Binary Numbers

Skills

Evaluate solution options.
Implement and apply an algorithm.
Explain how abstraction manages complexity.

Essential Knowledge :

Learning Objective: Explain how data can be

represented using bits.

DAT-1.A.1

Data values can be stored in variables, lists

of items, or standalone constants and can be

passed as input to (or output from) procedures.

Computing devices represent data digitally,

meaning that the lowest-level components of

any value are bits.

Bit is shorthand for binary digit and is either

0 or 1.

A byte is 8 bits.

Abstraction is the process of reducing

complexity by focusing on the main idea.

By hiding details irrelevant to the question at

hand and bringing together related and useful

details, abstraction reduces complexity and

allows one to focus on the idea.

Bits are grouped to represent abstractions.

These abstractions include, but are not limited

to, numbers, characters, and color.

DAT-1.A.7

The same sequence of bits may represent

different types of data in different contexts.

DAT-1.A.8

Analog data have values that change smoothly,

rather than in discrete intervals, over time.

Some examples of analog data include pitch

and volume of music, colors of a painting, or

position of a sprinter during a race.

DAT-1.A.9

The use of digital data to approximate realworld analog data is an example of abstraction.

DAT-1.A.10

Analog data can be closely approximated

digitally using a sampling technique, which

means measuring values of the analog signal at

regular intervals called samples. The samples

are measured to figure out the exact bits

required to store each sample.

Learning Objective: Explain the consequences of

using bits to represent data.

DAT-1.B.1

In many programming languages, integers

are represented by a fixed number of bits,

which limits the range of integer values and

mathematical operations on those values. This

limitation can result in overflow or other errors.

DAT-1.B.2

Other programming languages provide

an abstraction through which the size of

representable integers is limited only by the

size of the computer’s memory; this is the

case for the language defined in the exam

reference sheet.

DAT-1.B.3

In programming languages, the fixed number

of bits used to represent real numbers limits

the range and mathematical operations

on these values; this limitation can result

in round-off and other errors. Some real

numbers are represented as approximations in

computer storage

Learning Objective:

For binary numbers:

a. Calculate the binary (base 2) equivalent of a positive integer (base 10) and vice versa.

b. Compare and order binary numbers.

DAT-1.C.1

Number bases, including binary and decimal,

are used to represent data.

DAT-1.C.2

Binary (base 2) uses only combinations of the

digits zero and one.

DAT-1.C.3

Decimal (base 10) uses only combinations of

the digits 0 – 9.

DAT-1.C.4

As with decimal, a digit’s position in the binary

sequence determines its numeric value. The

numeric value is equal to the bit’s value (0 or 1)

multiplied by the place value of its position.

DAT-1.C.5

The place value of each position is determined

by the base raised to the power of the position.

Positions are numbered starting at the

rightmost position with 0 and increasing by 1

for each subsequent position to the left.

Binary Numbers

Skills

Essential Knowledge :

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