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Math for CS > Module 1 - 3 > Flashcards

Module 1 - 3 Flashcards

1
Q

How many variables do OR gates need?

A

Boolean addition corresponds to the logical function of an OR gate and also involves two or more variables. (e.g)

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2
Q

A hexadecimal system which has a base of 16, uses which digits?

A

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

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3
Q

How do you convert decimal into hexadecimal? (The non decimal side “whole numbers”)

A

You do successive division of the base (16) until we reach 0 the remainder then gets multiplied by the base (16) and whats left over from each time is the next hexadecimal digit.

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4
Q

When dealing with two’s complement, If you need to increase the bits in the register how is it done?

A

We can just copy the MSB into the other previous bits.

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5
Q

Is the accuracy or precision of an approximate number given by the place value of the least significant digit?

A

Precision

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6
Q

How do you convert Hexadecimal to Octal?

A

First you convert Hexadecimal to binary, and then convert binary into Octal.

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7
Q

What does inverse mean?

A

Oposite of or to flip.

The inverse of True is False

The inverse of 1 is 0

the inverse of A is∀

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8
Q

What do truth tables look like?

A

The following list of Universal tables represents 2, 3 and 4 variables.

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9
Q

What does “Precision” show?

A

Precision shows how closely we can read the measurement scale. e.g. “If we have some measurement tool that is very finely graded we can determine our numbers with a very high precision. HOWEVER, if this tool was wrongly designed our measured numbers will be very inaccurate.”

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10
Q

A Binary system has a base of ___ ?

A

2

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11
Q

What would a conversion between positive and negative numbers look like using two’s complement?

A
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12
Q

When the MSB is 1 is the number positive or negative?

A

Negative

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13
Q

How to convert decimal to binary? (the decimal side “fraction”)

A

You do successive multiplication of the base (2) to the specified decimal point.

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14
Q

How do you convert a binary number given in one’s compliment if the number is negative? (MSB=1)

A

If the number is negative we add the column weights that contain 0’s, but we keep a negative sign in the front.

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15
Q

What is the range an engineering format number can have?

A
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16
Q

How do you convert decimal into binary? (The non decimal side “whole numbers”)

A

You do successive division of the base (2) until we reach 0 the remainder left over from each time we divide is the next binary digit.

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17
Q

How do you convert decimal into octal? (The non decimal side “whole numbers”)

A

You do successive division of the base (8) until we reach 0 the remainder then gets multiplied by the base (8) and whats left over from each time is the next octal digit.

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18
Q

What are the 4 rules of subtraction of binary numbers?

A
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19
Q

How do you convert octal into decimal?

A

You multiply the octal digit by the systems base (8) to the power of the position of the digit.

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20
Q

An octal system which has a base of 8, uses which digits?

A

0, 1, 2, 3, 4, 5, 6, 7

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21
Q

As the numbers stored in a computer have a previously specified digit size, the calculations will provide results with what two types of answers?

A

truncated or rounded

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22
Q

What is another word of saying base 10?

A

Radix 10

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23
Q

How do you convert a binary number given in one’s compliment if the number is positive (MSB=0)

A

you follow the same steps as converting binary to decimal just like you would any other time.

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24
Q

What are Postulates?

A

Postulates are self - evident truths

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25
Q

How do you convert binary into decimal?

A

You multiply the binary digit by the systems base (2) to the power of the position of the digit. **you can skip the zeros from binary because 0 x (x) = 0**

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26
Q

What is the leftmost significant digit called?

A

The Most Significant Digit (MSD)

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27
Q

What is the best way to deal with fractions?

A

The best way to deal with fractions in an equation is as fractions. When you try converting fractions into decimals you start losing accuracy, and depending on the equation even a little accuracy in one place could make a MASSIVE difference in the final answer. (e.g.) the difference of finding 2 thirds of a million dollars is $600,000 when truncating the .66. $700,000 when rounding up to .70. And $666,666.67 when you turn the million into a fraction making the equation 2/3 x 1000000/1.

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28
Q

What does MSB stand for?

A

Most significant bit, and the left most bit. (MSB)

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29
Q

How many digits can you have before the decimal point and what is the range of the digit(s) in engineering format?

A

You can have up to 3 digits in engineering format and that number can’t be smaller than 1 and has to be smaller than 1000

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30
Q

What is a Fixed-Point number?

A

Also referred to as integers, a Fixed-Point number is where the decimal point is fixed after the rightmost digit.

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31
Q

What is a Floating-Point number?

A

Real numbers that contain the decimal point.

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32
Q

What are the steps in writing the scientific notation for a number greater than 1?

A
  1. We move the decimal point to the left in order to have only one non-zero digit to the left of it. 2. We count the number of places that the decimal point was moved. That number represents the power of 10 in the scientific notation. 3. We write the product of the number and the power of 10 to express the scientific notation.
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33
Q

What is another way AND is reffered as and what are their respected symbols?

A

LOGICAL PRODUCT - AND can also be reffered to as a Conjunction - (x) or (.)

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34
Q

What are the rules to rounding off?

A

We examine the key digit in the next place to the right of the digit where we want to round off out number, then we round down if the key digit is less than 5, and round up if the key digit is greater or equal to 5.

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35
Q

How do you convert Octal to Hexadecimal?

A

First you convert octal to binary, and then convert binary into hexadecimal.

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36
Q

What depics the output of an OR operation?

A

The ouput of an OR operation is 1 if at least one input is 1; the output of an OR operation is 0 only if all inputs have a value of 0. (e.g)

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37
Q

Express .00000094 in engineering format.

A

First move decimal in blocks of 3 (.000|00094) then (.000000|94) then as you can see you can’t move 3 more decimal places, but the number you are at is less than one, so you have to add 0’s to the end accordingly, (.000000940) making the engineering notation 940E-9

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38
Q

What does the Engineering format table look like?

A
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39
Q

What is the rightmost significant digit called?

A

The Least Significant Digit (LSD)

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40
Q

How do you convert Octal to Binary?

A

You separate each octal number, and write the corresponding 3 bit binary equivalent to the octal digit using the octal/binary truth table.

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41
Q

What does LSD stand for?

A

Least significant digit, and the right most digit.

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42
Q

What is truncating?

A

Truncating is when you just cut the number as the desired digit, we do not look at the digits after it, we simply just cut the rest off.

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43
Q

How do we add or subtract numbers given in engineering format?

A

To add and/or subtract numbers written in engineering format we convert all numbers to the same E power; the numbers are then added/subtracted as given. At the end we write the final answer in engineering format by shifting the decimal point right or left and adjusting the E power.

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44
Q

What number systems fall under the non-positional category?

A

Roman Numerals and Egyptian Hieroglyphics

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45
Q

What are the three steps in determining the two’s compliment of a binary number?

A
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46
Q

What is scientific notation?

A
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47
Q

What are Properties that are VALID in boolean math?

A
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48
Q

A decimal system which has a base of 10, uses which digits?

A

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

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49
Q

How does addition work with binary numbers?

A
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50
Q

A decimal system has a base of ___ ?

A

10

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51
Q

Using the sign magnitude number system, how would you display a positive 12 and a negative 12 in binary.

A

00001100(2) = +12 10001100(2) = -12

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52
Q

A binary system which has a base of 2, uses which digits?

A

0, 1

53
Q

How does multiplication with with binary numbers?

A
54
Q

What are the basic principals of Boolean algebra?

A
55
Q

What is the NEGATION (NOT - inverse) symbol?

A
56
Q

How do you represent a negative decimal number in two’s complement notation?

A
57
Q

How do you convert hexadecimal to decimal?

A

You multiply the Hexadecimal digit by the systems base (16) to the power of the position of the digit.

58
Q

What are the 4 rules of multiplication of binary numbers?

A
59
Q

What is the relation between the truth values of A and ∀?

A
60
Q

What is the base of a Hexadecimal number?

A

16

61
Q

Whats the first thing you do when converting binary to octal or vice versa?

A

you build an octal/binary truth table.

62
Q

Regardless if the number we are writing scientific notation for is greater or less than 1, how do we know when to stop moving the decimal place?

A

We stop moving the decimal point when we have have one non-zero digit to the left of it when dealing with numbers greater than 1 & one non-zero digit to the right when dealing with numbers less than 1.

63
Q

00000001, which digit is the LSB?

A

the 1

64
Q

What is a truth table?

A

A truth table is a sequential list of the values of a Boolean expression for each possible combination of values of the variables. It is made up of a Universal table on the left side and the values of the expression on the right side. The Universal table contains all possible input combinations of the values (0 or 1) of the variables in an expression. For n-variables the Universal table has n-columns and 2n rows. An extended truth table is used in determining the truth table values of an expression.

65
Q

How do we multiply numbers given in engineering format?

A

To multiply numbers written in engineering format we multiply the numbers separately and we add the E powers; at the end we write the answer in engineering format as well.

66
Q

Does the MSB still represent the sign bit in two’s compliment notation?

A

Yes MSB = 0 is still positive and MSB = 1 is still negative

67
Q

How does division work with binary numbers?

A
68
Q

What is engineering format?

A
69
Q

How is the place of a digit in a decimal system correspond to it’s specified weight?

A
70
Q

How to convert decimal to octal? (the decimal side “fraction”)

A

You do successive multiplication of the base (8) to the specified decimal point.

71
Q

How do we divide numbers given in engineering format?

A

To divide numbers written in engineering format we divide the numbers separately and we subtract the E powers; at the end we write the answer in engineering format as well. (opposite of subtracting)

72
Q

In Roman numerals, what are the symbols and their corresponding values?

A

I (1), V (5), X (10), L (50), C(100), D(500), M(1000)

73
Q

How do we find the accuracy of an approximate number?

A

The accuracy of an approximate number is given by the number of significant digits it has, it represents how close the measured value is to the real accurate number.

74
Q

What are the steps in writing the scientific notation for a number smaller than 1?

A
75
Q

When the MSB is 0 is the number positive or negative?

A

Positive

76
Q

When is an expression in SUM form? (SUM-OF-PRODUCT)

A

If an expression is made up of several smaller expressions (term) OR-ed together, the expression is in sum form. (e.g)

C + AB + A( B + C) is in sum form where C, AB, and A (B + C) are terms that are OR-ed.

77
Q

What are the 3 rounding rules to follow when dealing with different types of operations?

A
  • When using addition and/or subtraction of approximate numbers we round our answer to the least precise number used in the calculations. - When using multiplication and/or division of approximate numbers we round our answer to the least accurate number used in the calculations. - When using a series of different operations (+ - x /) we will not round any intermediate answers, but we will retain the significant digits based on the operations following BEDMAS. The final answer will be rounded off according to the last operation used. (keep track of every step using the first two methods but only make the actual round at the end, always take the least of the two rounding options.)
78
Q

How do you convert Binary to octal?

A

You group bits of 3 from the binary number, and write the corresponding octal digit using the octal/binary truth table.

79
Q

if 1100(2) is the same as 12 in our decimal system, how would it be displayed if it was in an 8-bit register?

A

00001100(2)

80
Q

Can you do binary subtraction with two’s compliment?

A

Yes, first you have to make sure to show the binary numbers in their proper bit register form, then you find the two’s compliment of the subtrahend and add it to the minuend. *Make sure you cut overflow if there is 9th bit*

81
Q

How do we divide numbers given in scientific notation?

A

To divide numbers written in scientific notation we divide the numbers separately and we subtract the exponent of 10; at the end we write the answer in scientific notation as well.

82
Q

What is the purpose of applying Boolean postulates, properties and/or theorems?

A

To simplify complex Boolean expressions and build a smaller logic block diagram (less expensive circuit)

83
Q

How do we find the precision of an approximate number?

A

The precision of an approximate number is given by the place value of the least significant digit. It represents the ability of the measurement tool to give close range of repeated measurements.

84
Q

How do Logic Block Diagrams work?

A

Take each input and trace through the diagram to a combination point (AND, OR or NOT gate); Combine inputs on other side of the gate and continue to next function point. (e.g) The output of this example is f (A, B, C, D) =

85
Q

10000000, which digit is the MSB?

A

the 1

86
Q

What are the two forms a Boolean expression can be presented in?

A

SUM and PRODUCT form

87
Q

What are used to illustrate Boolean Expressions?

A

To illustrate Boolean Expressions we use Logic Block Diagrams and Truth Tables.

88
Q

An octal system has a base of ___ ?

A

8

89
Q

How many variables do AND gates need?

A

AND gates involve two or more variables. (e.g)

90
Q

What number systems fall under the positional category?

A

Binary, Octal, Decimal & Hexadecimal

91
Q

What is the minimum and maximum values of decimal numbers that can be represented using two’s complement method in 8 bit register & how do you find them?

A

Minimum = -128 Maximum = 127 (e.g) For the lowest value, obviously what we want to do is set the one negative bit (the MSB, or ‘sign bit’) corresponding to −128−128 to 11 and all other bits corresponding to positive values to zero. MSB —–> (1) 0 0 0 0 0 0 0 ^ ^ ^ ^ ^ ^ ^ ^ - (2**7) 0 0 0 0 0 0 0 ^ -128 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = -128 For the highest value, we obviously want to zero the MSB and set all other bits to 1. MSB(0) 1 1 1 1 1 1 1 ^ ^ ^ ^ ^ ^ ^ ^ 0 2**6 + 2**5 + 2**4 + 2**3 + 2**2 + 2**1 + 2**0 ^ ^ ^ ^ ^ ^ ^ ^ 0 + 64 + 32 + 16 + 8 + 4 + 2 + 1 =127

92
Q

Does does subtraction work with binary numbers?

A

(0)(1) (1)(10)

93
Q

When using two’s complement notation, when converting a positive binary number in given number of bits register to its decimal equivalent what needs to be done?

A

You convert binary to decimal just like you would using any other method. Only when the binary number is negative does the process change.

94
Q

How is the place of a digit in a binary system correspond to it’s specified weight?

A

1 0 1 0 . 1 0 1 ^ ^ ^ ^ ^ ^ ^ 3 2 1 0 -1 -2 -3 1 = 2**3 0 = 2**2 1 = 2**1 0 = 2**0 1 = 2**-1 0 = 2**-2 1 = 2**-3

95
Q

What are the two type of number systems?

A

Positional and non-positional

96
Q

How many digits can you have before the decimal point and what is the range of the digit(s) in scientific notation?

A

You can only have one digit before the decimal in scientific notation and that number can’t be smaller than 1 and has to be smaller than 10.

97
Q

What does it mean to write a decimal representation?

A

To write a decimal representation means to express the number in terms of its digits and powers of 10. (The base of decimal number system. (e.g.) 6543 = 6 x 10**3 + 5 x 10**2 + 4 x 10**1 + 3x 10**0

98
Q

What are the 4 rules of addition of binary numbers?

A

0 0 1 1 +0 +1 +0 +1 —- —- —- —- 0 1 1 10

99
Q

How many variable do XOR gates need?

A

Just as many as OR gates, 2 or more

100
Q

How do we multiply numbers given in scientific notation?

A

To multiply numbers written in scientific notation we multiply the numbers separately and we add the exponents of 10; at the end we write the answer in scientific notation as well.

101
Q

What is a key digit?

A

a Key Digit is the digit directly to the right of the digit we are analyzing when we want to round off

102
Q

What is another way OR is refferred as and what is their respected symbol?

A

LOGICAL SUM - OR can also be called disjunction : ( + )

103
Q

What is the step in getting the one’s compliment of a binary number?

A

you invert all of the digits; changing the 1’s to 0’s and the 0’s to 1’s.

104
Q

What does a LOGIC BLOCK DIAGRAM look like?

A
105
Q

What depicts the output of an AND operation?

A

The ouput of an AND operation is 1 only if all inputs are 1; the output of an AND operation is 0 is at least 1 input is 0. (e.g)

106
Q

What does LSB stand for?

A

Least significant bit, and the right most bit.

107
Q

What are the 4 rules in significant digits?

A
  • All non-zero digits are significant - All zeros between two significant digits are significant - Leading zeros are not significant - Trailing zeros are significant only if after the decimal point
108
Q

What is PRODUCT form? (PRODUCT-OF-SUM)

A

If an expression is made up of several smaller expressions AND-ed together, the expression is in product form. (e.g)

(A + B)(A +C)D is in the product form where A + B, A + C and D are AND-ed together.

109
Q

How are Boolean expressions built?

A

Boolean expressions are built from Boolean statements (A,B…) by a finite number of Boolean operations (x, +, ⊕….) resul in a Boolean value True or False ( 1 or 0 )

110
Q

What does MSD stand for?

A

Most significant digit, and the left most digit.

111
Q

What is the full range of representable decimal numbers using the sign magnitude method?

A

The range goes from +127 to -127, with dual representations of 0, giving a full range of 256 representable decimal numbers.

112
Q

How are Theorems defines in Boolean Algebra?

A
113
Q

What are the two types of numbers a computer can store?

A

Fixed-Point (integers) where the decimal point is fixed after the rightmost digit, and Floating-Point numbers (real numbers) that contain the decimal point.

114
Q

Is the accuracy or precision of an approximate number given by the number of significant digits it has?

A

Accuracy

115
Q

What does “Accuracy” show?

A

Accuracy shows how close the measured (and recorded) number is to the actual true number. e.g. “If we have some measurement tool that is very finely graded we can determine our numbers with a very high precision. HOWEVER, if this tool was wrongly designed our measured numbers will be very inaccurate.”

116
Q

What happens when we try to subtract a bigger binary number from a smaller one when dealing with two’s complement?

A

You can subtract a bigger binary number from a smaller one, but we will expect to receive a negative answer. (e.g) “Minuend”00000011(2) - “Subtrahend” 00001100(2) Subtrahend ———>0 0 0 0 1 1 0 0 Inverting ———>1 1 1 1 0 0 1 1 Adding 1 + 1 ——————– 1 1 1 1 0 1 0 0 “two’s complement” Adding Minuend > + 0 0 0 0 0 0 1 1 ———————- NO overflow bit -> 1 1 1 1 0 1 1 1 Therefore the answer is 1 1 1 1 0 1 1 1 which is negative, and note that there is no overflow bit to cut.

117
Q

What happens to the answer displayed on a calculator screen?

A

Our calculator displays 8 digits and store few more afterwards for further calculations. The answer shown on the screen is rounded off

118
Q

How to convert decimal to Hexadecimal? (the decimal side “fraction”)

A

You do successive multiplication of the base (16) to the specified decimal point. (eg) . | 48x16 = 7.68 7 | 68x16 = 10.88 A | 88x16 = 14.08 E | 08x16 = 1.28 1 | 28x16 = 4.48 4 | 48x16 = 7.68 7 | 68 Therefore .48(dec) = .7AE147 (hex)

119
Q

What are the drawbacks to using the sign magnitude number system?

A
  • there are two ways to represent zero (00000000, 10000000) - this method requires more complex hardware - the range of decimal numbers that can be represented with is small as we are losing one bit to the sign.
120
Q

What is another way XOR is reffered as and what is its respected symbol?

A

EXCLUSIVE OR - XOR can also be called exclusive disjunction - ( ⊕ )

121
Q

How do we add or subtract numbers given in scientific notation?

A

To add and/or subtract numbers written in scientific notation we convert all numbers to the same power of 10; the numbers are then added/subtracted as given. At the end we write the final answer in scientific notation by shifting the decimal point right or left and adjusting the exponent.

122
Q

Define Significant Digits.

A

Significant digits in an approximate (measured) number are digits that are known with certainty to be accurate.

123
Q

What depicts the output of an XOR operation?

A

The output of an XOR operation is true whenever an off number of the inputs are true and is false whenever an even number of inputs are true.

124
Q

When using two’s complement notation, when converting a negative binary number in given number of bits register to its decimal equivalent what needs to be done?

A

To convert a negative binary number in given number of bits register to its decimal equivalent we convert like we normally would, with the exception of leaving the MSB (sign bit that makes it negative) as a negative power of 2, while the rest continues positive as normal. (e.g) Converting 10010011(2) to its negative decimal equiv. (causing negative) MSB —–> (1) 0 0 1 0 0 1 1 ^ ^ ^ ^ ^ ^ ^ ^ - (2**7) + 2**4 + 2**1 + 2**0 ^ ^ ^ ^ -128 + 16 + 2 + 1 = -109 Therefore 10010011(2) = -109(10) **when dealing with two’s complement**

125
Q

How do you convert Hexadecimal into binary?

A

You separate each hexadecimal number, and write the corresponding 4 bit binary equivalent to the hex digit using the hex/binary truth table. (eg) Hex | Bin —————- 0 | 0000 1 | 0001 2 | 0010 3 | 0011 4 | 0100 5 | 0101 6 | 0110 7 | 0111 8 | 1000 9 | 1001 A | 1010 B | 1011 C | 1100 D | 1101 E | 1110 F | 1111 6437(hex) = 6 4 3 7 ^ ^ ^ ^ 0110 0100 0011 0111 = 110010000110111 (bin)

126
Q

Whats the first thing you do when converting binary to hexadecimal or vice versa?

A

you build an hexadecimal/binary truth table. (e.g) Hex | Bin —————- 0 | 0000 1 | 0001 2 | 0010 3 | 0011 4 | 0100 5 | 0101 6 | 0110 7 | 0111 8 | 1000 9 | 1001 A | 1010 B | 1011 C | 1100 D | 1101 E | 1110 F | 1111

127
Q

How is the place of a digit in a decimal system correspond to it’s specified weight?

A

1 2 3 4 . 5 6 7 ^ ^ ^ ^ ^ ^ ^ 3 2 1 0 -1 -2 -3 1 = 8**3 2 = 8**2 3 = 8**1 4 = 8**0 5 = 8*-1 6 = 8**-2 7 = 8**-3

128
Q

How do you convert Binary into Hexadecimal?

A

You group bits of 4 from the binary number, and write the corresponding hexadecimal digit using the hex/binary truth table. (eg) Hex | Bin —————- 0 | 0000 1 | 0001 2 | 0010 3 | 0011 4 | 0100 5 | 0101 6 | 0110 7 | 0111 8 | 1000 9 | 1001 A | 1010 B | 1011 C | 1100 D | 1101 E | 1110 F | 1111 110010000110111 (bin) = 0110 0100 0011 0111 ^ ^ ^ ^ 6 4 3 7 = 6437(hex)

129
Q

What is the difference between the exponent in scientific notation and engineering format?

A

In scientific notation the exponent can be any number positive or negative, greater than or equal to 1, and engineering format can be any number positive or negative, greater than 1, but has to be a multiple of three.. e.g 10**3, 10**6, 10**9…..etc

Math for CS (3 decks)

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