Even and Odd Integers in Arithmetic

Even and Odd Integers in Arithmetic

Introduction

Let n and m be integers, where:  

n = ..., -3, -2, -1, 0, 1, 2, 3, ...

and 

m = ..., -3, -2, -1, 0, 1, 2, 3, ...

An even integer is any integer that can evenly divided by 2, without remainder.  Hence, if p is an even integer, then p = 2 ∙ n

An odd integer is any integer that can not be evenly divided by 2  (remainder 1).  In this case:  p = 2 ∙ n + 1

Addition

Adding two integers will result in an integer.  

even + even = even

2 ∙ n + 2 ∙ m  

= 2 ∙ (n + m)

odd + odd = even

(2 ∙ n + 1) + (2 ∙ m + 1)

= 2 ∙ n + 2 ∙ m + 2

= 2 ∙ (n + m + 1)

even + odd = odd

(2 ∙ n) + (2 ∙ m + 1)

= 2 ∙ n + 2 ∙ m + 1

= 2 ∙ (n + m) + 1

Multiplication 

Multiplying two integers will result in an integer.  

even × even = even

(2 ∙ n) ∙ (2 ∙ m)

= 2 ∙ (n ∙ m)

odd × odd = odd

(2 ∙ n + 1) ∙ (2 ∙ m + 1)

= 4 ∙ m ∙ n + 2 ∙ n + 2 ∙ m + 1

= 2 ∙ ( 2 ∙ m ∙ n + n + m) + 1

even ×  odd = even

(2 ∙ n + 1) ∙ (2 ∙ m)

= 4 ∙ m ∙ n + 2 ∙ m

= 2 ∙ (2 ∙ m ∙ n + n)

even^2 = even

(2 ∙n)^2 

= 4 ∙ n^2

= 2 ∙ (2 ∙ n ∙ n)

odd^2 = odd

(2 ∙ n + 1)^2

= 4 ∙ n^2 + 4 ∙ n + 1

= 2 ∙ (2 ∙ n ∙ n + 2 ∙ n) + 1

 

Eddie 

All original content copyright, © 2011-2022.  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. 

Calculos: Mathematics Calculation Game (Kickstarter)

Calculos:  Mathematics Calculation Game  (Kickstarter)

The Card Game

Xavier Wilain created the card game Calculos, which is based on quick, mental mathematics dealing with the four arithmetic operations:  addition, subtraction, multiplication, and division.  There is a mixed version that has all four arithmetic operations.  

Players start with a card and try to find a calculation that matches any of the calculations on the center card.  

For example, if the center card has:

2 × 5

3 × 6

5 × 5

4 × 4

If you have any of the calculations on your card, say 4 × 4, you shout "4 times 4 equals 16" and claim the card.  The object is for find a calculation faster than their opponents.   

Here is a video of the card game and how to play it:

https://www.youtube.com/watch?v=-PEfEVTV4wI

The card game comes in four levels: each level has a different set of calculations on the card, from four to eight.   On the XW_SeriousGames website, each level can be purchased separately or altogether for each operation.

Link:  https://www.xwseriousgames.com/

The Mobile App (Kickstarter)

Wilain is running a Kickstarter campaign to adapt the card game into a mobile app that contains all 20 levels (4 four levels for each arithmetic operation plus one for mixed) with plans to add levels in the future.  Wilain has goal to raise 33'000 CHF (Swiss Francs, which is $35,897.63 in US Dollars on 6/28/2021 - conversion provided by Morningstar).  If successful, the mobile app is set to begin development in October 2021.  The target fund date is July 27, 2021.  

Kickstarter page:

https://www.kickstarter.com/projects/xw-serious-games/calculos/description

Please check this game out, it promises to be a great way to strength our mental calculation skills.  

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. 

Sum of Increasing Series

 Sum of Increasing Series

The Sum From 1 to N - Increasing by 1

It is well known that sum from 1 to N, which each term increasing by 1 is:

1 + 2 + 3 + 4 + ... + N = ∑ x from x = 1 to N = (N + 1) * N / 2

The derivation is fairly easy.  Let S be the sum:

S = 1 + 2 + 3 + ... + N-1 + N

Add S to both sides:

2 * S = 1 + 2 + 3 + ... + N-1 + N + 1 + 2 + 3 ... + N-1 + N

Thanks to the commutative property of addition, we can arrange terms, and with a clever and creative way of arranging terms:

2 * S = ( 1 + N ) + ( 2 + N - 1 ) + ( 3 + N - 2 ) + ... + ( N - 1 + 2 ) + ( N + 1 )

Note that:

2 * S = 

1   +    2   +   3   + ... +  N-1  + N +

N  +  N-1 + N-2  + ..  +   2   +  1

Written this way, there are N "pairs".  Hence:

2 * S = ( 1 + N ) + ( N + 1 ) + ( N +1 ) + ... + ( N + 1 ) + ( N + 1 )

2 * S = ( N + 1 ) * N

Solving for S:

S =  ( N + 1 ) * N / 2

The Sum From 1 to N - Increasing by 2

Now lets consider the sum:

1 + 3 + 5 + 7 + ...

Fun fact: Adding odd numbers in this fashion will always total perfect squares.

1 + 3 = 4 = 2^2

1 + 3 + 5 = 9 = 3^2

1 + 3 + 5 + 7  = 16 = 4^2

1 + 3 + 5 + 7 + 9  = 25 = 5^2

... and so on

Let's define S as the sum:

S = 1 + 3 + 5 + ... + N-4 + N-2 + N

Note that

1 = 2 * 0 + 1

3 = 2 * 1 + 1

5 = 2 * 2 + 1

and so on...

For integer q,

N = 2 * q + 1

Then:

S = 1 + 3 + 5 + ... + ( 2*(q-2) + 1 ) + ( 2*(q-1) + 1) + ( 2*q + 1 )

Using the same strategy as last time:

2 * S = 1 + 3 + 5 + ... + ( 2*(q-2) + 1 ) + ( 2*(q-1) + 1) + ( 2*q + 1 ) 

           + 1 + 3 + 5 + ... + ( 2*(q-2) + 1 ) + ( 2*(q-1) + 1) + ( 2*q + 1 )

2 * S =

     1         +          3            +         5            + ... + (2*(q-2) + 1) + (2*(q-1) + 1) +   (2*q + 1) +

(2*q + 1) + (2*(q-1) + 1)  + (2*(q-2) + 1) +  ... +             5        +            3        +     1

Combine each pair as such:

2  * S = (2*q + 2) + (2*q + 2) + (2*q + 2) + ... + (2*q + 2) + (2*q + 2) + (2*q + 2)

Since q starts at q = 0, there are q+1 "pairs".  

2 * S = (q + 1) * (2*q + 2)

S = (q + 1) * (2*q + 2) / 2

To show that the sum is a perfect square:

S = (q + 1) * (2*q + 2) / 2

= (2*q^2 + 2*q  + 2*q + 2) / 2

= (2*q^2 + 4*q + 2) / 2

= (q^2 + 2*q + 1)

= (q + 1)^2

Example:

Calculate 7^2 (albeit the "longer" way):

7 = q+1;  q = 6:

S = (6 + 1) * (2*6 + 2) / 2 = 7 * 14 / 2 = 7 * 7 = 49

The Sum From 1 to N - Increasing by 3

Now add the series:

S = 1 + 4 + 7 + 10 + ... + N - 3 + N

where each term increases by 3.  Letting q be a positive integer and noting that

1 = 3*0 + 1

4 = 3*1 + 1

7 = 3*2 + 1

10 = 3*3 + 1

and so on...

Then:

S = (3*0 + 1) + (3*1 + 1) + (3*2 + 1) + .... + (3(q-1) + 1) + (3q + 1)

2* S 

= (3*0 + 1) + (3*1 + 1) + (3*2 + 1) + ... + (3(q-1) + 1) + (3q + 1) 

+ (3*0 + 1) + (3*1 + 1) + (3*2 + 1) + ... + (3(q-1) + 1) + (3q + 1)

2 * S 

= (3*0 + 1) + (3*1 + 1)     + (3*2 + 1)     + ... + (3(q-1) + 1) + (3q + 1) 

+ (3*q + 1) + (3(q-1) + 1) +(3(q-2) + 1) + ...  +  (3*1 + 1)    + (3*0 + 1)

There are q+1 pairs.

2 * S 

= (3*q + 2)  + (3*q + 2) + (3*q + 2) + ... + (3*q + 2) + (3*q + 2)

2 *S = (q + 1) * (3*q + 2)

S = (q + 1) * (3*q + 2) / 2

Example:

q = 5  (terms from 0 to 5)

S = (5 + 1) * (3*5 + 2) / 2 = 6 * 17 / 2 = 51

1 + 4 + 7 + 10 + 13 + 16 = 51 

The Sum From A for q terms, increasing by D

Let A ≥ 0, D > 0 and q be a positive integer, and A is a starting term, let the sum S be:

S = A + (A + D) + (A + 2*D) + (A + 3*D) + ... + (A + (q-2)*D) + (A + (q-1)*D) + (A + q*D)

Since we start at q = 0, there are q+1 terms.

2*S =

A + (A + D) + (A + 2*D) + (A + 3*D) + ... + (A + (q-2)*D) + (A + (q-1)*D) + (A + q*D) +

A + (A + D) + (A + 2*D) + (A + 3*D) + ... + (A + (q-2)*D) + (A + (q-1)*D) + (A + q*D)

2*S = 

A + (A + D) + (A + 2*D) + ... + (A + (q-2)*D) + (A + (q-1)*D) + (A + q*D) +

(A + q*D) + (A + (q-1)*D) + (A + (q-2)*D) + ... + (A + 2*D) + (A + D) + A

2*S = (2*A + q*D) + (2*A + q*D) + (2*A + q*D) + ... + (2*A + q*D) + (2*A + q*D) + (2*A + q*D) 

2*S = (q + 1) * (2*A + q*D)

S = (q + 1) * (2*A + q*D) / 2

Example:

A = 10, D = 7; q = 4  (5 terms, increase by 7, initial term is 10)

(4 + 1) * (2*10 + 4*7) / 2 = 5 * (20 + 28) / 2 = 120

Note:  10 + 17 + 24 + 31 + 38  = 120

Source:

Knott, Dr. Ron "Proving that 1+2+3+...+n is n(n+1)/2"  February 12, 2003.  Accessed December 6, 2020.  http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/runsums/triNbProof.html

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. 

Sines and Cosines: Adding and Subtracting Angles

 Sines and Cosines:   Adding and Subtracting Angles

Note:   

π/2 radians = 90°,   π radians = 180°

Sine

sin(x + π/2) = sin(x) cos(π/2) + cos(x) sin(π/2) = cos(x)

sin(x - π/2) = sin(x) cos(π/2) - cos(x) sin(π/2) = -cos(x)

sin(π/2 - x)  = sin(π/2) cos(x) - cos(π/2) sin(x) = cos(x)

sin(x + π) = sin(x) cos(π) + cos(x) sin(π) = -sin(x)

sin(x - π) = sin(x) cos(π) - cos(x) sin(π) = -sin(x)

sin(π - x)  = sin(π) cos(x) - cos(π) sin(x) = sin(x)

Cosine

cos(x + π/2) = cos(x) cos(π/2) - sin(x) sin(π/2) = -sin(x)

cos(x - π/2) = cos(x) cos(π/2) + sin(x) sin(π/2) = sin(x)

cos(π/2 - x) = cos(π/2) cos(x) + sin(π/2) sin(x) = sin(x)

cos(x + π) = cos(x) cos(π) - sin(x) sin(π) = -cos(x)

cos(x - π) = cos(x) cos(π) + sin(x) sin(π) = -cos(x)

cos(π - x) = cos(π) cos(x) + sin(π) sin(x) = -cos(x)

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. 

Sumat 6

 Sumat 6

Introduction

The Addimult Sumat 6 is a mathmatical aid designed to aid the user in addition and subtraction calculations.  Addimult was primarily based in Germany and Liechtenstein, which produced mechanical slides of allowing the user to add and subtract numbers of different amounts of digits (usually from 5 to 9).    Addimult produced sliders from the 1930s to the 1960s.  

This summer, I found and purchased a Sumat 6 in antique shop in Manhattan Beach, California.  It was fortunate that the Sumat 6 came with a nice pouch, which was still intact and instructions.   It is a very classy machine which held up well for at least a 50 year old machine.

The Sumat 6 presented today has a six digit capacity.  The slider is made of metal and has two sides:

The Addition Side  (slider bar on top)

The Subtraction Side (slider bar on the bottom)

The Sumat 6, like all the mechanical slides produced at the time, has a metal stylus.  The stylus is used to push columns up and down in holes.  On top of each column has a circle that shows its value.  The markers in each column are either metallic silver or painted red.  For example, on the Addition side, a column that shows three red markers followed by seven silver markers gives the column a value of 3.   On the Subtraction side, a column that shows seven red markers followed by three silver markers shows a value of 3.  

Addition and Subtraction

The value changes depending on what direction you push the slide:

Operation	Slide Down	Slide Up
Addition	value increases	value decreases
Subtraction	value decreases	value increases

There are two additional markers, shown in red:  the up arrow and the down arrow.

Red Up Arrow ( ↑ ) : Push the next left column down 1.   Scroll the current column all the way up (0 on the Addition side, 9 on the Subtraction side)

Red Down Arrow ( ↓ ) : Push the column down one spot (to 0 on the Addition side, 9 on the Subtraction side)  

Updating one column does not automatically update other columns, so you need to keep your addition and subtraction processes in your head as you are calculating.  

Examples

Example 1:  425 + 784

Use the Addition side.   Set the sliders to 000425.  

Start with the ones digit, which in this case is the right-most column.  Slide it down four notches.  So far, so good. Readout should have 000429.

Now the tens, slide the column down 8. Note that after 8 notches we hit the red up arrow.   Slide the hundred column down 1  and reset the tens column to 0.  You are done in the tens column since 8 notches are slid.  Readout should have 000509.

Finally the hundreds column, slide it down 7 notches.  Note after 5 notches, we hit the red up arrow again.   Time to slide the thousands column down 1 and reset the the hundreds column to 0.   We need to slide the hundreds column 2 more notches (5 of the 7 have been used).    The readout now has 001209.

The answer:  425 + 784 = 1209

Example 2:  1365 - 746

Use the Subtraction side.  Set the sliders to 001365.  

In this problem the right-most column represent the ones digit.

Start with the ones digit and slide it down 6.  Note after six notches until the red up arrow is reached.   Slide the tens column down 1 and reset the ones column to 9.  Readout now should read 001359.

Now slide the tens column down 4.  Readout:  001319.

Now the hundreds, slide it down seven notches.  After four notches, the red up arrow is reached.   Slide down the thousands column and reset the hundreds column to 9.  We have three notches left, so slide the hundreds column down 3.  Readout:  000619.

We are done.

The answer:  1365 - 746 = 619

Sources

International Slide Rule Museum "Mechanical Slide & Rotary Adders" 2013-2018.  https://www.sliderulemuseum.com/Adders.htm   Retrieved September 5, 2020

Riches, David M.  "Addimult" Mathematical Instruments: A private collection.  August 2020.  http://www.mathsinstruments.me.uk/page25.html   Retrieved September 5, 2020.  

Please visit or support the resources above!

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. 

A Look at Some 1st Grade Common Core Problems

A Look at Some 1st Grade Common Core Problems

Common Core… Time to Dive In

My goddaughter is entering first grade this year.  We all heard of Common Core Mathematics.  My introduction to Common Core was an unfortunate one: a Facebook post where a student was marked wrong because a drawing wasn’t done correctly.  However, since Common Core in the United States is a reality and since I anticipate being asked for help with the math homework, I’m going to need to check out what is being taught in first grade. 

I’m going to highlight some types of problems that might be encountered in a first grade class. 

I want to thank A+ Plus Math Coach, link: http://www.aplusmathcoach.com for posting worksheets that cover Common Core math from grades K-5.  Most of the type of problems I will talk about in today’s blog post is based of these worksheets. 

Math Skills Emphasized

* Place Value, Tens and Ones

* Addition

* Subtraction

* Simple Graphs

* Counting

It would help if the student knows all the addition facts involving 0 to 10.  Yes, that addition table we memorized as kids is still helpful.

For a full list of skills (it’s a long one), click here:  http://www.corestandards.org/Math/Content/1/OA/

Counting Problems

Sample Problem 1:  How many balloons are in the box?

 This is fairly simple, we should count 12 balloons.

Other counting problems have students arranging objects into groups of ten when possible. 

Sample Problem 2:  Rhonda has six pieces of candy and Rita has eight pieces of candy.  How many pieces of candy do they have together?

Illustrated is the candy Rhonda (dark brown pieces) and Rita (tan pieces) have.  One exercise to have the student group the objects in tens when possible.  For this problem, we can have one group of ten pieces while four pieces are left over.  1 ten and 4 ones make 14. 

Adding and Subtracting

Most adding and subtracting problems should be straight forward.  As I mentioned before, it will help greatly if the student knows their addition tables.

Algebra, Without the Symbols

Some problems are presented as they come from first level algebra.  Instead of “Solve for X”, it’s “Solve for the blank space”.

Sample Problem 3:  Fill in the blank:  5 + ___ = 8. 

If the student knows their addition, the student would come up with 3 as the answer.  Sometimes, the problem states “You have 5 units.  How many units do you need to make 8 units?”

Sample Problem 4:  Think of the problem 8 + __ = 10 to solve 10 - ___ = 8.

“You have 8 units.  How many units do you need to make 10 units?”

“You have 10 units, how many units do you need to give away to have 8 units left?”

The answer to both questions is 2.  If someone finds a better way to explain this, please post this in the comments. 

Sample Problem 5:  The following sentence is false:  5 + 7 + 9 = 16.  Remove one of the numbers on the left to make the sentence true.

The goal here is to find the two addends in the sentence that add up to 16.  From the three possibilities:

5 + 7 = 12, no

5 + 9 = 14, no

7 + 9 = 16, yes

Since 7 and 9 are required, the 5 needs to be removed.

Doubles

One concept that may be introduced is the concept of additive doubles.  Simply put, the doubles are:

1 + 1 = 2	2 + 2 = 4	3 + 3 = 6	4 + 4 = 8	5 + 5 = 10
6 + 6 = 12	7 + 7 = 14	8 + 8 = 16	9 + 9 = 18	10 + 10 = 20

How can this come into play?  Some adding problems can be labeled as double plus one and double minus one. 

Sample Problem 6:  7 + 8

This can be seen as a double plus one problem. Note we can break the 8 into 7 + 1, then we have 7 + 7 + 1 (a double addition of 7). 

The thought process:

If 7 + 7 = 14 (double 7 fact)

Since 8 is 1 more than 7, add 1 to 14:

14 + 1 = 15.

An alternate strategy is the doubles minus one strategy. 

Start by recognizing that 8 + 8 = 16 (double 8 fact)

Since 7 is 1 less than 8, subtract 1 from 16:

16 – 1 = 15

This a strategy that facilitates mental math.

For video explanation of doubles, click on this link from Stephanie K:   https://www.youtube.com/watch?v=mbKkasLm5DY or from Bob Kowalec:  https://www.youtube.com/watch?v=elj4aup0wJk 

Graphs and Polls

I saw several problems that would require students to read graphs. 

Sample Problem 7:  Look at the table below, as a classroom of students in Mrs. Roberts said what their favorite toy is.  Mrs. Roberts tallies the results in the box below:

What is the most popular toy?  How many students participated in the poll?  (and similar questions)

This type of problem encourages counting and reading graphic representation of polls.  (Answers:  video game, 3 + 6 + 7 = 16)

Final Remarks

Other problems that I saw included requiring students to mentally add and subtract 10, and compare numbers between 10 and 99 using place value (compare tens digit first, then if they are the same compare the ones digit first). 

This is not going to be every problem that could be presented in a first grade Common Core math class, but I wanted to get an idea of what is taught in the classrooms.  I don’t know how much different first grade math is from today from when I was in first grade, which is 35 years ago. 

Eddie

All original content copyright, © 2011-2018.  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.  Please contact the author if you have questions.

HP 12C: Complex Arithmetic (Addition, Multiplication, and Division)

HP 12C:   Complex Arithmetic (Addition, Multiplication, and Division)

Let A and B represent the two complex numbers

A = R1 + R4*i

B = R2 + R5*i

Where i = √-1

Store real components in R1 and R2, and imaginary parts R4 and R5.  Choose the arithmetic calculations:

Enter 0 for addition:  A + B

Enter 1 for multiplication:  A * B

Enter 2 or otherwise for division:  A ÷ B

Result:  C = R3 + R6*i

Program:

STEP	CODE	KEY
01	44, 0	STO 0
02	43, 35	X=0
03	43, 33, 09	GTO 09
04	1	1
05	30	-
06	43, 35	X=0
07	43, 33, 18	GTO 18
08	43, 33, 35	GTO 35
09	45, 1	RCL 1
10	45, 2	RCL 2
11	40	+
12	44, 3	STO 3
13	45, 4	RCL 4
14	45, 5	RCL 5
15	40	+
16	44, 6	RCL 6
17	43, 33, 63	GTO 63
18	45, 1	RCL 1
19	45, 2	RCL 2
20	20	*
21	45, 4	RCL 4
22	45, 5	RCL 5
23	20	*
24	30	-
25	44, 3	STO 3
26	45, 1	RCL 1
27	45, 5	RCL 5
28	20	*
29	45, 2	RCL 2
30	45, 4	RCL 4
31	20	*
32	40	+
33	44, 6	STO 6
34	43, 33, 63	GTO 63
35	45, 2	RCL 2
36	36	ENTER
37	20	*
38	45, 5	RCL 5
39	36	ENTER
40	20	*
41	40	+
42	44, 0	STO 0
43	45, 1	RCL 1
44	45, 2	RCL 2
45	20	*
46	45, 4	RCL 4
47	45, 5	RCL 5
48	20	*
49	40	+
50	45, 0	RCL 0
51	10	÷
52	44, 3	STO 3
53	45, 4	RCL 4
54	45, 2	RCL 2
55	20	*
56	45, 1	RCL 1
57	45, 5	RCL 5
58	20	*
59	30	-
60	45, 0	RCL 0
61	10	÷
62	44, 6	STO 6
63	45, 3	RCL 3
64	31	R/S
65	45, 6	RCL 6
66	43, 33, 00	GTO 00

Example:

A = 4.25 + 3.16*i, B = -2.27 + 1.04*i

Option 0 (+):  1.98 + 4.20*i

Option 1 (*):  -12.93 – 2.75*i

Option 2 (÷): -1.02 – 1.86*i

Eddie

This blog is property of Edward Shore, 2017