From 0befc43f226bd6541c5b77ce7a5fbc247b7e4535 Mon Sep 17 00:00:00 2001 From: TzeMingHo Date: Thu, 12 Mar 2026 00:06:31 +0000 Subject: [PATCH 1/3] answering number-systems --- number-systems/README.md | 43 ++++++++++++++++++++++++++++------------ 1 file changed, 30 insertions(+), 13 deletions(-) diff --git a/number-systems/README.md b/number-systems/README.md index 77a3bde94..9f13ae116 100644 --- a/number-systems/README.md +++ b/number-systems/README.md @@ -5,43 +5,60 @@ 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 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 because 1000 is 8 and 0111 is 7 Which is larger: 00100 or 01011? -Answer: +Answer: 01011 because 01011 is 11 and 00100 is 4 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 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 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 starts with 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*$/.test(String(binaryNumber)) + } Convert the decimal number 14 to hex. -Answer: +Answer: 10 = A, 11 = B, 12 = C, 13 = D, 14 = E Convert the decimal number 386 to hex. -Answer: +Answer: F = 15, +386 / 15 = 25.7333 +386 - (F * 25) = 11 +So, 386 = F*25+B Convert the hex number 386 to decimal. Answer: From b20a8c4d2c05d80a024cbc21669b176a11ae972e Mon Sep 17 00:00:00 2001 From: TzeMingHo Date: Thu, 12 Mar 2026 12:06:53 +0000 Subject: [PATCH 2/3] completed answers for number-systems --- number-systems/README.md | 37 ++++++++++++++++++------------------- 1 file changed, 18 insertions(+), 19 deletions(-) diff --git a/number-systems/README.md b/number-systems/README.md index 9f13ae116..68b7380bd 100644 --- a/number-systems/README.md +++ b/number-systems/README.md @@ -11,10 +11,10 @@ Convert the binary number 101101 to decimal: Answer: 2^5 + 2^3 + 2^2 + 2^0 = 32 + 8 + 4 + 1 = 45 Which is larger: 1000 or 0111? -Answer: 1000 because 1000 is 8 and 0111 is 7 +Answer: 1000 is larger because 1000 has more places Which is larger: 00100 or 01011? -Answer: 01011 because 01011 is 11 and 00100 is 4 +Answer: 01011 is larger because it has more places What is 10101 + 01010? Answer: 11111 @@ -34,8 +34,8 @@ Answer: 2 bits as 00 = 0, and 11 = 3 How many bits would you need in order to store the numbers between 0 and 1000 inclusive? Answer: 2^9 = 512 2^8 = 256 - 2^7 = 128 - 2^6 = 64 +2^7 = 128 +2^6 = 64 2^3 = 8 512 + 256 + 128 + 64 + 32 + 8 = 1000 b1111101000 = 1000 @@ -44,39 +44,38 @@ we need 10 bits How can you test if a binary number is a power of two (e.g. 1, 2, 4, 8, 16, ...)? Answer: 1, 10, 100, 1000, 10000 1, 2, 4, 8, 16 -If the binary starts with 1 or 1 followed by 0s, it will be a power of two. +If the binary number is 1 or 1 followed by 0s, it will be a power of two. We can use regexp to test the binary number. - function testPowerOfTwo(binaryNumber) { - return /^0*10*$/.test(String(binaryNumber)) - } +function testPowerOfTwo(binaryNumber) { +return /^0*10*$/.test(String(binaryNumber)) +} Convert the decimal number 14 to hex. Answer: 10 = A, 11 = B, 12 = C, 13 = D, 14 = E Convert the decimal number 386 to hex. -Answer: F = 15, -386 / 15 = 25.7333 -386 - (F * 25) = 11 -So, 386 = F*25+B +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 From bcfd1ff366327466b7ef2701ab9a1a4a41834c54 Mon Sep 17 00:00:00 2001 From: TzeMingHo Date: Sun, 15 Mar 2026 09:46:45 +0000 Subject: [PATCH 3/3] updated answers --- number-systems/README.md | 12 ++++++++---- 1 file changed, 8 insertions(+), 4 deletions(-) diff --git a/number-systems/README.md b/number-systems/README.md index 68b7380bd..c149b1628 100644 --- a/number-systems/README.md +++ b/number-systems/README.md @@ -5,7 +5,11 @@ 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: 1000 + 0100 + 0010 = 1110 = 8 + 4 + 2 = 14 +Answer: 2^4 = 16, +so I should start with 2^3 = 8 (1000), +14 - 8 = 6, +6 - 2^2 (0100)= 2 (0010) +14 = 1000 + 0100 + 0010 = 1110 Convert the binary number 101101 to decimal: Answer: 2^5 + 2^3 + 2^2 + 2^0 = 32 + 8 + 4 + 1 = 45 @@ -14,7 +18,7 @@ Which is larger: 1000 or 0111? Answer: 1000 is larger because 1000 has more places Which is larger: 00100 or 01011? -Answer: 01011 is larger because it has more places +Answer: 01011 is larger because the leading number is 01 instead of 00, and they both have 5 bits. What is 10101 + 01010? Answer: 11111 @@ -23,10 +27,10 @@ What is 10001 + 10001? 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: 1111 +Answer: 1111 = 15 How many bits would you need in order to store the numbers between 0 and 255 inclusive? -Answer: 2^8 = 256. So, we need 7 bits. b0000000 = 0, and b1111111 = 255 +Answer: 2^8 = 256. So, we need 8 bits. b00000000 = 0, and b11111111 = 255 How many bits would you need in order to store the numbers between 0 and 3 inclusive? Answer: 2 bits as 00 = 0, and 11 = 3