The Best Non Graphing Calculator
Professor won't let you use a graphing calculator? Here's our roundup of 5 scientific calculators that will help you get through class.
* This post may contain affiliate links and I will be compensated if you make a purchase through my links.
As an Amazon Associate I earn from qualifying purchases.
You’ll need a calculator you can bring with you to classes, work, or your next big test. During special engineering tasks, standardized testing, or ordinary classes, the right calculator is a must.
Types of Calculators
Before you choose a non-graphing calculator, it’s important to understand what sets them apart. Basic calculators are only capable of division, multiplication, addition, and subtraction. While some models can also calculate percentages and square roots, they tend to have limited functionality outside of that.
One benefit to a typical, basic calculator is simplicity. You won’t have to struggle to use one of these tools. Graphing calculators are for people who must program customized functionalities into the calculator. They’re also useful for plotting points on a graph.
People who are taking calculus, geometry, or majoring in engineering use these calculators. Note that these tools cost significantly more than non-graphing models. Non-graphing calculators (also called scientific calculators) are for more advanced math problems. If you’re about to start a physics, chemistry, or trigonometry class, you’ll need one.
Now that you have a basic understanding of the most common types of calculators out there, it’s time to choose a non-graphing calculator. Below, we’ve selected a few different models for different needs.
Best Non-Graphing Scientific Calculator
The HP 35s Scientific Calculator
The HP 35s Scientific Calculator has an LCD display and is useful for medicine, science, surveying, and engineering uses. The tool uses RPN (reverse polish notation), a signature feature of the model. Note that this model allows normal algebraic entry, too.
The HP 35s Scientific Calculator can be used for the SAT Reasoning Test, ACT, Levels 1 and 2 in SAT Subject Mathematics tests, AP Physics, and AP Chemistry. You may also use it for NMSQT and PSAT.
The calculator has a library of over 100 constants and built-in functions. It’s pre-loaded and ready to use immediately. The HP 35s Scientific Calculator comes with a 1-year warranty, but double check with Amazon for specifications.
- Choose between RPN and algebraic entry
- Large display for viewing entries
- Adjustable contrast for outdoor use
- Great memory capacity
- Wide shape feels awkward to hold
- Takes non-standard batteries
Best HP Alternative
The Casio FX-5800P-N
The Casio FX-5800P-N is a good alternative to the HP we just reviewed. It can calculate complex numbers, comes with 26 memories, and performs trigonometric functions. There are 128 stored formulae and 40 physical constants on this tool, along with a repetition function.
It has flash memory as a back-up function and calculates using irrational values and fractions. The Casio FX-5800P-N will keep its memory when you put new batteries in, as long as you follow the instructions. It’s easy to program and uses a simple programming language. You won’t have to learn a complex language to use this calculator.
One very convenient feature that users will appreciate is that the Casio FX-5800P-N takes a regular AAA battery. It also has an ergonomic shape that makes it easy to hold for long periods of time.
- Comfortable to hold
- Multi-replay function included
- Hard protective case included
- Easy-to-understand programming language
- Doesn’t connect to a computer with USB
Best Budget Pick
The Texas Instruments Ti-36x Pro
The Texas Instruments Ti-36x Pro offers a lot of graphing calculator features but doesn’t have the graphing component. It has an affordable price and is suitable for use on exams. Engineering majors should find the unit both helpful and intuitive to use.
Most users will notice that it’s simple to use for calculating angle unites in trigonometry and performing fractional equations. The calculator shows stacked fractions, symbols, and math expressions the way they appear in textbooks. You can see multiple calculations displayed on the screen at once.
You may choose between three solvers: system of linear equations, polynomial, and numeric equation. The buttons are made with hard plastic and color-coded for your convenience.
- Comes with a sliding protective case
- Shuts off automatically
- Color-coded keys
- Affordable price
- Lightweight design
- Doesn’t have full complex number support
- Has only one key for accessing 8 variables
Best Lightweight Pick
The Sharp EL-W535TGBBL Calculator
The Sharp EL-W535TGBBL Calculator allows you to enter calculations and displays them as they’re shown in your textbook. It has 422 functions and works for both students and professionals in biology, physics, chemistry, trigonometry, algebra, science, and more.
The Sharp EL-W535TGBBL Calculator uses solar power with a battery backup and has a durable design with a protective case. The LCD display shows 4 lines and the tool has 7 modes including normal, complex, drill, and more. The home key lets you begin fresh as needed from any screen.
- Nice display
- and compact
- Attractive design
- Solar and battery power
- May not last as long as more expensive models
Best for Calculus Students
The Casio FX-115ES
The Casio FX-115ES has been designed to be user-friendly regardless of your level of knowledge. It has enhanced features such as finding derivatives, advanced inequality functionality, and product notation templates. The Casio FX-115ES does matrix inversion, quadratics, and cubics.
It’s super easy to use and has an affordable price. If you’re taking Pre-Calculus, Trigonometry, Statistics, Pre-Algebra or Algebra I or II, General Math, Chemistry, Engineering, or Physics, this tool will support you. It may be used on NCEES exams, AP tests, ACT, SAT, PSAT, and NMSQT tests. The tool comes with battery backup and solar power, so you can use it no matter what. Its functions are intuitive and it has a natural textbook display for easy reading.
The FX-115ES Calculator has more than 280 functions including linear regression, computer science, base arithmetic, complex number calculations, and more. It also has a replay function that allows you to backtrack and recalculate your answers.
- Useful for multiple purposes
- Backtracking feature
- User-friendly functions
- Doesn’t feel very durable
Unless you require one for a very specific purpose, you don’t need to spend a ton on a graphing calculator. Instead, choose a non-graphing (scientific) calculator like one of the options on our list.
The HP 35s Scientific Calculator is our favorite overall due to value for the price and the functions it offers. Maybe you’re a bit intimidated by RPN (although the 35s supports algebraic notation as well. Here’s a good video that explains the HP 35s RPN system.
Non-graphing calculators are in high demand for engineers and students who need to take certification tests that restrict other calculators. They’re useful for physics, trigonometry, and other studies, as well.
Hopefully, this article helped you select the best one for your needs.
Frequently Asked Questions
What is the difference between a CAS graphing calculator and a non-CAS graphing calculator?
A non-CAS calculator solves for numeric results whereas a CAS calculator can give you algebraic answers – integrals, derivatives, simplify equations.
What is the best non-graphing calculator?
That depends on your needs, but the HP 35s is a solid choice. If you’re on a budget then check out the Ti-36x Pro from Texas Instruments.
What is a non-programmable calculator?
A simple numeric calculator without macros or programming functions. Used for basic math calculations like addition, subtraction, multiplication, division, square root etc.
What is RPN?
RPN stands for Reverse Polish Notation. It is an alternate mathematical notation where the operands are entered before the operators (multiplication, division etc.). Some calculators (particular HP) use this method.


1. The finite-infinite equivalence principle
According to Pythagorean and Platonic mathematics, everything is simultaneously governed by two principles: The finite and the infinite.1 Composing these coexisting principles into one comprehensive principle, we postulate that everything is governed by the finite-infinite equivalence principle such that everything is both finite and infinite, or, to put it in another way, both one and infinitely many.
If this is true and governs the ultimate reality of everything, then the ultimate and real language of the universe (everything) is not analytic, but instead synthetic, and the numbers that express the complex finite-infinite nature of the universe are not the natural numbers, which are uniquely finite, but the real numbers, which are both finite and infinite. These are some examples of real numbers:
i) The number one: 1 = 0,99999…
ii) The pi number: π = 3.14159…= 3.14
iii) Euler’s number: e = 2.71828… = 3
iv) The Pythagorian number: √2 =1.41421… = 1.4
What they have in common is that each is the limit of an unlimited number of parts. If the limit is denoted by one digit or by a finite number of digits and the unlimited number of parts by an infinite number of digits, we may then conclude that all real numbers are both finite and infinite in agreement with the finite-infinite equivalence principle.
Now, through the compression of the infinite number of digits into one digit or a finite number of digits, our finite neuronal brain and its finite tools “grasp and enclose the infinite, thus bringing infinity to finite beings.”2 Compressing the infinite number of digits into one digit or into a finite number of digits is what we term the finitization process of the infinite undertaken by our finite brain’s faculty of analytic reason (understanding).
The converse process — that is, infinitely expanding the one digit or the finite number of digits, is the infinitization process of the finite undertaken by our infinite mind’s faculties of imagination (intellectual intuition) and infinite synthetic reason. Mathematicians and scientists continuously and unconsciously perform these two processes. It is the way that science progresses in its effort to observe the unobservable and realize the impossible. Based on the above line of synthetic reasoning, let us examine the nature of space and its significance in comprehending the real language of the universe.
The diagonal of the Pythagorean square is the geometric topos of the complex whole
All things are related to one another by the ratio of their numbers.Philolaos, the Pythagorean
Consider the Pythagorean unit square acbd whose sides are equal to one unit length, as the two-dimensional representation (model) of our universe. We will demonstrate that the square’s diagonal ab is the geometric topos of anything existing as a complex whole or universe possessing the geometric properties of i) complexity, ii) proportionality between its parts, iii) indeterminacy, iv) neutrality (impartiality) v) balance, vi) freedom, vii) free, constant motion at a constant or accelerating speed, and viii) problem-solving action such as resolving conflicts.
There are two complementary ways for the body m to move from a to b. The first is the direct way, along with the diagonal distance ab involving a single displacement; the second is, the indirect way, along with the horizontal distance ad,and the vertical distance db involving two displacements (figure 1.1). The equivalence between these coexisting ways of movement from a to b is expressed by the following synthetic formula:
1.1. ab = ad + db,
which holds that motion along the diagonal ab is equivalent to motion along the horizontal distance ad and the vertical distance db. Because two unequal distances ab and the sum ad + db are equal despite their inequality, we may recast the above synthetic formula in this manner:
1.2. (ab < ad + db)(ab = ad + db).
Formula 1.2 holds that the diagonal distance ab is both unequal and equal to the sum ad + db without contradiction or paradox. In fact, in Euclidean metric geometry, the diagonal ab is less than the sum ad + db, whereas in non-Euclidean, non-metric geometry, ab is equal to the sum ad + db.
Based on the foregoing considerations, we conclude that the diagonal ab is a complex whole or universe receiving simultaneously equal and opposite determinations (parts): for example, the horizontal and vertical parts ad and db, which form a sum to which the complex diagonal ab is both unequal and equal. Moreover, anything existing on the complex diagonal ab,such as the moving body m, is a complex whole or universe traveling simultaneously two different paths, the diagonal path ab and the sum of horizontal and vertical paths ad + db without contradiction or paradox.3
Because the parts ad, db of the complex whole ab form both a sum of separated parts and a totality of conjoined and unified parts as a consequence of the sum’s equality and simultaneity with the single diagonal ab, we may define the diagonal complex whole ab as the sum total of all its parts ad and db:
1. 3. ab = (ad + db)(ad × db),
where + is the sign of addition and × is the sign of conjunction.
If we count each part as 1, then replacing the parts by 1 in the above formula, we obtain the following:
1.4. ab = (1+1)(1×1) = (2 = 1).
This formula states that the complex ab is both many things and one thing ⎯ the sum of at least two things ad and db, which conjoined form one thing, namely the whole ab. Thereby, the diagonal ab verifies the one − many synthetic equivalence principle, which governs the complex whole or universe. The word universe derives from the Latin, uni (“one”) and versus (past participle of the verb vetere, “to turn” or the inverse). The universe is, then, complex and all-embracing, simultaneously one and the inverse, or many without contradiction or paradox. However, if we consider the universe improperly (falsely), that is, as a simple individual, then its opposite determinations are necessarily contradictories (opposites which cannot exist simultaneously) and not contraries (opposites which exist simultaneously) and the idea of the all-embracing universe becomes self-contradictory or paradoxical (see also note 3).
We can also show the equivalence between the diagonal ab and the sum of its parts ad + db by using the one-to-one correspondence method of set theory. If every point m of the diagonal ab corresponds to a unique point n on the horizontal part ad and at the same time to a unique point n′on the vertical part db, then the diagonal ab has the same number of points with the sum of its parts ad + db. Because they have the same number of points, we conclude that ab and ad + db are equal (see formula 1.1).
Another way to show the comprehensive and complex nature of the diagonal ab is this way: Let the square’s horizontal side ad represent one unit of space on the space axis x and the square’s vertical side ac represent one unit of time on the time axis t — the diagonal ab on the axis z represents then, the equality of the opposite parts, namely of space and time, which defines the complex two-dimensional universe or whole (figure 1.2). Thus, when a body m travels at a constant speed, or at a constant acceleration, one unit of space ad on the horizontal space axis x at one unit of time ac on the vertical time axis t, it necessarily travels on the square’s diagonal ab, which is the equality of space and time.
Because the body m travels simultaneously equal units of space ad and time ac, we conclude that their ratio is constant and equal to 1 and that they are equivalent or directly proportional:
1.5. ab =ad/ac = 1 ↔ ad = ac.
Grounded in the formula 1.5, we assert that the square’s diagonal ab is the geometric topos of opposite quantities ad and ac, which are equivalent or directly proportional and have a constant ratio equal to 1. Thus, if one quantity, say ad, is doubled, the other quantity, say ac,is also doubled. Synthetic Logos is the term we use for the unifying link between opposite determinations (parts) ad and ac that obliges them to vary proportionally so that their ratio remains constant. In this way, variation symmetrized by synthetic logos does not disrupt the permanence of the complex two-dimensional universe (space-time) whose geometric topos is the square’s diagonal ab.
The synthetic logos of the square’s diagonal ab
Synthetic logos as a general principle becomes in its modern version the synthetic equivalence principle, which states the equality and simultaneity of opposites; it was formulated by the Greek Ionian philosopher Heraclitus (sixth to fifth century BCE) — the farther of synthetic or dialectic thought. Relative to the formula 1.5, the general synthetic equivalence principle becomes the specific space-time equivalence principle, which stipulates the equivalence and simultaneity of opposites, of the horizontal space distance ad and the vertical time distance ac.
Accordingly, based on the formula 1.5, we state that anything existingin the universe is in its ultimate reality a complex whole m lying on the diagonal ab of the square acbd and therefore lying both on the horizontal space distance ad, more specifically on the point n of ad, and on its equal opposite vertical time distance ac, more specifically on the moment n′ of ac. The complex whole m occupies simultaneously both positions n and n′ in the respective space and time distances ad and ac. A one-dimensional representation of the complex whole is the complex line, which because of its extension occupies both positions n and n′ without contradiction or paradox.
On the other hand, synthetic logos as a mental faculty becomes in its modern version the faculty of synthetic reason, which is the power of our infinite mind (the Greek infinite nous) to conceive the object in its complete and maximum reality (ens realissimum), namely as a real complex whole or universe defined as the sum total of all its determinations (parts) and governed by synthetic logos. The three actions of synthetic logos are thesis, represented by the horizontal part ad; antithesis, represented by the vertical part ac; and synthesis, represented by the square’s diagonal ab comprehending, reconciling and equalizing the orthogonal opposites ad and ac (figure 1.3).
2. The indeterminate logos of the square’s diagonal ab
Another fundamental property of the square’s diagonal ab is its logical and metrical indeterminacy. Because the diagonal ab is equidistant from its opposite sides, it is a third intermediate alternative negating (transcending) both opposite sides (parts- determinations) and in this sense is free from any particular determination. The diagonal ab verifies, therefore, the synthetic principle of the included third, which is a principle of the absolute indetermination of the diagonal ab and which we formalize as follows:
2.1. ab = (ad + ac)′ = (ad)′(ac)′,
where the sign of addition (+) designates either/or and the sign of negation (′) designates not.
Based on formula 2.1, we state that anything existing is an indeterminate whole lying on the diagonal ab and therefore lying neither (not-either) on the horizontal space distance ad (thesis) nor on the vertical time distance ac (anthithesis). The indeterminate whole is neutral and impartial as it lies beyond the partial determinations, i.e. space and time but at the same time synthetic as it embraces them all.
The principle of the included third is the negation of the analytic principle of the excluded third — the principle of the absolute determination of everything — according to which anything existing in the universe is a simple individual, lacking parts or having unequal parts, for example unequal space and time distances. In this case, the simple individual is lying on the diagonal ab of the rectangle acbd whose horizontal side representing the space distance ad and the vertical side representing the time distance ac are unequal. However, such a simple individual exists falsely or incompletely because any being existing in the universe is necessarily in its ultimate reality, a complete whole reflecting the wholeness of the universe. As a complete whole it lies on the diagonal ab of the square acbd whose horizontal side representing the space distance ad and the vertical side representing the vertical time distance ac are equal. Ultimately, we define the diagonal ab as the mutual position (coexistence) and mutual negation (co-absence) of its equal and opposite determinations (parts) ad and ac:
2.2. ab = (ad = ac) = (ad)(ac) = (ad)′(ac)′.
The diagonal space-time distance ab is therefore the geometric topos of any being in the universe, which is in its ultimate nature a complex and indeterminate whole free of all determinations — for example of space and time — by virtue of comprising the totality of determinations.
As a complex whole or universe, the diagonal ab of the unit square acbd verifies the synthetic principle of equivalence, which postulates the equality and coexistence of opposites ad and ac. As an indeterminate whole or universe, the diagonal ab verifies the principle of the included third — the principle of the absolute indetermination of everything — which postulates the equality and co-absence of opposites ad and ac. Insofar as the indeterminate whole ab is equidistant from the opposites ad and ac, it is a balanced being, whereas insofar as it transcends their analytic (contradictory) opposition to find a solution outside and beyond their perennial conflict, it is a problem-free being, in free and constant motion at a constant or accelerated speed.
The circle C is equivalent to the diagonal ab of the Pythagorean square
Before we move on to part 3 of this work, we will demonstrate that the unit circle C, circumscribing the unit square acbd is equivalent to the diagonal ab, which is at the same time the diameter d = 2r (figure 2.1). In other words, we will show that the circle C, which is π times greater than its diameter ab = 2r is equivalent to the diameter ab = 2r. To demonstrate the equivalence between C and ab, we will use the one-to one-correspondence method of set theory. Accordingly, if every point m on ab corresponds to a unique point n on the semicircle C/2= adb and at the same to a unique point n′on the semicircle C/2= acb, then the diagonal ab has the same number of points with the sum of semi-circles C/2 + C/2 defining the unit circle C:
Because the diagonal ab and the unit circle C have the same number of points, we conclude that they are equivalent regardless of their Euclidean metrical inequality:
2.3. ab = C and C = ab,
and that they have due to their equivalence the same intrinsically complex and indeterminate nature. Moreover, as equivalent complex wholes, they constitute the geometric topoi of anything exiting as a complex whole or universe, verifying synthetic principles of existence such as the synthetic equivalence principle, which stipulates the equality and coexistence of opposites, and the synthetic principle of the included third, which stipulates the equality and co-absence (neither/nor) of opposites.
3. The metrical indetermination of the square’s diagonal ab
Let us now examine the metrical indetermination of the diagonal ab of the unit square acbd representing two-dimensionally the universe. Having assumed that each side of the square is equal to one unit length, we measure its diagonal ab by means of the Pythagorean theorem, which we found to be the real finite-infinite number ab = √(ad)2+(db)2 = √12 + 12 = √2 = 1.414….Rounding (finitizing) the real number’s infinite decimal expansion to two significant digits, we write 1.414…= 1.4.
If we consider √2 as a simple point lacking extension, then it necessarily lies somewhere between the points 1.4 and 1.5 on the number line. However, because of its infinite decimal expansion, we do not know where it is exactly located in the interval (1.4, 1.5).4 If we consider the number √2 as a complex line, then it has a continuous extension, which means that it lies everywhere, both on the point 1.4 and on the point 1.5 of its line segment or interval (1.4, 1.5) by passing at once through the intermediate infinity of points; therefore the real number √2 can be equal to any point from 1.4 to 1.5. It follows that the diagonal radius ab = √2 has an indeterminate magnitude, that is, any magnitude ranging from 1.4 to 1.5. In other words the magnitude of the diagonal ab is the line segment or interval (a,b), where a = 1.4 and b = 1.5.
From the computational perspective of analytic reason, the impossibility of determining a unique magnitude to the diagonal ab is interpreted negatively as a sign of our absolute ignorance of its measure. We call negative indeterminacy this absolute ignorance of the real and true magnitude of the diagonal ab, assumed to be simple and one — namely, a simple individual. As we have argued, this negative indeterminacy is the consequence of the impossibility of determining the unique position and value of the number √2 in the line segment or interval (a, b) = (1.4, 1.5).
However, from the non-computational perspective of synthetic reason, the impossibility of determining a unique magnitude to the diagonal ab is interpreted positively — namely as a sign of our complete knowledge of its measure. We call positive indeterminacy this absolute knowledge of the real and true measure of the diagonal ab assumed to be a complex whole having at least two equally real and true magnitudes 1.4 and 1.5 and thus an infinite totality of equally real and true magnitudes occurring within the interval (a, b) = (1.4, 1.5). This positive indeterminacy reflects the real nature of the diagonal ab, conceived as a complex indeterminate whole and at the same time as the geometric topos of anything existing in the universe as a complex indeterminate whole. If anything lies on the diagonal ab of the unit square acbd, then it is necessarily a complex indeterminate whole possessing synthetic and indeterminate logos and conversely, if anything possesses synthetic and indeterminate logos, it necessarily lies on the diagonal ab of the unit square acbd representing our universe (space-time) two-dimensionally.
4. Geometric solution to the mystery of the cosmic expansion rate
Having revealed the intrinsic properties of the square’s diagonal ab, we subsequently proceed to use this diagonal structure of the universe, endowed with synthetic and indeterminate logos, to solve geometrically the mystery of its expansion rate presenting two different values at the same time. To refresh the reader’s memory, the universe’s current expansion rate is called the current Hubble constant H = v/d, which the cosmologists use to connect the universe’s current expansion speed v with its distance from us d.5 It implies, thus, the Hubble law
4.1. v = Hd, where H = v/d,
stating that the recession speed v of a galaxy is proportional to the distance from us d. The farther away the galaxy is from us, the faster it appears to be moving away. If we inverse the Hubble constant H, we obtain the current age t of the Hubble universe:
4.2. 1/H = d/v = t.
The Hubble constant is therefore a cosmological parameter that sets the scale — the age and size — of the universe. Now having defined the Hubble constant, we examine the epistemological question of different measurements assigning different values to it.
Let the unit square acbd circumscribed by a circle of center a and radius ab represent the Hubble universe two-dimensionally. If the square’s horizontal side ad measures the size of the Hubble universe, which is of the order of 1010 = 10 × 109 LY (10 billion light-years) taken as one unit of space, the vertical side ac measures the time (age) of the Hubble universe, which is of the order of 1010=10 × 109 Y(10 billion years) taken as one unit of time, then necessarily the space-time radius of the Hubble universe, which is the square’s diagonal ab, by means of the Pythagorean theorem is ab = √(ad)2+(ac)2 = √12 + 12 = √2 = 1.414…= 1.4 . This measure applied to cosmological dimensions gives relative to space ad the value 1.4 × 1010 = 14 × 109LY (14 billion light-years) and relative to time ac the value 1.4 × 1010 = 14 × 109Y (14 billion years). Their space-time ratio ab = ad/ac =1.4 × 1010LY /1.4 × 1010Y defines the constant speed of light c along the diagonal ab, which we take as a constant of proportionality connecting the space distance ad with the time distance ac. (figure 4.1). As a constant of proportionality unifying opposite (orthogonal) parts, we consider c as equal to unity:
4.3. c = ad/ac =1.4 × 1010LY /1.4 × 1010Y = 1.4×1026m/1.4×3×1017s = 3.333…×108m/s, or, rounding (finitizing) to one significant digit, c = 3×108m/s =1.
As we can see, the diagonal space-time radius ab is measured by the number √2, which insofar as we conceive it as a continuous and extended line, lies everywhere — both on the point 1.4 and on the point 1.5 — by passing at once through the intermediate infinity of points of its line segment or interval (1.4, 1.5) and hence by being equal to any of them (see part 3). In terms of cosmological dimensions, we conclude in an analogical manner that the Hubble universe, whose space-time radius is the complex and indeterminate diagonal ab,has simultaneously the size and age of 1.4 × 1010LY= Y and the equally true size and age of 1.5 × 1010LY = Y by passing at once through the intermediate infinity of equally true sizes and ages greater than 1.4 × 1010LY = Y and smaller than 1.5 × 1010LY =Y. The current Hubble constant H = v/d , which corresponds to the Hubble universe’s current age of 1.4 × 1010 = 14 × 109Y (14 billion years) is 73.5km/s/Mpc, or, rounded to two significant digits, 74km/s/Mpc (74 km per second per megaparsec ); it is obtained from the astrophysical measurements of the late universe (Adam Riess, 2019).6 If we inverse this Hubble constant, we obtain the current age t of the Hubble universe:
4.4. 1/H = d/v = t,
where d = Mpc (one megaparsec) = 106 × 3.09 × 1016 m = 3.09 × 1022 m and v =73.5km/s = 7.35 × 104m/s, and whose value is:
4.5. t = d/v = 3.09 × 1022/7.35 × 104 = 4.20 × 1017s/ 3 × 107s = 1.4 × 1010= 14 × 109 Y (14 billion years).
Similarly, the current Hubble constant H = v/d, which corresponds to the Hubble universe’s equally true current age of 1.5 × 1010 = 15 × 109Y (15 billion years) is 68.6/s/Mpc, or rounded to two significant digits, 69 km/s/Mpc (69 km per second per megaparsec) and is obtained from CMB (cosmic microwave background) measurements of the early universe (Hart, Luke, and Jens Chluba, 2020).7 If we inverse this Hubble constant, we obtain the current age t of the Hubble universe:
4.6. 1/H = d/v = t,
where d = Mpc (one megaparsec) = 106 × 3.09 × 1016 m = 3.09 × 1022 m and v =68.6 km/s = 6.86 × 104m/s, and whose value is:
4.7. t = d/v = 3.09 × 1022/6.86 × 104 = 4.50 × 1017s/ 3 × 107s = 1.5 × 1010= 15 × 109 Y (15 billion years).
The difference between the values 74 and 69km/s/Mpc and therefore between the values 14 and 15 billion years and light-years reflects the different but equivalent ways of measuring the Hubble constant and the age and size of the universe. The inductive method starts with astrophysical measurements in the late or local (here present) universe at a(tnow) and proceeds backward to the origin of the universe at b(t0). From these astrophysical measurements, we derive a current Hubble constant H = v/d equal to 74km/s/Mpc and a cosmic age and size of 1. 4 × 1010 = 14 × 109Y = LY (14 billion years and light-years). The symmetrical opposite deductive method starts with CMB measurements in the early or global (distant past) universe occurring near the origin b(t0) and proceeds forward to the local (hear present) universe at a(tnow). From these CMB measurements performed by the Planck satellite, we derive a current Hubble constant H = v/d equal to 69km/s/Mpc and a cosmic age and size of 1.5 × 1010 = 15 × 109Y = LY (15 billion years and light-years).We have, thus, the space-time interval ab = (a, b), where a = a(tnow) = 1.4 × 1010 and b = b(t0) = 1.5 × 1010 (figure 4.2).
The difference between the two values 74 and 69km/s/Mpc of the Hubble constant must not be interpreted negatively as an unsolvable conflict caused by the impossibility of determining and knowing, which of the alternatives is its only real and true value. This indeterminacy indicates the intrinsically complex and indeterminate nature of the current Hubble constant replicating the complex and indeterminate nature of the universe, whose geometric topos is the square’s diagonal ab. In this sense, the Hubble constant is not a simple determinate quantity— an extensionless point — having at any given time a unique true value, namely either the value a = a(tnow) = 74 or the value b = b(t0) = 69, as the finite analytic paradigm of science claims, but instead a complex indeterminate quantity— an extended line — having both true values a = a(tnow) =74 and b =b(t0) = 69 and therefore an infinite number of equally true values occurring within the line segment or interval ab = (a, b), where a =74 and b = 69.
Similarly, we claim that the age and size of the Hubble universe are in their ultimate realities complex indeterminate quantities- –that is, extended complex lines having both true values a = a(tnow) = 1. 4 × 1010 and b = b(t0) =1.5 × 1010 and therefore an infinite number of equally true values occurring within the line segment or interval ab = (a, b), where a = 1.4 × 1010 and b = 1.5 × 1010.
Based on these considerations, we conclude that the infinite number of values of the Hubble constant setting the infinite number of values of the cosmic age and size generates, in turn, an infinite number of equally true models (versions) of the same Hubble universe having the diagonal radius ab = √2 = (1.4, 1.5). These different models of the Hubble universe have different space-time radii — namely, different sizes and ages occurring within the same space-time line segment or interval ab = (a, b) = (1.4 × 1010, 1.5 × 1010) — and different Hubble constants occurring within the same line-like Hubble constant H = ab = (74, 69). Each model describes a particular value of the Hubble constant that determines a particular value of its cosmic age and size, but is constrained by the same Hubble order of magnitude, namely 1010, and by the same speed of light c =1 unifying space with time.
Because our current detectors (neuronal brain and tools) are finite and computational ruled by analytic principles of existence, we arbitrarily select at any given time one and only one true value out the infinite totality of values of the line-like Hubble constant H = ab = (74, 69) which we take as the current or actual Hubble constant. This actual Hubble constant determines, in turn, one and only one true age and true size out of the infinite totality of ages and sizes occurring in the line-like Hubble universe U = ab = (1.4 × 1010, 1.5 × 1010) and thereby one and only one true model of the Hubble universe out of the infinite totality of models, which we take as the actual Hubble universe. Subsequently, grounded in Leibniz’s principle of sufficient reason, we proceed to explain why we have chosen this particular value rather than another of the line-like Hubble constant ab = (74, 69).8 In reality, however, it is immaterial what value we chose to be the current value of the Hubble constant since according to the principle of the equivalence of synthetic logos or reason, all values are equally true and therefore consistent with the universal laws, fundamental principles, and properties of the Hubble universe.
Thus, the Hubble constant is not a simple individual — a point — having at any given time a unique value, say either the value a = 74km/s/Mpc or the value b = 69km/s/Mpc, but a complex whole— a line segment or an interval— having at any given time both true values a = 74 and b = 69 and hence an infinite totality of equally true values occurring within the line-like Hubble constant H = ab = (74, 69). In this sense, Wendy Freedman’s calculation of the Hubble constant assigning to it the value 69.8km/s/Mpc, or rounded to two significant digits, the value 70km/s/Mpc (70kilometers per second per megaparsec) and determining a cosmic age of 1.48 billion years is equally true as it occurs within the line-like Hubble constant roughly midway between the true values a = 74 (Riess team) and b = 69 (Planck team).9 This shows us that ultimately what is significant for rational cosmology is not the measurement of the Hubble constant, which is observer-dependent having any value from 74km/s/Mpc to 69km/s/Mpc but the discovery of an observer-independent unifying link (proportion) between different things — between the expansion speed v of the galactic universe and its distance from us d, which we call synthetic logos.
From the analytic to the synthetic ontology of existence
We use the term positive indeterminacy to mean the indetermination produced by the complete and maximal knowledge of the real object — for example, of the actual Hubble constant — assumed to be a complex indeterminate whole, that is, a line segment or interval ab = (a, b) possessing simultaneously different determinations a = 74 and b = 69 and therefore an infinite totality of determinations occurring within the interval (a, b). Because all determinations of the real object qua complex whole are equally true, we conclude together with the Greek Ionian philosopher Leucippus (fifth century BCE) that nothing is such rather than such — that no Hubble constant has the value 74km/s/Mpc rather than the value 69km/s/Mpc.10 We consider this egalitarian statement as a principle of synthetic logos, which we use to transform the analytic and conflicting opposition between a =74 and b = 69, where either one of the alternatives is true, into a synthetic and complementary opposition where both alternatives are true or false. Through the synthesis of these opposite and equally accurate and true measurements, we obtain the complete and maximum reality of the Hubble constant determining the complete age and size of the Hubble universe from all viewpoints.
At this point, we embrace a fundamental paradigm transformation of the finite analytic paradigm of cosmology into its negation, the infinite synthetic paradigm regarded as the true language of the physical universe, which will help us to correctly and comprehensively interpret the sense-data. The passage from the analytic to the synthetic paradigm proceeds deductively by using the modus ponens and modus tollens rules: If anything that exists — for example, the Hubble constant— is a simple individual, then it has at any given time, taken as the current time, a unique value, which is either a = 74 or b =69. However, measurements taken from maximally different viewpoints a(tnow) and b(t0) of the universe show that the current Hubble constant simultaneously receives both values a = 74 and b = 69. It follows that it is false that the Hubble constant is a simple individual — a simple determinate quantity verifying analytic principles of existence and represented geometrically by the point. If the proposition “the Hubble constant is a simple individual” is false, then its negation “the Hubble constant is a complex whole” must be true. We then conclude that the Hubble constant is a complex whole — a complex indeterminate quantity verifying synthetic principles of existence and represented geometrically by the line.
The analytic paradigm of science has originated in the analytic logic of the Greek philosopher Aristotle (fourth century BCE) equally called organon (tool). Analytic logic reflects the way our finite analytic detectors — our neuronal brain and tools — operate. By observing them, Aristotle postulated the analytic principles that govern their analytic behavior. As Aristotle put it “No senses applied to the object tells us that the same object is simultaneously so and not so”.11 If we project these principles outward they become the principles that govern the physical universe and anything existing in the universe. However, we commit an epistemological error when we consider these projected analytic principles as if they were the real and true principles of the universe in itself, whereas in reality, they are the principles of the universe submitted to our analytic perception.
A fundamental principle of existence is the principle of contradiction, which stipulates that nothing is simultaneously (both) a and a′(not-a) called b:
4.8. (a = b)′ or (ab)′,
where (′) stands for negation, falsity or impossibility, and = stands for equality, unity, and simultaneity. It follows that everything is either a or b, that is, either a is true or b is true:
4.9. a + b = (a =1) + (b = 1)
where + stands for either/or and 1 stands for true.
This formula constitutes the analytic principle of the excluded third and shows the object as an individual made of contradictory, isolated parts a and b, which are discontinuous points. Both analytic principles assume that anything existing in the physical universe is an individual — a simple point — deprived of extension and therefore incapable of containing both determinations a and b within the same point-object. However, if we would like to have the complete and maximal knowledge of the object, that is to say, to see the object from all viewpoints (parts-determinations), and hence attain its ultimate reality and truth, we must then assume that the object is a complex whole or universe — a complex line — having an extension and therefore capable of containing at least two viewpoints a and b and therefore an infinite totality of viewpoints occurring within the line-object ab = (a, b).
We then replace the analytic principle of contradiction, which is a principle of existence as a simple individual, with the synthetic principle of equivalence, which is a principle of existence as a complex whole. According to it, everything is simultaneously a and a′(not-a) called b — that is, both a and b are true relative to the same thing:
4.10. a = b or ab =(a =1) (b =1),
where = stands for equality, unity, and simultaneity and 1 stands for true.
This formula shows the object as a continuous line ab made of equal and coexisting parts a and b, which are continuous lines.
The synthetic paradigm of science has originated in the synthetic logos of the Greek Ionian philosopher Heraclitus (sixth century BCE) who considered it as the principle of unity of opposites occurring beneath or beyond the surface of phenomena of the universe. As a unifying principle, synthetic logos assigns an invisible unity, intelligibility, and balance to the universe. Because our analytic senses distinguish and separate, the unity of the universe is realized by synthetic logos, which is not only a unifying principle but also, as we have already claimed in part 1, a mental faculty of our infinite mind aiming at the complete and maximum knowledge of the universe — of every object — defined as the sum total of its infinite number of determinations (parts). Because we have access to infinity through its finite part to know completely and maximally the universe ⎯ that is, any object within it, it is necessary and sufficient to know through different measurements its finite number of determinations a and b and through these finite determinations to determine and know, by the finite-infinite equivalence principle, its infinite number of determinations occurring within the line-object ab = (a, b). Similarly, to know completely and maximally the line-like Hubble constant H = ab = (a =74, b = 69), it is necessary and sufficient to know through different measurements its finite number of values, namely the two values a = 74 and b =69 —in order to determine and know its infinite totality of values occurring within its line segment or interval ab.
Based on these considerations, we conclude that the Hubble tension between the values a = 74 and b =69 exists because we tacitly assume an analytic ontology of the Hubble constant that generates an unsolvable conflict between them. Relative to a unique observer who starts his measurements either from the local viewpoint a(tnow) and hence from the here and present universe, or the global viewpoint b(t0) and hence from the distant and past universe, the Hubble constant H is a simple individual— a simple quantity having at any given time, considered as the current time, a unique value that verifies analytic principles of existence. The conflict between the alternative viewpoints a(tnow), giving the value 74, and that of b(t0), giving the value 69, remains unsolvable within the analytic paradigm of the unique observer. However, once we introduce the pair of equal and opposite observers, namely the observer a measuring H from the local viewpoint a(tnow) and the observer b measuring H from the global viewpoint b(t0), the analytic (contradictory) opposition between these alternative viewpoints becomes a conflict-free synthetic opposition where they are simultaneous and equally true or false.
The sum total of all observations ranging from a = 74 to b = 69 by passing through the infinite totality of potential observations occurring within the line segment or interval ab =(a, b) defines our complete and maximum measurement and knowledge of the Hubble constant whose ultimate reality is that of being a complex indeterminate whole.
If the finite Hubble universe and any of its correlated finite quantities such as the Hubble constant determining its age and size are in their ultimate realities equivalent to the infinite physical whole or universe, which we defined as the sum total of its infinite number of parts, we then conclude that any finite part of the infinite universe is a replica of the infinite universe and conversely the infinite universe is equal to any of its finite parts. This constitutes the non-Euclidean synthetic principle of self-similarity, which governs the self-contained infinite universe being both a finite part, namely the Hubble universe, and an infinite physical whole in itself. If the infinite physical universe is a continuous line whose finite parts, for example, the Hubble universe, the Hubble constant, and the Hubble age and size are in their ultimate realities complex infinite wholes — that is, continuous lines — then the real continuum of the physical universe is the infinite whole of infinite wholes represented by the line of lines and governed by synthetic principles of existence.
The passage from analytic ontology favoring the analytic paradigm of dogmatic science to synthetic ontology favoring the synthetic paradigm of non-dogmatic science grounded in synthetic logos is analogous to the passage from natural numbers, which are uniquely finite point-numbers verifying analytic principles of existence, to real numbers, which are both finite and infinite line-numbers verifying synthetic principles of existence and constitute the ultimate reality of natural numbers. The real physical universe is not, therefore, a continuum of natural point-numbers — a line of points — in isolation between them, as dogmatic science claims, but instead a continuum of real line-numbers — a line of lines — in immediate contact between them that replicate the infinite physical whole.
To put it all in perspective and solve the Hubble cosmic crisis, what we need is not a “new physics,” that is, the discovery of new phenomena — of new sense-data — but rather the intelligible reinterpretation of the given sense-data in the light of a holistic and synthetic ontology of existence.
© Ion Soteropoulos, Paris, March 15, 2021
1 Aristotle, Metaphysique A, 5
2 Aristotle, Physique III(4) 203 a10
3 According to Aristotle, anything possessing simultaneously opposite determinations, which he calls contraries, belongs to the class of universals, whereas anything possessing non-simultaneously, that is, successively opposite determinations is an individual (Metaphysique I, 10, 1058b, 30-35). To the Aristotelian definition of the individual, we add that the successive determinations of the individual are necessarily contradictory because of their impossible coexistence.
4 Let us assume the following: If there is the square root of two (√2), then there is a number a such that a × a = 2. Grounded in this assumption let us try to locate √2 on the number line. If we suppose that √2 lies on the point 1.4, then 1.4 × 1.4 = 1.96, which is less than 2. If we suppose that √2 lies on the point 1.5, then 1.5 × 1.5 = 2.25, which is greater than 2. It follows that the number √2 lies neither on 1.4 nor on 1.5 but instead somewhere between them. However, because of the infinite decimal expansion of √2 we do not know at which point it is located between 1.4 and 1.5. This impossibility of determining the location of √2 in the interval (1.4, 1.5) indicates our ignorance of the determinate and true position and value of √2.
5 To express the current value of the Hubble constant the cosmologists assign the subscript 0 to H: H0.
6 To learn about Adam Riess’ s calculations of the Hubble constant, visit: https://www.nasa.gov/feature/goddard/2019/new-hubble-constant-measurement-adds-to-mystery-of-universe-s-expansion-rate. Last updated: July, 2019. Editor Rob Garner.
7 Hart, Luke and Jens Chluba. “Updated fundamental constant constraints from Planck 2018 data and possible relations to the Hubble tension” in ArXiv: 1912. 03986V2 [astro-ph], May 20, 2020.
8 Leibniz’s principle of sufficient reason poses the analytic question Why is something such rather than such? This question aims at explaining the arbitrary choice of one alternative against the other in agreement with the analytic principle of the excluded third. Thus, the sufficient reason principle is a consequence of the finite analytic behavior of our neuronal brain. An object intuited by our infinite mind as a complex whole or universe — that is, as a complex extended line simultaneously receiving contrary (equal opposite) determinations a and b, is perceived by our finite neuronal brain and its tools as if it were a simple individual — that is, as an extensionless point possessing at any given time a unique determination, for example, either determination a or determination b. As a simple individual having at any given time a unique determination, we ask ourselves Leibniz’s analytic question. Now the impossibility of determining which of the alternative determinations is the true determination of the real object defines the state of negative indeterminacy expressing our absolute ignorance of the real object assumed by our finite brain and its faculty of analytic reason to be a simple determinate individual, that is to say an extensionless point (see part 3).
9 For Wendy Freedman’s calculations of the Hubble constant, visit https://www.nasa.gov/feature/goddard/2019/new-hubble-constant-measurement-adds-to-mystery-of-universe-s-expansion-rate. Last updated: July, 2019. Editor Rob Garner.
By inversing the Hubble constant H = v/d = 69.8km/s/Mpc, we derive the corresponding equally true current age of the Hubble universe, which is 1/H = d/v =t = 3.09 × 1022 m /6.98 × 104m/s = 4.426 × 1017s/3 × 107s = 1.48 × 1010 = 14.8 × 109Y (14.8 billion years).
10 The Greek Ionian atomist philosopher Leucippus and his associate Democritus (fifth century BCE) postulated an infinite number of atoms in perpetual motion and held that their shapes and sizes are infinite in number, based on the egalitarian principle that “nothing is such rather than such” (Kirk, Raven and Schofield. The Presocratic Philosophers. Cambridge, UK: Cambridge University Press, 1988, 416). In this way, they linked infinity with the principle of the uniformity and equality of atoms. Accordingly, if there is an infinite number of atoms, then by virtue of the principle of the uniformity and equality between the number of the atoms and the number of their shapes and sizes, there is necessarily an infinite number of shapes and sizes that are equally real and true.
11 Métaphysique. Trad. J.Tricot. Paris: Librairie Philosophique J. Vrin, 1984, Γ4, 10106, 15-20.


McGill provides Common Core math through its Houghton Mifflin Expressions (TK-3rd) and Go Math (3rd-5th) programs. Math manipulatives and school-to-life connections are saught for the internalization of math concepts.
To support the various mathematical concepts from number sense, to algebraic thinking and problem solving, we offer online math programs for students to practice their daily math skills. Our TK/K and 1st grade teachers use Sumdog.com, our second grade teacher uses Turtlediary.com, and 3rd-5th use Aleks and LeverEd.
Recognizing the dilemma of multiple programs, McGill is in the process of selecting one math curriculum to create a TK-5th grade continuum.
McGill assesses students with Renaissance STAR assessments four times a year to keep track of learning progress and need for Response to Intervention supports.


- What is principle of duality give an example?
- What is duality principle?
- What do you mean by duality?
- What is the basic principle of Boolean algebra?
- What is duality in spirituality?
- Who invented Boolean logic?
- What is truth table definition?
- WHAT IS A in Boolean algebra?
- What is the duality of life?
- What is Boolean duality principle?
- What is duality in Buddhism?
- Why logic gates are used?
- What is universal gate with example?
- What is De Morgan’s Law in logic?
- Why do we use K map?
- What is duality in sociology?
- What does duality mean in art?
- What is duality principle in digital electronics?
- What is De Morgan’s theorems?
- What are the 3 basic logic gates?
- What is duality in psychology?
- What are universal logic gates?
- What does duality principle state in computer?
- What is duality of self?
What is principle of duality give an example?
In this case, 1+1=1 is the true statement.
Its dual is 0.0=0 , which is also a true statement.
The statement is not 1+1 but rather 1+1=1 .
What the duality principle says is that “if you exchange every symbol by its dual in a formula you get the dual result”..
What is duality principle?
Answer: The principle of duality in Boolean algebra states that if you have a true Boolean statement (equation) then the dual of this statement (equation) is true. The dual of a boolean statement is found by replacing the statement’s symbols with their counterparts.
What do you mean by duality?
: the quality or state of having two different or opposite parts or elements : dualism That duality—sophistication paired with authenticity on the wine list, simplicity spiced with creativity on the menu—gives Marea energy and distinctive character.—
What is the basic principle of Boolean algebra?
The basic operations of Boolean algebra are as follows: AND (conjunction), denoted x∧y (sometimes x AND y or Kxy), satisfies x∧y = 1 if x = y = 1, and x∧y = 0 otherwise. OR (disjunction), denoted x∨y (sometimes x OR y or Axy), satisfies x∨y = 0 if x = y = 0, and x∨y = 1 otherwise.
What is duality in spirituality?
The idea of nondualism is typically contrasted with dualism, with dualism defined as the view that the universe and the nature of existence consists of two realities, such as the God and the world, or as God and Devil, or as mind and matter, and so on.
Who invented Boolean logic?
George BooleGeorge Boole, (born November 2, 1815, Lincoln, Lincolnshire, England—died December 8, 1864, Ballintemple, County Cork, Ireland), English mathematician who helped establish modern symbolic logic and whose algebra of logic, now called Boolean algebra, is basic to the design of digital computer circuits.
What is truth table definition?
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables.
WHAT IS A in Boolean algebra?
Boolean algebra is a division of mathematics that deals with operations on logical values and incorporates binary variables. Boolean algebra traces its origins to an 1854 book by mathematician George Boole. … Most commonly Boolean variables are presented with the possible values of 1 (“true”) or 0 (“false”).
What is the duality of life?
Duality teaches us that every aspect of life is created from a balanced interaction of opposite and competing forces. Yet these forces are not just opposites; they are complementary. … Let’s apply this concept to one of the most consequential aspects of our existence: life and death.
What is Boolean duality principle?
According to this principle, if we have postulates or theorems of Boolean Algebra for one type of operation then that operation can be converted into another type of operation (i.e., AND can be converted to OR and vice-versa) just by interchanging ‘0 with 1’, ‘1 with 0’, ‘(+) sign with (.)
What is duality in Buddhism?
Dualism and nondualism (or non-duality) are words that come up frequently in Buddhism. … Dualism is a perception that something — or everything, including reality itself — can be sorted into two fundamental and irreducible categories.
Why logic gates are used?
Computers need a way to manipulate those 1s and 0s, so that they can eventually do more complicated operations like calculating the 50th digit of π. Computers use logic gates to transform the 1s and 0s from input wires. A logic gate accepts inputs and then outputs a result based on their state.
What is universal gate with example?
A Universal gate is a logic gate from which all other types of logic gates can be made. This is done by wiring together more than one of that gate in a certain way. Because of this, any Boolean logic expression can also made using only universal gates. Examples include the NAND gate and the NOR gate.
What is De Morgan’s Law in logic?
In formal logic: Validity in PC. …… nor …,” the first De Morgan law can be read as “both p and q if and only if neither not-p nor not-q”; thus it expresses the principle that two propositions are jointly true if and only if neither of them is false.
Why do we use K map?
Karnaugh maps are used to simplify real-world logic requirements so that they can be implemented using a minimum number of logic gates. A sum-of-products expression (SOP) can always be implemented using AND gates feeding into an OR gate, and a product-of-sums expression (POS) leads to OR gates feeding an AND gate.
What is duality in sociology?
Structuration theory, concept in sociology that offers perspectives on human behaviour based on a synthesis of structure and agency effects known as the “duality of structure.” Instead of describing the capacity of human action as being constrained by powerful stable societal structures (such as educational, religious, …
What does duality mean in art?
”Duality” is a word that comes up frequently in connection with Miss Walkingstick, and she has used it to describe her process of painting. … Next, the word denotes the character of her paintings, which also seem like objects because of their box-like proportions and thickly layered surfaces.
What is duality principle in digital electronics?
Duality Theorem This theorem states that the dual of the Boolean function is obtained by interchanging the logical AND operator with logical OR operator and zeros with ones. For every Boolean function, there will be a corresponding Dual function.
What is De Morgan’s theorems?
De Morgan’s Theorem, T12, is a particularly powerful tool in digital design. The theorem explains that the complement of the product of all the terms is equal to the sum of the complement of each term. … According to De Morgan’s theorem, a NAND gate is equivalent to an OR gate with inverted inputs.
What are the 3 basic logic gates?
All digital systems can be constructed by only three basic logic gates. These basic gates are called the AND gate, the OR gate, and the NOT gate. Some textbooks also include the NAND gate, the NOR gate and the EOR gate as the members of the family of basic logic gates.
What is duality in psychology?
In short we have ‘minds’. Typically humans are characterized as having both a mind (nonphysical) and body/brain (physical). This is known as dualism. Dualism is the view that the mind and body both exist as separate entities.
What are universal logic gates?
A universal gate is a gate which can implement any Boolean function without need to use any other gate type. The NAND and NOR gates are universal gates. In practice, this is advantageous since NAND and NOR gates are economical and easier to fabricate and are the basic gates used in all IC digital logic families.
What does duality principle state in computer?
The principle of duality states that starting with a Boolean relation another Boolean relation can be derived by : 1. Changing each OR sign+ to an AND sign.. 2. … Principle of duality is use in Boolean algebra to complement the Boolean expression.
What is duality of self?
The classic duality of self-subject and self-object is related to the linguistic duality of self as a pronoun of the first and the third person. … The results add to our understanding of the role of objective self-awareness in self-other comparisons and in causal attributions from actors’ and observers’ perspectives.


There was a time when Sudoku was the game of the moment. A printed game, to be solved with the pen or better with the pencil. A grid made up of 9 large squares, each containing 9 squares. The goal is to ensure that each large box contains the numbers from 1 to 9 without repetitions.
If you are facing your Sudoku puzzle and you can’t get out of it, there is an app that helps you get some suggestions. Magic Sudoku uses ARKit to calculate the exact contents of all the boxes.
To solve the game, therefore, you just need to frame the page to immediately obtain the calculation of all the large boxes and their contents. In practice, solving a sudoku puzzle will be child’s play.


NOTE – Applications will be accepted to the STEM BUILD Opportunity for best consideration through
Friday, April 7, 2017
The STEM Living Learning Community joined the campus as UMBC’s tenth Living and Learning Community (LLC) in Fall 2015 and expanded to 75 students in Fall 2016 to become its largest LLC. It is typically for first year students (Freshmen and Transfer) and BTP Trainees, with some students applying to continue in the STEM LLC for a second year.
The goals of the STEM LLC are to:
- develop intentional connections and relationships among students majoring in Science, Technology, Engineering, and Mathematics
- expose students to other STEM majors,
- expand perspectives,
- encourage peer mentoring,
- foster study group participation,
- promote career exploration, and
- provide opportunities to develop leadership skills. All students with STEM majors are eligible to apply, but preference will be given to participants in the STEM BUILD at UMBC initiative.


There are over 2,000 free patterns on LionBrand.com that use CYC #4/Worsted Weight (medium) gauge yarn. That’s a lot of patterns to choose from! However, only 7 of those patterns and 15 kits are written specifically for Jeans® yarn. This doesn’t mean you need to feel restricted by only these projects.
Worsted weight is very versatile and Jeans®, being a smooth plied yarn, lends itself to being substituted in many of the patterns designed for this weight. You can get a good idea of if a yarn will work well as a substitution by determining the yards per gram. This isn’t a foolproof method as you will see some variance by fiber content, but it does provide a good place to start.
To determine the yards per gram you’ll need to do a bit of simple math. Divide the total yards of the skein by the total weight. For example, Jeans® yarn is 246 yards and 100 grams. You would then do:
246/100 = 2.46 (yards per gram)
Working with this information it is possible to determine patterns suggesting Vanna’s Choice might not make the transition between yarns (unless you wanted to tweak the design). Vanna’s Choice comes in at 170 yards per 100 grams (or 1.7 yards per gram), which means it is heavier than Jeans®.
Jeans® yarn falls on the lighter weight end of the #4 scale, so you can compare some DK/#3 weight yarns as well. LB Collection Cotton Bamboo is near in yardage per gram coming in at 2.4 using the math outlined above. Browse pattern using Cotton Bamboo.
The math discussed above is a handy calculation to remember for swapping yarn and can be applied to any weight. This information paired with knowledge of the texture and fiber content will have you changing yarns in patterns like a pro!
You must be logged in to post a comment.


Hi, thanks for the reply.
Armoury Crate Version 18.104.22.168, no updates seem available
All the points can dragged around and have their positions saved, but it doesn't seem like any of them consistently affect the fan speed, other than the leftmost point, which will always cause a change when adjusted, though not necessarily proportional to the point position.
After some more messing around with the fan curves, this time starting with one of the default curves and adjusting it, I was able to get better behavior, but still not consistent. Note the reported fan speed on the right versus the designed curve on the left.
Here's an example of a fan curve completely not working as expected, the fan speed should be 100% at 62 C, but instead it takes the value of the leftmost point (0%).
Here's another example. The fan speed should still be 100%, but it's not. It's also not 50%, suggesting some kind of calculation error when interpolating the points on the curve.
Strangely, this fan curve seems to produce more normal results, though I haven't the time to test under full load and see if the settings continue to work.
It beginning to seem more like the underlying issue is that the curve interpretation for fan control can't handle steep changes in speed with relation to heat, and the math biases the fan speed low.


High precision calculator for very large numbers
The scientific pocket calculator handles the calculation with very large numbers and with high precision.
For this enter a mathematical formula or equation into the calculator, like for example
The floating-point numbers have an adjustable precision from 5 to 10000 decimal digits and in the scientific notation,
such as "1.5e+6", extremely large exponents of up to 4.6 quintillions are possible.
15*(6.7+sqrt(7))^3 + 7*ln(2)
After pressing the equality key on the calculator you will get the result of the formula with high precision
and accuracy. The input of a formula can be done by the software keys of the calculator or the PC keyboard.
The following number types are available in the calculator:
The scientific calculator makes the following arithmetic operations as mathematical functions
- The 4 basic arithmetic operations addition, subtraction, multiplication and division
- Exponentiation, also for fraction numbers
- Logarithm calculation
- Trigonometric functions sine, cosine and tangent and the corresponding inverse functions
- Hyperbolic functions
- Factorial calculation
- Binomial coefficient "n choose k" with a real number "n" and a natural number "k"
- Greatest common divisor gcd
- Least common multiple lcm
- Modular exponentiation x^y mod z
- Multiplicative inverse
- Absolute value of complex numers
- Angle of complex numers


Interesting Sample Resume
We choose to provided in this article because this may be one of great resource for any Texas Staar Algebra 2 Released Test Questions ideas. Don’t you come here to learn some new unique Texas Staar Algebra 2 Released Test Questions ideas? We really hope you can easily accept it as one of your reference and many thanks for your effort for exploring our website. Make sure you distribute this image for your loved friends, family, society via your social networking such as facebook, google plus, twitter, pinterest, or any other social bookmarking sites.
Download now Texas Staar Algebra 2 Released Test Questions. Our image gallery has huge collection of pictures. You can find Texas Staar Algebra 2 Released Test Questions etc. Visit our image gallery to find another Resume and images for your computer’s desktop, tablet, android and laptop background widescreen image. We have collected full screen and high resolution images for Resume lovers. Just right click on the images and save on computer.
© Interesting Sample Resume
6th Grade Math Terms and Definitions
, Algebra Write Equation In Standard form
, Algebra Pizzazz Factoring
, Christmas Subtraction Worksheets Second Grade
, 8th Grade History the American Journey
, Algebra Clep Prep
, Matrix Algebra In Excel Vba
, Best Books for Fourth Graders
, Third Grade Teacher T Shirts
, Algebra 2 Practice sol Problems From Vdoe


- Program Admission Information for 2021-2022
- Honors Math Parent Sample Letter
- Letter for Honors Math Teachers
- Registration for Online Algebra Testing
- Scantron Spreadsheet (excel)
- Honors Math Forms (excel)
All questions regarding Honors Math can be referred to:
Sister Edward Quinn, Asst. Superintendent of Curriculum, Instruction and Assessment
Program admission for the 2021-2022 School Year
- Since we do have standardized testing results for this year, the protocol for accepting students into this program will return to the previous protocol:
- Standardized testing grades:
- Math: NP of NCE—90% or above
- CSI: In the range of 125 (118 or above)
- Reading: NP of NCE—85% or above
- Final grades in math:
- From third grade entering the program in fourth grade: VG or O
- From Grade 4 entering the program in fifth grade: 93 or above.
- Final grades in reading: 85 or above for all students
- Effort grades—4
- Teacher Recommendation
- Students beginning the program in Grade 5 should follow the same Terra Nova criteria.
- Students falling below—but within a five point range of—the above criteria (except CSI) can be considered for probationary status. Once accepted, a letter explaining probationary status is to be sent to the parents, signed, and returned to the school.
- Honors math forms for students being considered for probationary status in the program should be completed and submitted to the Elementary Math committee as has been done in the past.
- All honors math forms are to be sent electronically to Rose Greskoff at firstname.lastname@example.org by June 30. Forms and Honors Math information can be found on the website: https://aopelementary.org/curriculum-assessment/honors-math/
- All students who have taken the Algebra I course should take the Scantron Algebra Achievement test. This test is taken several times throughout the year as part of the Performance series. This test is not just for students entering diocesan high schools, but is our documentation that the student has officially completed a secondary Algebra course at the elementary level. Students are not excused because they are not entering a diocesan high school.
- It is not necessary for Honors Math teachers to send the results to the High Schools. Rigby is able to access the reports electronically and forward them to the high schools. What we do need is simply a list of the Honors Math students and the names of the high school they are attending. Dr. Rigby will transfer the scores to the high schools.


Estamos recebendo a visita de Bjarne Grimstad, pesquisador da NTNU (Universidade de Ciência e Tecnologia da Noruega) que estará defendendo o doutorado em breve. Ele vai estar em visita ao DAS na semana de 21 a 25/Setembro.
Deixo o convite para um seminário em 21/09 (segunda-feira) das 14 h às 15 h na sala PPGEAS-2. Detalhes abaixo,
Prof. Eduardo Camponogara
Title of talk: Modelling and global optimization with splines
Abstract: This talk addresses the use of splines to model and optimize black-box functions. Splines, being piecewise-defined by polynomials, are highly flexible functions that may accurately approximate most shapes. A spline can be used to approximate any sampled function, also when the sample is noisy, in which case a smoothing spline should be applied. Splines have many properties attractive for optimization: a high degree of smoothness, analytical derivatives, and a readily available convex hull. These properties are utilized in the recently developed solver CENSO for global optimization with spline constraints. CENSO contains a spatial Branch-and-Bound algorithm that accepts spline constraints modeled with the function approximation software SPLINTER. In this talk, we will introduce the open-source software SPLINTER and CENSO. Examples will be given that demonstrate the flexibility of modeling optimization problems with spline constraints.
Autobiography: Bjarne Grimstad is a PhD candidate at the Department for Engineering Cybernetics, Norwegian University of Science and Technology. He has worked on the topics of function approximation, integer programming, black-box optimization, and global optimization. The motivation behind his work has been to optimize the production from multiphase flow networks by applying various combinations of tools from the above topics. Mr. Grimstad was awarded a PhD scholarship from IBM in 2012 and has since developed the open-source software SPLINTER and CENSO.


Page 5 of 6
Posted: Sat Jan 27, 2007 2:29 pm
om het niveau een beetje te verlagen :
"-Jantje, wat is de helft van 8 ?
-heel goed, en de helft van 2 ?
-schitterend ! en weet je dan ook wat de helft is van 1 ?
-2, antwoordt Jantje
-2 ? hoe kom je daarbij ?
-als ik een appel in 2 verdeel heb ik 2 stukken."
Posted: Mon Jan 29, 2007 5:24 pm
Ok, dat haalde het niveau inderdaad naar beneden...
Een paar uitspraken van bekende mensen:
Mathematicians are like Frenchmen: whatever you say to them, they translate it into their own language, and forthwith it means something entirely different.
A mathematician is a blind man in a dark room looking for a black cat which isn't there.
He who can properly define and divide is to be considered a god.
Posted: Mon Jan 29, 2007 5:25 pm
A mathematician and a physicist agree to a psychological experiment. The (hungry) mathematician is put in a chair in a large empty room and his favorite meal, perfectly prepared, is placed at the other end of the room. The psychologist explains, "You are to remain in your chair. Every minute, I will move your chair to a position halfway between its current location and the meal." The mathematician looks at the psychologist in disgust. "What? I'm not going to go through this. You know I'll never reach the food!" And he gets up and storms out.
The psychologist ushers the physicist in. He explains the situation, and the physicist's eyes light up and he starts drooling. The psychologist is a bit confused. "Don't you realize that you'll never reach the food?" T he physicist smiles and replies: "Of course! But I'll get close enough for all practical purposes!"
Posted: Mon Jan 29, 2007 5:28 pm
1. A physicist and a mathematician are sitting in a faculty lounge. Suddenly, the coffee machine catches on fire. The physicist grabs a bucket and leap towards the sink, filled the bucket with water and puts out the fire. Second day, the same two sit in the same lounge. Again, the coffee machine catches on fire. This time, the mathematician stands up, got a bucket, hands the bucket to the physicist, thus reducing the problem to a previously solved one.
2. A mathematician and an engineer are on desert island. They find two palm trees with one coconut each. The engineer climbs up one tree, gets the coconut, eats. The mathematician climbs up the other tree, gets the coconut, climbs the other tree and puts it there. "Now we've reduced it to a problem we know how to solve."
Posted: Mon Jan 29, 2007 5:28 pm
Wie kan het bewijzen?
Several scientists were asked to prove that all odd integers higher than 2 are prime.
Mathematician: 3 is a prime, 5 is a prime, 7 is a prime, and by induction - every odd integer higher than 2 is a prime.
Physicist: 3 is a prime, 5 is a prime, 7 is a prime, 9 is an experimental error, 11 is a prime. Just to be sure, try several randomly chosen numbers: 17 is a prime, 23 is a prime...
Engineer: 3 is a prime, 5 is a prime, 7 is a prime, 9 is an approximation to a prime, 11 is a prime,...
Programmer (reading the output on the screen): 3 is a prime, 3 is a prime, 3 a is prime, 3 is a prime....
Biologist: 3 is a prime, 5 is a prime, 7 is a prime, 9 -- results have not arrived yet,...
Psychologist: 3 is a prime, 5 is a prime, 7 is a prime, 9 is a prime but tries to suppress it,...
Chemist: What's a prime?
Politician: "Some numbers are prime.. but the goal is to create a kinder, gentler society where all numbers are prime... "
Programmer: "Wait a minute, I think I have an algorithm from Knuth on finding prime numbers... just a little bit longer, I've found the last bug... no, that's not it... ya know, I think there may be a compiler bug here - oh, did you want IEEE-998.0334 rounding or not? - was that in the spec? - hold on, I've almost got it - I was up all night working on this program, ya know... now if management would just get me that new workstation that just came out, I'd be done by now... etc., etc. ..."
Posted: Mon Jan 29, 2007 5:33 pm
Een beetje een flauwe deze:
New York (CNN). At John F. Kennedy International Airport today, a high school mathematics teacher was arrested trying to board a flight while in possession of a compass, a protractor and a graphical calculator. According to law enforcement officials, he is believed to have ties to the Al-Gebra network. He will be charged with carrying weapons of math instruction. It was later discovered that he taught the students to solve their problem with the help of radicals!
Posted: Mon Jan 29, 2007 5:35 pm
Iemand zin om een loterij te organizeren?
A mathematician organizes a lottery in which the prize is an infinite amount of money. When the winning ticket is drawn, and the jubilant winner comes to claim his prize, the mathematician explains the mode of payment: "1 dollar now, 1/2 dollar next week, 1/3 dollar the week after that..."
Posted: Mon Jan 29, 2007 5:44 pm
Iemand een vriendin?
"Do you love your math more than me?"
"Of course not, dear - I love you much more."
"Then prove it!"
"OK... Let R be the set of all lovable objects..."
Posted: Mon Jan 29, 2007 5:48 pm
En deze is zo ongelooflijk waar:
Relations between pure and applied mathematicians are based on trust and understanding. Namely, pure mathematicians do not trust applied mathematicians, and applied mathematicians do not understand pure mathematicians.
Posted: Mon Jan 29, 2007 5:54 pm
Variante op eentje die eerder gepost werd:
A constant function and e^x are walking on Broadway. Then suddenly the constant function sees a differential operator approaching and runs away. So e^x follows him and asks why the hurry. "Well, you see, there's this differential operator coming this way, and when we meet, he'll differentiate me and nothing will be left of me...!" "Ah," says e^x, "he won't bother ME, I'm e to the x!" and he walks on. Of course he meets the differential operator after a short distance.
e^x: "Hi, I'm e^x"
diff.op.: "Hi, I'm d/dy"
Posted: Mon Jan 29, 2007 5:58 pm
Misschien een programmapunt voor het Vlaams Belang?
Q: What is a compact city?
A: It's a city that can be guarded by finitely many near-sighted policemen.
Posted: Mon Jan 29, 2007 6:04 pm
En eentje voor mij:
"What's your favorite thing about mathematics?" "Knot theory." "Yeah, me neither."
Posted: Mon Jan 29, 2007 6:38 pm
slaapverwekkend... Dat lees ik zelfs niet
Posted: Mon Jan 29, 2007 6:43 pm
Zijn best wel goei vind ik.
Posted: Thu Feb 01, 2007 3:57 pm
Wiskunde is niet grappig, wiskunde is BLOEDserieus
(bloed in rood en vet gezet om Math Wolf een plezier te doen/nog wat meer te ergeren.)


Please find the online curriculum for this half term. Students are to consolidate and recap lessons as well as practice past papers. All work will be assessed by your subject teacher.
In the links below, you will find work assigned by your teachers for the core subjects – Mathematics, English and Science.
You are expected to use this time to study these key areas to ensure that you are up to date with your studies in preparation for the upcoming exams or assignments.
Only the core subject areas have been uploaded here, your vocational subject areas will set you work via other platforms, such as Google Classroom.
To access Google Classroom, you need to login to Google using your school email address (email@example.com) which will take you to the RMUnify page where you can access Google Classrooms.
If you have forgotten your password or have never logged in before, please contact firstname.lastname@example.org to reset your password.
If any students or parents need to get in touch with the pastoral team, please contact email@example.com.
Also, the online resource twinkl has offered a one month ultimate membership free of charge to help support further learning and revision for Mathematics, English and Science, which you can find here www.twinkl.co.uk/offer using the code UKTWINKLHELPS
You can also now stream an incredible collection of stories without having to pay for a subscription through audible
You can access a range of educational resources here
Here is a full list of Google Classroom codes for you to join:
Health and Social Care
Year 10: rgabvzw
Year 11: jaj7g4y
Year 10: ijskiid
Year 11: bd2ntrs
Year 12: s6kikuz
Year 13: 6gfeejh
Y 10 Maths, Anne: wnwojx6
Y 10 Maths, Heather: hme2odk
Y10 Maths, Adam: l42jf5z
Y11 Maths, Adam: 6qxppob
Y11 Maths, Anne: epz5sxq
Y 11 Maths, Heather: 7daleyq
Y12 Core Maths, Anne: glb7btg
Y13 Core Maths, Heather: kkgsqjs
Key Stage 5 GCSE Resit Maths, Adam: iyb3qmw
Key Stage 5 Functional Maths Level 2, Anne: buqw2mv
Key Stage 5 Functional Maths Level 1, Anne: 3smfoea
Year 10 English Language: oil7jg6
Year 10 English Literature: tjhztik
Year 11 English Language code: 3ljhdas
Year 11 English Literature code: tga6fvs
Resit code: sehjh3v
Functional skills code: iugzqw7
A Level Code: tjw7u4v
Year 10 Biology: fw3ovla
Year 10 Chemistry: apsrtho
Year 10 Physics: 6pv7qje
Year 11 Biology: fazhipf
Year 11 Chemistry: rp3xs57
Year 11 Physics: 2olw6v4
Year 10: kpohqhd
Year 12/13: wdgi2s5
Year 10: u5fafxn
Year 11: vxtidg3
Year 10: tjujb5f
Year 11: 25pd6re
Year 12: 53ikdvy
Year 13: nzy2tvx
Year 10: uk2f7tj
Year 11: eruef33


this is how you actually reform education. Not corporatizing or privatizing.
Children are intrinsically eager and able to learn. If we step back from our limiting preconceptions about education, we discover learning flourishes when we facilitate it rather than try to advance it through force, intimidation, and coercion.
Over 85 years ago a pioneering educator proved that delaying formal instruction, in this case of mathematics, benefits children in wonderfully unexpected ways. Louis P. Benezet, superintendent of the Manchester, New Hampshire schools, advocated the postponement of systematic instruction in math until after sixth grade. Benezet wrote,
I feel that it is all nonsense to take eight years to get children thru the ordinary arithmetic assignment of the elementary schools. What possible needs has a ten-year-old child for knowledge of long division? The whole subject of arithmetic could be postponed until the seventh year of school, and it could be mastered in two years’ study by any normal child.
View original post 877 more words