Spotlight: Sharp EL-520L

 Spotlight: Sharp EL-520L

Quick Facts

Model: EL-520L

Company: Sharp

Type: Solar Scientific Algebraic

Power: Solar with battery backup, 2 x LR 44

Case: Slide case

Memory: 7 memory registers: A, B, C, D, X, Y, M (M has memory addition and subtraction)

Years in Production: 1998

Display: 2 line, with results up to 10 digits

Shout out to Spaceboy Jeffy’s Vintage Emporium from where I purchased the calculator.

Scientific Calculations in Two Lines

The Sharp EL-520L is an early two-line scientific calculator. The top line is where mathematical expressions are entered, where the bottom line is where the results are displayed.

Expressions are entered as written, as noted by Sharp’s ADVANCED D.A.L. (Direct Algebraic Logic.

Example (degrees mode, float mode):

(top | bottom)

0.5; Screen: DEG | 0.5

×; Screen: DEG 0.5*_ |

e^; Screen: DEG 0.5*e^_ |

(sin 10°); Screen: DEG 0.5*e^(sin10) |

=; Screen: DEG 0.5*e^(sin10)= | 0.594818475

Modes

The modes of the EL-520L are controlled by the MODE keys and two toggle keys:

Toggle keys:

[ DRG ]: Change mode through the degrees, radians, and grads cycle. This is how the angle mode is changed on scientific calculators from the 1980s.

[ 2ndF ] [ DRG ] ( DRG> ): A variation of angle change. In this variation, a conversion of angle measurement.

[ 2ndF ] [ . ] (FSE): This the display format toggle: Floating point (no indicator), fixed point (FIX), scientific notation (SCI), engineering notation (ENG). The TAB function, accessed by [ 2ndF ] [ +/- ], set the number of decimal points in the fixed point, scientific notation, and engineering notation format.

There is also a modify (MDF) function that internally rounds the result to the fixed decimal point settings. The modify function does not affect anything stored to variables, however. I wish there was a round function (Round(n, decimal points)) instead, to be honest.

The Mode Key:

The [ MODE ] key offers three modes:

0: Normal. Normal calculator mode.

1: STAT x: Single variable statistics. The [ STO ] key becomes the comma key and memory plus key [ M+ ] becomes the data entry key.

2: STAT xy: Linear regression mode. The data is fit to the equation y = a + bx where a is the y-intercept and b is the slope.

In the statistic modes, the variables (A -D, X, Y, M) are not operable.

Replay and Editing Keys

In Normal mode, there are three editing functions:

[ DEL ]: The standard delete character.

[ 2^nd ] [ ← ] (⟲): Takes the cursor to the beginning of the last expression

[ 2^nd ] [ → ] ( ? ): Takes the cursor to the next number after the cursor’s position

There is no insert/replace mode. It seems like Sharp was experimenting with editing commands.

Order of Operations (Quirks?)

In a lot of algebraic calculators, the square function has higher priority than negation. Entering -4² returns -16.

Implied multiplication gets a higher priority than division and regular multiplication. Therefore if we enter the infamous expression:

6÷2(1+2) returns 1 instead of 9.

This is known as PEJMDAS where J is juxtaposition, or implied multiplication. For more information, please see The How and Why of Mathematics’ video from 2019:

https://www.youtube.com/watch?v=4x-BcYCiKCk

I’ll leave the debate at this point. Of course, when it doubt, use extra parenthesis or insert an additional multiplication sign ( × ).

One quirk: To insert variables (A, B, C, D, X, Y, M) into expressions, press [ 2ndF ] [ RCL ] (ALPHA).

Final Thoughts

The Sharp EL-520L is nice, basic level calculator with the ability to enter expressions. The EL-520L has a nice amount of the features that will satisfy most needs.

Source

Flow Simulation, Ltd. “Sharp EL-520 (ADVANCED D.A.L.)” calculator.org https://www.calculator.org/calculators/Sharp_EL-520L.html

Eddie

All original content copyright, © 2011-2026. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Order of Operations: The Pesky Math Problem from the Internet [ 6 ÷ 2(1 + 2) ]

Order of Operations: The Pesky Math Problem from the Internet

Perhaps the World’s Most Problematic Math Problem

You’ve seen this, we all seen this, it’s the math problems that’s put all calculators in jeopardy, it’s the dreaded:

6 ÷ 2(1 + 2)

Is it 1 or is it 9? No matter what we answer, the debate still rages on and on. Why? Because not everyone agrees with the order of operations. And the problem lies with implied multiplication.

Seeing that this issues comes up multiple times of the HP Museum of Calculators (latest can be seen at [Source 5]), it’s time to input my two dollars into the subject. (It was two cents but the price went up due to inflation.)

What is Implied Multiplication?

Implied multiplication occurs when two or more quantities, often with or without a numeric coefficient, are combined with one or more variables without explicit multiplication symbols (*, ⋅, or ×). It is implied that when this notation is used, the quantities are to be added.

Examples:

xy is implied to mean x * y

2x is implied to mean 2 * x

3rs is implied to mean 3 * r * s

If there are two or more quantities and at least one of them is surrounded by parenthesis, this is also implied multiplication.

Examples:

The following are implied to mean 2 * 6:

2(6)

(2)6

(2)(6)

4(x)y is implied to mean 4 * x * y.

Implied multiplication is also known as juxtaposition. [Source 2]

Infix multiplication is multiplication with symbols (*, ⋅, or ×).

Where is implied multiplication in the order of operations?

PEMDAS vs PEJMDAS

PEMDAS and PEJMDAS are the two major order of operations.

In PEMDAS, also known as BODMAS in some parts of the world, put implied multiplication (juxtaposition) on the same level as infix multiplication and division. The video by The How and Why of Mathematics serves an excellent video to explain the difference and the potential problems that can be encountered. [Source 1]

PEMDAS

PEMDAS stands for:

P: parenthesis

E: exponents (and roots) (one-argument functions such as sin, int, abs and log are included)

MD: all multiplication and division, left to right

AS: all addition and subtraction, left to right

Similarly, BODMAS stands for:

B: brackets (serves as the same function as parenthesis)

O: operations (exponents, roots, one-argument functions such as sin, int, abs, and log)

DM: all division and multiplication, left to right

AS: all addition and subtraction, left to right

For this blog post, I will refer to this sequence as PEMDAS.

PEJMDAS

PEJMDAS moves implied multiplication to a higher priority:

P: parenthesis

E: exponents (and roots) (one-argument functions such as sin, int, abs and log are included)

J: multiplication by juxtaposition (implied multiplication), left to right

MD: infix multiplication and division, left to right

AS: all addition and subtraction, left to right

Examples

Problem 1: 48 / 2 * 3

PEMDAS:

48 / 2 * 3

24 * 3

72

PEJMDAS:

48 / 2 * 3

24 * 3

72

Here there is no difference because the multiplication is infix.

Problem 2: 48 / 2 (3)

PEMDAS:

48 / 2 (3)

24 (3)

72

PEJMDAS:

48 / 2 (3)

48 / 6

8

Notice in PEJMDAS the implicit multiplication, as marked by two numbers juxtaposed next to each other separated by parenthesis, takes priority.

Problem 3: 100 – 5 (2 + 3)

PEMDAS:

100 – 5 (2 + 3)

100 – 5 (5)

100 – 25

75

PEJMDAS:

100 – 5 (2 + 3)

100 – 5 (5)

100 – 25

75

Here we arrive at the same problem because subtraction has the lower priority than implication in both cases.

Problem 4: 100 / 5 (2 + 3)

PEMDAS:

100 / 5 (2 + 3)

100 / 5 (5)

20 (5)

100

PEJMDAS:

100 / 5 (2 + 3)

100 / 5 (5)

100 / 25

4

In PEJDMAS, implied multiplication has priority over division.

Conclusion – Which Method Reigns Supreme? Should we Even use Implied Multiplication?

I’m not usually a fan of the phrase “pick a lane”, but for the sake of consistency, I’ll make an exception. I would prefer the world to pick either PEMDAS or PEJMDAS and stick to it as a universal rule. It seems like PEJMDAS might get the favor since more mathematicians, scientists, and professionals prefer it. I view the PEMDAS vs PEMJDAS as similar to the way the world views which are the standard scientific units:

PEMDAS: ft, sec lbs (United States, especially in the education field)

PEJMDAS: m, sec, kg

Having grown up in the United States, it will take me a bit to adjust to PEJMDAS from PEDMAS.

I’m also in favor of just using additional multiplication, parenthesis, and perhaps using a fraction bar to make problems much clearer. To borrow a suggestion from Tony Barlow, a mathematician who tested the ill-fated never-released TI-88:

“Kill Implied Multiplication. Kill Implied Multiplication. Kill Implied Multiplication.” [2]

Disallowing implied multiplication allows us to avoid two potential problems:

1. In calculators with CAS capabilities and in Python, we are allowed variable names with more than one character (rate, chg, pts). If implied multiplication is allowed with variable names with more than one character, and “ratepts” is typed, do I mean “rate * pts”, the entire variable “ratepts”, or something else like “r * a * t * e * p * t * s”?

2. This makes it clear when we mean function calls, which thankfully is usually understood. But to someone who is unfamiliar with functions, something like “sin(x)” may mean “sin * x” or “ s * I * n * x” to them. “f(x)” may be misinterpreted as “f * x”.

Calculators with the classic AOS (algebraic operating system, postfix system), RPN (Reverse Polish Notation), or Chain operation do not deal with implied multiplication:

AOS: In all cases, the multiplication and parenthesis keys must be explicitly pressed. (Examples: TI-30 series, Casio fx-260 series)

RPN: RPN is a parenthesis-less operating system. (Examples: Almost all HP calculators, all Swiss Micros calculators)

Chain: Everything entered in the Chain operating system is done how you enter the keys. Therefore, these calculators don’t even deal with the order of operations, accuracy is completely up to you. (Examples: all basic four-function calculators)

I hope you find this blog entry helpful, and hopefully one day, we have a universal solution.

This is not the only issue with the order of operations, another issue is where to place the unary operator of negation, but that’s for another day.

Eddie

Sources

[1]

The How and Why of Mathematics. “The Problem with PEMDAS: Why Calculators Disagree” August 5, 2019. Accessed April 15, 2024. Video. https://www.youtube.com/watch?v=4x-BcYCiKCk

[2]

Wright, Gene. “HHC 2022: TI-88 Part 1: History and Go / No Go Decision” September 14, 2022. Accessed April 15, 2024. https://www.youtube.com/watch?v=wl16wzmn3wA Video. (Refer to time stamp 13:24).

[3]

“Multiplication” Wikipedia. Edited April 17, 2024. Accessed April 21, 2024. https://en.wikipedia.org/wiki/Multiplication

[4]

“Order of Operations” Wikipedia. Edited April 19, 2024. Accessed April 19, 2024. https://en.wikipedia.org/wiki/Order_of_operations

[5]

“What is the correct result?” Museum of HP Calculators. Thread started on March 18, 2024. https://www.hpmuseum.org/forum/thread-21474.html

Next Post: June 8, 2024

All original content copyright, © 2011-2024. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Retro Review: Casio fx-55 Fraction Mate

 Retro Review: Casio fx-55 Fraction Mate

Quick Facts

Model: fx-55 Fraction Mate

Company: Casio

Years: probably the 1990s or early 2000s, predecessor to the fx-55 Plus* 

Memory Register:  1 independent memory

Battery:  Solar with battery back up (LR44)

Screen:  8 digits, 4 digit numerator, 4 digit denominator, function indicator (+, -, ×, ÷, ^), equals and parenthesis indicators 

Logic:  Algebraic

Comes with hard case and reference card  

In 2018, I reviewed the fx-55 Plus:  https://edspi31415.blogspot.com/2018/08/review-casio-fx-55-plus.html

Right now there are plenty of fx-55 (original keyboard) for sale on auction sites for eBay at really inexpensive prices, sometimes as low as about $2.00.

Target Audiences

*  Elementary and middle school students

*  Anyone who wants a few more functions than a regular four-function calculator, but not a scientific calculator

*  You want a calculator with an emphasis on fractions

Keyboard

The fx-55 Fraction Mate does not have a shift key.   Some of the keys are:

[ + ], [ - ], [ × ], [ ÷ ]:  Arithmetic keys

[ ( ], [ ) ]:  Parenthesis keys

[ x^y ]:  Power key with a caret indicator on the screen

[ 1/x ]:  Reciprocal

[ RAN# ]:  3 digit random number between 0 and 1

[ FIX ]:  Fix setting.  For float setting, press [ FIX ] [ . ]

[ RND ]:  Round number to fixed setting internally

[ 10^x ]:  Antilog function

[ X ←→ Y ]:  Switch operators

[ X ←→ M]:  Exchange number in the display with number in memory

[ % ]: Percent key, see below

[ > ]: Backspace key

[ a ], [ b/c ]:  Fraction entry keys

[ F ←→ D ]:  Fraction/decimal conversion

[ a b/c ←→ d/c ]:  Improper/proper fraction conversion

[ SIMP ]:  Fraction simplification key

[ ÷R ]:  Integer division, with quotient and remainder.  If both numbers are positive, then remainder acts as a modulo operator

[ √ ]: square root

[ x^2 ]: square function

Integer Division

Pressing [ ÷R ] will return both the quotient and remainder of a division calculation.  If both numbers are positive, then the remainder is the modulo.  

Example:  55 [ ÷R ] 12 returns 4 R 7.  Then 55 = 7 MOD 12.

Percent Key

To add % to n, press:

n [ × ] p [ % ] [ + ]

To subtract % from n, press:

n [ × ] p [ % ] [ - ]

Simplify Key

The [ SIMP ] key simplifies fractions when possible.   The factor used to simplify the fraction is briefly displayed.  All fractions on the fx-55 FRACTION MATE are simplified manually.

Example:   1450/2900 = 

1450 [ b/c ] 2900 [ = ]

Display:  = 1450/2900  Simp

The Simp indicator means the fraction can be simplified.

[ SIMP ]

2 brifely displays before 725/1450 Simp

Repeated presses of [ SIMP ]:

5,  145/290 Simp

5,  29/58  Simp

29,  1/2

Order of Operations

The fx-55 follows the order of operations.  This is great for students in algebra.  

Example: 

3 [ + ] 9 [ × ] 11 [ = ] returns 102     (multiplication/division first, then addition/subtraction) 

Verdict

I like the fx-55 Fraction Mate.  It has a lot of additional functions that beyond the traditional four-function calculator:  parenthesis, powers, reciprocal, fractions.  The display and the numbers are big.  I like how the arithmetic and power functions are displayed.   Worth the buy.

Casio Ledudu's page for fx-55, including the manual:  https://casio.ledudu.com/pockets.asp?type=1176&lg=eng

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Review: Ativa 2-Line Calculator DD-2361

 Review:  Ativa 2-Line Calculator DD-2361

Just the Facts:

Model:  DD-2361 (I was stated in the manual)

Company: Ativa (Office Depot)

Type: Four Function

Battery:  Solar with backup battery LR1130

Logic: Algebraic

Memory Registers: None

Price:  $8.99

A Two Line Screen

The Ativa DD-2361 calculator has a two lines: the top line shows the entered expression.  Each calculation can have up to 79 characters.   The bottom line shows answers, which can have up to 12 characters.  

Functions include the percent function which, works like four function calculators.  

Example:  

Top:  7 x 8 - 2 x 9

Bottom:  38

The order of operations are used.  

The keys on the keyboard are huge and responsive.  Expressions can be edited using left and right arrow keys.  Characters are replaced in editing.  

Verdict

I like the big keys.  The DD-2361 does lack a few key features: square root, an independent memory register, and due to the calculator following the order of operations, parenthesis.  This is one of few two-line four-function calculators where you can buy it a store, and not just online.  

Eddie

All original content copyright, © 2011-2020.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.