Counting – Binary, Hexadecimal, Etc

Question 1

How do you convert from Base 2 to Base 8?

Answer 1

You convert straight from base 2 to base 8 by selecting the base 2 number in threes from right to left and changing to base 10:

110111
Becomes 110 – convert into base ten
111 – convert into base ten
Bring the answers together.
If it can’t be evenly broken into threes, then add ‘0’s to the left most side until it can.

Question 2

How do you convert from Base 8 to 2?

Answer 2

Convert each digit singly into base 2, then bring them together.
For example, 67₈ to base 2 is done by converting ‘6’ first to base 2, then ‘7’ to base 2, and after this bring the answers together.

Question 3

How do you convert base 2 to base 16?

Answer 3

Pick the digits of the number in fours, then make as if you want to convert each pair to base 10.

Bring the answers together, and you have your base 16.

Question 4

What are the magic fingers?

Answer 4

It’s a way to count in base 2. Label your fingers from your right thumb (it’s easier this way) to your right pinky 1, 2, 4, 8, 16. When counting (left to right) your folded fingers count as ‘0’, and your raised ones as ‘1’.

So if you want to get ‘3’ in base 2, fold all your fingers, raise the one labelled ‘2’ and the one labelled ‘1’. So counting from left to right, you get: ‘11’.

Question 5

How are binary numbers represented in digital computers?

Answer 5

They are represented via binary storage devices (e.g. flip-flops)

Question 6

What is sign bit?

Answer 6

Sign bit is a way to represent positive and negative numbers. A bit is fixed in front of the number in question, with ‘0’ representing positive and ‘1’ being negative.

Question 7

What are the compliment forms?

Answer 7

1’s complement and 2’s complement.

Question 8

What is the 1’s complement form?

Answer 8

It is a form that works by changing each ‘1’ to a ‘0’, and each ‘0’ to a ‘1’.
Thus 101011 becomes 010100.

Question 9

What is the 2’s complement form?

Answer 9

The 2’s complement form works by first taking the 1’s complement of the given number, then adding ‘1’ to its Least Significant Bit position (LSB). So to get the 2’s complement of 101011:

1’s complement – 010100
2’s complement – 010101

Question 10

What is BCD?

Answer 10

It means Binary Coded Decimal and it is a format that assigns each of the ten decimal digits (0-9) a standard equivalence in binary. Numbers >9 are gotten not by standard conversion, but by the combination of each digit’s corresponding BCD value.

E.g., 12 becomes 00010010, which is 1 (0001) and 2 (0010) in BCD written together.

Question 11

What are the invalid codes in BCD?

Answer 11

1010, 1011, 1100, 1101, 1110, and 1111 are invalid codes in BCD. They are the standard conversions of 10-15 respectively.

10000 is not included cause that’s already >4 bits.

Question 12

How do you convert positive numbers to 2’s complement?

Answer 12

Positive numbers are left in their normal form; you leave them as they are.

Question 13

How do you convert signed negative numbers in 1’s complement to decimal?

Answer 13

Do regular conversion to decimal, taking note of the sign bit, then add ‘1’ to the final result (i.e. -24 becomes -24 + 1 = -23

Question 14

How do you convert signed negative numbers in 2’s complement to decimal?

Answer 14

Regular decimal conversion and take note of the sign.

Question 15

What is a byte?

Answer 15

A byte is 8-bits, two bytes is 16-bits, on and on…

Question 16

What is the formula for total numbers that can be represented in n-bit?

Answer 16

2ⁿ

Question 17

What is the formula for the range of possible signed 2’s complement numbers for a given n-bit?

Answer 17

-(2^{n - 1}) to (2^{n - 1} - 1)

The -1 in the positive arm shows that we are counting ‘0’ as a positive number.

Question 18

What is the Floating Point Binary Number System?

Answer 18

It is a system of a expressing very large or very small positive or negative numbers in, binary without an increase in the number of bits, in two parts: mantissa and exponent.

Question 19

What are is a Floating Point Number?

Answer 19

It (also known a a real number) consists of a mantissa (which is an integer and a fraction) and an exponent.

Question 20

What are the mantissa and the exponent?

Answer 20

The mantissa describes the magnitude of the number, while the exponent informs us where the radic point (decimal or binary point) will be on the number.

Question 21

What is the FPU?

Answer 21

Floating point Unit: computers have a coprocessor which performs calculations concerning floating point numbers, the coprocessor is called FPU.

Question 22

What is the Single-Precision Floating Point Binary Number?

Answer 22

It is a number format which consisting of 32-bits: one bit is the sign-bit which appears as its most significant bit (left-most bit); the next 8-bits holds its biased exponent; and the mantissa takes up the last 23-bits.

Question 23

What is meant by the “biased exponent” in a Single-Precision Floating Point Binary Number?

Answer 23

It is biased because the number’s exponent is actually added to 127, allowing a wider range of exponent to be represented without the need for an extra sign bit denoting the exponent’s positive or negative nature.

So, the exponent 12 in 2¹² becomes 12+127 = 139 in this system.