Numbers

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Numbers you should know.

We assume you know your multiplication facts up to $12\times 12$ . Because you are math teachers, it helps to have some other numbers on the tip of your tongue—or at least be able to figure them out quickly in your head.

Cubes:

(a): $2^3 = \answer {8}$
(b): $3^3 = \answer {27}$
(c): $4^3 = \answer {64}$
(d): $5^3 = \answer {125}$
(e): $6^3 = \answer {216}$
(f): $7^3 = \answer {343}$
(g): $8^3 = \answer {512}$
(h): $9^3 = \answer {729}$
(i): $10^3 = \answer {1000}$
(j): $11^3 = \answer {1331}$

Factorials: $4!$ means $4\cdot 3\cdot 2\cdot 1$ .

(a): $1!= \answer {1}$
(b): $2!= \answer {2}$
(c): $3!= \answer {6}$
(d): $4!= \answer {24}$
(e): $5!= \answer {120}$
(f): $6!= \answer {720}$
(g): $7!= \answer {5040}$

Powers of 2:

(a): $2^4 = \answer {16}$
(b): $2^5 = \answer {32}$
(c): $2^6 = \answer {64}$
(d): $2^7 = \answer {128}$
(e): $2^8 = \answer {256}$
(f): $2^9 = \answer {512}$
(g): $2^{10} = \answer {1024}$

Computers and digital electronics are built from transistors, which can function as electronic switches. Because a switch can be either off or on, switch positions can interpreted as the digits $0$ and $1$ , respectively, in a base $\answer [format=string]{two}$ numeration system. This is why the powers of two turn out to be very important in computer science. In particular, because $2^{10}=\answer {1024}$ , which is approximately $1000$ , computer science borrowed some base-ten prefixes from the metric system.

Computer memory and storage is typically measured in bytes, where one byte can store one “character,” such as a letter, punctuation mark, or space in this sentence. With the metric prefixes, we approximate powers of two with powers of ten. For example:

(a): $1$ Kilobyte (1 KB) $= 2^{10}\textrm { bytes }\approx \answer {10^3}$ bytes = one thousand million billion trillion bytes.
(b): $1$ Megabyte (1 MB) $= \answer {2^{20}} \textrm { bytes } \approx \answer {10^6}$ bytes = one thousand million billion trillion bytes.
(c): $1$ Gigabyte (1 GB) $= \answer {2^{30}} \textrm { bytes } \approx \answer {10^9}$ bytes = one thousand million billion trillion bytes.
(d): $1$ Terabyte (1 TB) $= \answer {2^{40}} \textrm { bytes } \approx \answer {10^{12}}$ bytes = one thousand million billion trillion bytes.

In computer storage, 1 byte is composed of 8 bits, each of which is a base-two digit, either 0 or 1. Today, internet speeds are usually given in Mb/sec, which means “Megabits per second.” So if your Internet speed is $100$ Mb/sec, that means about $\answer {100\cdot 10^6}$ bits per second. (Type your answer without commas.)

Computer processor speeds are typically given in GHz, where 1 Hz = 1 cycle per second. Many processor operations require 1 cycle to complete, so we can imagine that a 2.4 GHz processor completes about $\answer {2.4\cdot 10^9}$ operations per second. (Most modern processors have several ‘cores’ which can perform operations at the same time. But never mind.)

The following comparisons benefit from scientific notation.

In 1984, a fully-equipped IBM PC/XT had $640$ KB of memory, a $10$ MB hard drive, and processor running at $4.77$ MHz. In those years, it was possible to connect to other computer users through an “online service provider” (the World Wide Web did not exist), with a modem operating at 300 bits per second, tying up the phone line, and at a cost of $\$10$ —or even as much as $\$30$ per hour. (A leading online service provider was Compuserve, based in Columbus, Ohio.)

In contrast, a typical 2021 laptop computer might have a 2.4 GHz processor, 16 GB of memory, a 512 GB (solid state) “hard” drive, and an internet connection operating at $100$ Mb/sec.

By what factors do the capacities of a 2021 computer improve upon the 1984 version?

(a): Processor speed: $\answer [tolerance=20]{2.4\cdot 10^9/(4.77\cdot 10^6)}$ .
(b): Disk storage: $\answer [tolerance=2000]{512\cdot 10^9/(10\cdot 10^6)}$ .
(c): Memory: $\answer [tolerance=1000]{16\cdot 10^9/(640\cdot 10^3)}$ .
(d): Online speed: $\answer [tolerance=17000]{100\cdot 10^6/300}$ .