 2 = 1 r 1
1 / 2 = 0 r 1

11111111110

In hex:

2046 / 16 = 127 r 14
127 / 16 = 7 r 15
7 / 16 = 0 r 7

15 = F
14 = E

7FE

173:


Binary:
173 / 2 = 86 reminder 1
86 / 2 = 43 reminder 0
43 / 2 = 21 reminder 1
21 / 2 = 10 reminder 1
10 / 2 = 5 reminder 0
5 / 2 = 2 reminder 1
2 / 2 = 1 reminder 0
1 / 2 = 0 reminder 1

Binary = 10101101

Hex:
173 / 16 = 10 reminder 13
10 / 16 = 0 reminder 10

Hex = AD

398:

Binary:
398 / 2 = 199 reminder 0
199 / 2 = 99 reminder 1
99 / 2 = 49 reminder 1
49 / 2 = 24 reminder 1
24 / 2 = 12 reminder 0
12 / 2 = 6 reminder 0
6 / 2 = 3 reminder 0
3 / 2 = 1 reminder 1
1 / 2 = 0 reminder 1
Binary = 110001110

Hex:
398 / 16 = 24 reminder 14
24 / 16 = 1 reminder 8
1 / 16 = 0 reminder 1

Hex = 18E

I used the integer ‘69’ for this practice:

Binary Conversion:

69 / 2 = 34 r 1
34 / 2 = 17 r 0
17 / 2 = 8 r 1
8 / 2 = 4 r 0
4 / 2 = 2 r 0
2 / 2 = 1 r 0
1 / 2 = 0 r 1

69 = 1000101

Hexadecimal Conversion:

69 / 16 = 4 r 5
4 / 16 = 0 r 4

69 = 45

Here’s my attempt at the binary and hexadecimal conversions. I started with the given value of 173, and then did my own of 117:

173 Binary:
173 / 2 = 86 r 1
86 / 2 = 43 r 0
43 / 2 = 21 r 1
21 / 2 = 10 r 1
10 / 2 = 5 r 0
5 / 2 = 2 r 1
2 / 2 = 1 r 0
1 / 2 = 0 r 1

Binary Result: 10101101

173 Hexadecimal:
173 / 16 = 10 r 13
10 / 16 = 0 r 10
(In Hexadecimal: 13 = D, 10 = A)

Hexadecimal Result: AD

117 Binary:
117 / 2 = 58 r 1
58 / 2 = 29 r 0
29 / 2 = 14 r 1
14 / 2 = 7 r 0
7 / 2 = 3 r 1
3 / 2 = 1 r 1
1 / 2 = 0 r 1

Binary Result: 1110101

117 Hexadecimal:
117 / 16 = 7 r 5
7 / 16 = 0 r 7

Hexadecimal Result: 75

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Here’s my solution for number 42

Binary
42/2 = 2 r 0
21/2 = 10 r 1
10/2 = 5 r 0
5/2 = 2 r 1
2/2 = 1 r 0
1/2 = 0 r 1

42 → 101010 in binary form

Hex solution
42/16 = 2 r A
2/16 = 0 r 2

42 → 2A in hex form

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I did it using C++. The answer is 10101101 (Binary), AD( = ‘10’ ‘13’ Hexadecimal)

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173 in binary:

173 / 2 =
86 r 1
43 r 0
21 r 1
10 r 1
5 r 0
2 r 1
1 r 0
0 r 1

So 173 in binary = 10101101

173 in hexadecimal:

173 / 16 =
10 r 13
0 r 10

So 173 in hexadecimal = AD

235 in binary:

235 / 2 =
117 r 1
58 r 1
29 r 0
14 r 1
7 r 0
3 r 1
1 r 1
0 r 1

So 235 in binary = 11101011

235 in hexadecimal:

235 / 16 =
14 r 11
0 r 14

So 235 in hexadecimal = EB

I chose 293.

Binary: 100100101
Hex: 125

In double-checking the Hex I forgot that for each place we multiple the previous by 16 (so the 3rd place is 256) so couldn’t figure out why I couldn’t re-convert it. Finally got it tho! This is fun stuff! I feel like I’m starting to get things that have long eluded me.

173 is

10101101
AD

my number is 420

110100100
44

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