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NW | 2026-mar-sdc | TzeMing Ho | Sprint 1 | number-systems-exercise #357

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answering number-systems
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56 changes: 36 additions & 20 deletions number-systems/README.md
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Expand Up @@ -5,61 +5,77 @@ Do not convert any binary numbers to decimal when solving a question unless the
The goal of these exercises is for you to gain an intuition for binary numbers. Using tools to solve the problems defeats the point.

Convert the decimal number 14 to binary.
Answer:
Answer: 1000 + 0100 + 0010 = 1110 = 8 + 4 + 2 = 14
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@DaryaShirokova DaryaShirokova Mar 14, 2026

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which method have you used here exactly?

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@TzeMingHo TzeMingHo Mar 15, 2026

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I was trying to subtract the power of 2 from 14 to get the binary numbers.
14 - 2^3 = 6
6 - 2^2 = 2
2 = 2^1


Convert the binary number 101101 to decimal:
Answer:
Answer: 2^5 + 2^3 + 2^2 + 2^0 = 32 + 8 + 4 + 1 = 45

Which is larger: 1000 or 0111?
Answer:
Answer: 1000 is larger because 1000 has more places

Which is larger: 00100 or 01011?
Answer:
Answer: 01011 is larger because it has more places
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@DaryaShirokova DaryaShirokova Mar 14, 2026

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it's more important what the leading digit is

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@TzeMingHo TzeMingHo Mar 15, 2026

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Yeah. I should have said the leading 1 is taking more places, and they are both 5 bits.


What is 10101 + 01010?
Answer:
Answer: 11111

What is 10001 + 10001?
Answer:
Answer: 100010

What's the largest number you can store with 4 bits, if you want to be able to represent the number 0?
Answer:
Answer: 1111
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@DaryaShirokova DaryaShirokova Mar 14, 2026

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In decimal?

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@TzeMingHo TzeMingHo Mar 15, 2026

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15


How many bits would you need in order to store the numbers between 0 and 255 inclusive?
Answer:
Answer: 2^8 = 256. So, we need 7 bits. b0000000 = 0, and b1111111 = 255
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@DaryaShirokova DaryaShirokova Mar 14, 2026

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So how would 0 be represented?

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@TzeMingHo TzeMingHo Mar 15, 2026

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Oh. I was 1 bit off. It should take 8 bits to store from 0 to 255.


How many bits would you need in order to store the numbers between 0 and 3 inclusive?
Answer:
Answer: 2 bits as 00 = 0, and 11 = 3

How many bits would you need in order to store the numbers between 0 and 1000 inclusive?
Answer:
Answer: 2^9 = 512
2^8 = 256
2^7 = 128
2^6 = 64
2^3 = 8
512 + 256 + 128 + 64 + 32 + 8 = 1000
b1111101000 = 1000
we need 10 bits

How can you test if a binary number is a power of two (e.g. 1, 2, 4, 8, 16, ...)?
Answer:
Answer: 1, 10, 100, 1000, 10000
1, 2, 4, 8, 16
If the binary number is 1 or 1 followed by 0s, it will be a power of two.
We can use regexp to test the binary number.

function testPowerOfTwo(binaryNumber) {
return /^0*10*$/.test(String(binaryNumber))
}

Convert the decimal number 14 to hex.
Answer:
Answer: 10 = A, 11 = B, 12 = C, 13 = D, 14 = E

Convert the decimal number 386 to hex.
Answer:
Answer: 386 - 16^2 = 130
130 - 16 x 8 = 2
hex: 182

Convert the hex number 386 to decimal.
Answer:
Answer: 3 x 16^2 + 8 x 16 + 6 x 1 = 902

Convert the hex number B to decimal.
Answer:
Answer: 11

If reading the byte 0x21 as a number, what decimal number would it mean?
Answer:
Answer: 2 x 16 + 1 X 1 = 33

If reading the byte 0x21 as an ASCII character, what character would it mean?
Answer:
Answer: !

If reading the byte 0x21 as a greyscale colour, as described in "Approaches for Representing Colors and Images", what colour would it mean?
Answer:
Answer: As the byte 0x21 is 33 in decimal, which is in the range of 0 full black to 255 full white. The color is close to black.

If reading the bytes 0xAA00FF as an RGB colour, as described in "Approaches for Representing Colors and Images", what colour would it mean?
Answer:
Answer: AA (170) means middle high red, 00 means no green, and FF (255) means full blue. The colour should be purple.

If reading the bytes 0xAA00FF as a sequence of three one-byte decimal numbers, what decimal numbers would they be?
Answer:
Answer: 170, 0, 255
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