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Il nostro sistema numerico può essere suddiviso in molti modi diversi. La forma più elementare di matematica è contare e quasi tutte le culture umane hanno parole per rappresentare i numeri (i Pirahã del Sud America sono un'eccezione notevole). Pertanto l'insieme di numeri più basilare è l'insieme dei numeri da contare rappresentato dal doppio barrato \(\mathrm{N}: \mathbb{N}=\{1,2,3,4,5,6,7, \ldots \}\) (metteremo da parte il dibattito sull'opportunità o meno di includere zero in questo insieme).
Se proviamo a sottrarre un numero di conteggio più grande da un numero di conteggio più piccolo, scopriamo che non ci sono membri nell'insieme dei numeri di conteggio per rappresentare la risposta in questa situazione. Questo estende l'insieme dei numeri naturali all'insieme degli interi: \(\mathbb{Z}=\{\ldots,-3,-2,-1,0,1,2,3, \ldots\} .\) Gli interi sono rappresentati dalla doppia sbarra \(Z\), per la parola tedesca per i numeri - "zahlen". Nelle prime apparizioni di numeri negativi nei sistemi matematici cinese e indiano, i valori negativi venivano spesso usati per rappresentare il debito. Poiché la matematica greca era basata sulla geometria, non usavano numeri negativi.
Passando alla moltiplicazione e alla divisione, se mettiamo in discussione il valore di \(8 \div 2=4\) rispetto a \(8 \div 3=?,\) dobbiamo ancora una volta espandere la nostra concezione dei numeri per consentire una risposta al seconda domanda \(8 \div 3=?\). Comprendere i rapporti di numeri interi o numeri razionali consente soluzioni a tali problemi. L'insieme dei numeri razionali è rappresentato dalla doppia barrata \(Q\), per rappresentare un quoziente:
\(\mathbb{Q}=\sinistra\{\frac{a}{b}: a, b \in \mathbb{Z}\destra\}\)
La comprensione greca dei numeri si fermava principalmente qui. Sentivano che tutte le quantità potevano essere rappresentate come il rapporto di numeri interi. La lunghezza della diagonale di un quadrato i cui lati sono di lunghezza 1 produsse una notevole costernazione tra i pitagorici come risultato di ciò. Usando il Teorema di Pitagora per la diagonale di un quadrato i cui lati sono di lunghezza 1 mostra che la diagonale sarebbe \(c^{2}=1^{2}+1^{2}=2,\) quindi \(c =\sqrt{2} .\) Questo numero non può essere rappresentato come un rapporto di numeri interi. Questa nuova classe di numeri aggiunge l'insieme
di numeri irrazionali all'insieme esistente di numeri razionali per creare i numeri reali, rappresentati con una doppia barrata \(\mathrm{R}\) : \(\mathbb{R}\).
Questa gerarchia di numeri è spesso rappresentata nel diagramma seguente:
Uno dei modi migliori per concettualizzare il sistema dei numeri reali è sulla linea dei numeri: ogni punto sulla linea dei numeri corrisponde a un numero reale univoco e ogni numero reale corrisponde a una posizione univoca sulla linea dei numeri reali.
Dopo lo sviluppo della stampa nel XV secolo, il Liber Abaci di Fibonacci fu tradotto in italiano dal latino e letto in tutta Italia. Di conseguenza, l'Italia divenne un fiorente centro della matematica fino al XVII secolo, quando il centro della matematica europea si spostò a nord, in Francia, Germania e Inghilterra.
Per tutto il 1500 matematici italiani come Girolamo Cardano, Raffaello Bombelli e Niccolò Fontana Tartaglia lavorarono per estendere le idee nel libro di Fibonacci. Hanno prodotto formule per risolvere equazioni di grado cubico ( \(x^{3}\) ) e quartico ( \(x^{4}\) ). Nel risolvere alcune di queste equazioni hanno scoperto che le loro formule a volte producevano valori negativi sotto una radice quadrata. Nessuno dei sistemi di numerazione conosciuti potrebbe accogliere questa possibilità. Nel libro di Cardano sull'algebra Ars Magna, incontra un problema che coinvolge la radice quadrata di un numero negativo. Dice: "È chiaro che questo caso è impossibile. Tuttavia lavoreremo così..." e procede a calcolare una valida soluzione complessa al problema. I matematici alla fine definirono l'unità complessa \(\sqrt{-1}=i\) e poi
ideato un sistema in cui tutti i numeri complessi sono una combinazione di una parte reale
\((a)\) e una parte "immaginaria" (bi).
I numeri complessi sono un sistema numerico bidimensionale rappresentato dalla doppia barrata \(C: \mathbb{C}=\{a+b i: a, b \in \mathbb{R}\},\) dove \(i\ ) è l'unità complessa definita come \(i=\sqrt{-1} .\) Per tutta la fine del 1700 e l'inizio del 1800 i matematici si spostarono gradualmente verso un'interpretazione geometrica dei numeri complessi bidimensionali. Quello che oggi è noto come "diagramma di Argand" rappresenta la porzione a valori reali di un numero complesso lungo un asse orizzontale e il multiplo dell'unità complessa lungo l'asse verticale.
Esprimere radici quadrate di numeri negativi
Le radici quadrate delle quantità negative sono generalmente espresse come multiplo di \(i\)
Esempio \(\PageIndex{1}\)
\[
\begin{array}{c}
\sqrt{-4}=2 e \\
\sqrt{-25}=5 e \\
\sqrt{-7} \circa 2,646 i
\end{array}
\]
Addizione, sottrazione e moltiplicazione con numeri complessi
Il calcolo con numeri complessi ha molte somiglianze con il lavoro con le variabili. La parte reale e la parte immaginaria sono trattate separatamente per l'addizione e la sottrazione, ma possono essere moltiplicate e divise.
Esempio \(\PageIndex{2}\)
Calcola quanto segue:
\[
\begin{array}{c}
(6-4 io)+(-2+7 io)=4+3 io \\
(-9+2 io)-(-4+6 io)=-9+2 io+4-6 io=-5-4 io \\
3(10+io)=30+3 io \\
-7i(-5+8i)=35i-56i^{2}
\end{array}
\]
Ora incontriamo un fatto interessante sui numeri complessi e, in particolare, sull'unità complessa \(i .\) Per definizione, \(i=\sqrt{-1} .\) Pertanto, se eleviamo al quadrato \(i\) otteniamo dovrebbe ottenere \(-1 .\) Nell'ultimo problema di esempio sopra, possiamo sostituire \(i^{2}\) con -1 per completare il problema.
\begin{allineato}
-7 i(-5+8 i) &=35 i-56 i^{2} \\
&=35 i-56(-1) \\
&=35 i+56 \\
&=56+35 i
\end{allineato}
Esempio \(\PageIndex{3}\)
Calcola quanto segue:
\[
\begin{allineato}
(8-5 i)(1-4 i) &=8-32 i-5 i+20 i^{2} \\
&=8-37 i+20(-1) \\
&=8-37 i-20 \\
&=-12-37 io \\
\\
(9+2 io)^{2} &=(9+2 io)(9+2 io) \\
&=81+18 i+18 i+4 i^{2} \\
&=81+36 i+4(-1) \\
&=81+36 i-4 \\
&=77+36 i
\end{allineato}
\]
Potenze di \(i\)
Le potenze di \(i\) seguono uno schema interessante basato sulla definizione che \(i^{2}=-1\)
Possiamo vedere che \(i^{1}=i\) e che \(i^{2}=-1,\) come risultato, \(i^{3}=i^{2} * i^ {1}=-1 * i=-i\)
In modo simile, \(i^{4}=i^{2} * i^{2}=(-1)(-1)=1\)
Ciò significa che \(i^{5}=i^{4} * i=1 * i=i\)
Se mettiamo insieme tutte queste informazioni otteniamo quanto segue:
\(i^{1}=i\)
\(i^{2}=-1\)
\(i^{3}=-i\)
\(i^{4}=1\)
\(i^{5}=i^{1}=i\)
\(i^{6}=i^{2}=-1\)
\(i^{7}=i^{3}=-i\)
\(i^{8}=i^{4}=1\)
In altre parole, ogni potenza di \(i\) è equivalente a \(i,-1,-i,\) o \(1 .\) Per determinare quale di questi valori è una potenza di \(i\) equivalente a, dobbiamo trovare il resto dell'esponente quando è diviso per 4
Esempio \(\PageIndex{4}\)
Semplifica \(i^{38}\)
Soluzione
poiché ogni \(i^{4}=1,\) allora \(i^{38}=i^{36} * i^{2}=\left(i^{4}\right)^{9} * i^{2}=1^{9} * i^{2}=i^{2}=-1\)
poiché 38 è 2 in più di un multiplo di \(4,\) allora \(i^{38}=i^{2}=-1\)
Esercizio \(\PageIndex{1}\)
Rappresenta graficamente i seguenti numeri complessi:
1) \(\quad 2+5 i\)
2) \(\quad 4-3 i\)
3) \(\quad -2+6 i\)
4) \(\quad -3-5 i\)
5)\(\quad 4\)
6) \(\quad -2 i\)
7) \(\quad 7-i\)
8) \(\quad -1+i\)
9) \(\quad -8+4 i\)
10)\(\quad 8+3 i\)
11) \(\quad 7 i\)
12) \(\quad -5-9 i\)
Esprimi ciascuna quantità in termini di \(i .\) Arrotonda i valori irrazionali al 1000esimo più vicino.
13) \(\quad \sqrt{-36}\)
14) \(\quad \sqrt{-81}\)
15) \(\quad \sqrt{-100}\)
16) \(\quad \sqrt{-49}\)
17) \(\quad \sqrt{-4}\)
18) \(\quad \sqrt{-25}\)
19) \(\quad \sqrt{-2}\)
20)\(\quad \sqrt{-6}\)
21) \(\quad \sqrt{-10}\)
22) \(\quad \sqrt{-31}\)
23) \(\quad \sqrt{-5}\)
24) \(\quad \sqrt{-3}\)
Eseguire l'operazione indicata e semplificare
25) \(\quad(6+7 i)+(5+3 i)\)
26) \(\quadrato(4-5i)+(3+9i)\)
27) \(\quad(9+8 i)-(1-2 i)\)
28) \(\quad(2+i)-(6-4 i)\)
29) \(\quad(7-4 i)-(5-3 i)\)
30) \(\quad(8+i)-(4+3 i)\)
31) \(\quad(7i)(6i)\)
32) \(\quad(4 i)(-8 i)\)
33) \(\quad(-2 i)(5 i)\)
34) \(\quad(12i)(3i)\)
35) \(\quad(1+i)(3+2 i)\)
36) \(\quad(1+5 i)(4+3 i)\)
37) \(\quadrato(6-5i)(2-3i)\)
38) \(\quadrato(8-3i)(2+i)\)
39) \(\quad(-3+4 i)(-1-2 i)\)
40) \(\quad(-7-i)(3-5 i)\)
41) \(\quad(4-2 i)^{2}\)
42) \(\quad(-5+i)^{2}\)
43) \(\quad(3+i)(3-i)\)
44) \(\quad(2+6i)(2-6i)\)
45) \(\quadrato(9-4i)(9+4i)\)
46) \(\quad(5+2i)(5-2i)\)
Esprimi come \(i,-1,-i,\) o 1
47)\(\quad i^{3}\)
48)\(\quad i^{7}\)
49) \(\quad i^{21}\)
50)\(\quad i^{13}\)
51) \(\quad i^{29}\)
52) \(\quad i^{56}\)
53) \(\quad i^{72}\)
54)\(\quad i^{35}\)
55) \(\quad i^{66}\)
56)\(\quad i^{103}\)
57)\(\quad i^{16}\)
58)\(\quad i^{53}\)
59)\(\quad i^{11}\)
60)\(\quad i^{42}\)
61) \(\quad i^{70}\)
62)\(\quad i^{9}\)
- Risposta
-
13) \(\quad 6 i\)
15) \(\quad 10 i\)
17) \(\quad 2 i\)
19) \(\quad 1.414 i\)
21) \(\quad 3.162 i\)
23) \(\quad 2.236 i\)
25) \(\quad 11+10 i\)
27) \(\quad 8+10 i\)
29) \(\quad 2-i\)
31) \(\quad-42\)
33) \(\quad 10\)
35) \(\quad 1+5 i\)
37) \(\quad-3-28 i\)
39) \(\quad 11+2 i\)
41)\(\quad 12-16 i\)
43) \(\quad 10\)
45) \(\quad 97\)
47)\(\quad -i\)
49)\(\quad i\)
51) \(\quad i\)
53) \(\quad 1\)
55) \(\quad -1\)
57) \(\quad 1\)
59) \(\quad -i\)
61) \(\quad -1\)
FAQs
What is the best climate model? ›
The corrected RCM2 and RCM9, which are the best climate models for reproducing historical meteorology and hydrology, were used to generate local potential scenarios of precipitation and temperature.
What is the resolution of the climate model? ›Climate Model Resolution
The results of processes modeled in each cell are passed to neighboring cells to model the exchange of matter and energy over time. Grid cell size defines the resolution of the model: the smaller the size of the grid cells, the higher the level of detail in the model.
Model projections rely on two things to accurately match observations: accurate modeling of climate physics and accurate assumptions around future emissions of CO2 and other factors affecting the climate.
Do climate models include clouds? ›There is most uncertainty about low clouds over the subtropical oceans. When you look at climate models, virtually all give a decrease in the amount of low clouds, but by how much varies considerably depending on the sensitivity of the model.
Are climate models 100% accurate? ›No model is 100% correct, as some degree of approximation is always needed when making projections. Still, models represent what is likely to happen based on our most advanced knowledge and certain assumptions about Earth processes that cannot be directly represented with data.
How accurate are climate models? ›Despite criticisms from climate change skeptics, global climate models have accurately predicted rising average sea surface temperatures, which are extremely important to predicting the intensity of climate change.
What are the three types of climate models? ›When creating climate models, scientists use one of three common types of simple climate models: energy balance models, intermediate complexity models, and general circulation models.
What are the 7 components of a climate model? ›- Atmosphere. ...
- Ocean. ...
- Sea ice. ...
- Land surface. ...
- Marine biogeochemistry. ...
- Ice sheets. ...
- Coupling between the components - Earth system models.
The Very Simple Climate Model is, as the name implies, very simple. In this model, average global temperature is determined entirely by the atmospheric carbon dioxide concentration via greenhouse warming of the atmosphere and uptake of carbon dioxide by the ocean and biosphere, which are kept constant through time.
How much hotter will it be in 50 years? ›Since 1880, average global temperatures have increased by about 1 degrees Celsius (1.7° degrees Fahrenheit). Global temperature is projected to warm by about 1.5 degrees Celsius (2.7° degrees Fahrenheit) by 2050 and 2-4 degrees Celsius (3.6-7.2 degrees Fahrenheit) by 2100.
Why is climate modeling so difficult? ›
The problem is that the higher the spatial resolution of your model, the more uncertainty there will be in the projections. Things like the amount and timing of precipitation in a particular area are highly variable.
What is the difference between weather and climate models? ›Weather models have a finer resolution in time and space than climate models and are run over a much shorter period (e.g., weeks), whereas climate models can be run for hundreds or even thousands of years.
What do climate models indicate? ›Climate models agree on many important facts about our climate. For instance, models reliably show that adding more greenhouse gases to the atmosphere will cause average temperatures to rise. Models also try to predict how climate change will affect rainfall, sea levels, ice cover, and other parts of the natural world.
What are the limitations of climate models? ›- an incomplete understanding of the climate system,
- an imperfect ability to transform our knowledge into accurate mathematical equations,
- the limited power of computers,
- the models' inability to reproduce important atmospheric phenomena, and.
Some models perform better than others for certain climate variables. No single model is clearly “the best”, while the multi-model average is the best. Despite the progress made, scientific uncertainty about the details of several processes (such as certain types of clouds) remains.
What is the biggest source of uncertainty for climate models? ›The uncertainty seen in climate modelling stems from two main sources: inter-model variability and intra-model variability.
When was the hottest year on record for Earth as a whole? ›| Rank | Year | Anomaly °F |
|---|---|---|
| 1 | 2016 | 1.80 |
| 2 | 2020 | 1.76 |
| 3 | 2019 | 1.71 |
| 4 | 2015 | 1.67 |
In 1990, scientists predicted the Earth would warm somewhere between 1.5ºC to 4ºC.
Who predicted global warming? ›Exxon Scientists Predicted Global Warming, Even as Company Cast Doubts, Study Finds. Starting in the 1970s, scientists working for the oil giant made remarkably accurate projections of just how much burning fossil fuels would warm the planet.
What are climate models missing? ›These types of experiments have since shown that an adequate description of basic processes like cloud formation, moist convection, and mixing is what climate models miss most.
What are the 3 largest factors determining climate? ›
- Atmospheric Circulation. The sun's rays provide both light and heat to Earth, and regions that receive greater exposure warm to a greater extent. ...
- Ocean Currents. ...
- Global Climate. ...
- Biogeography.
Burning fossil fuels, cutting down forests and farming livestock are increasingly influencing the climate and the earth's temperature.
What are 4 determinants of climate? ›The main factors influencing temperature include latitude, altitude, distance from oceans and lakes, and ocean currents.
What are the 4 parts of climate? ›The four connected components of Earth's climate system: the atmosphere, hydrosphere, geosphere, and biosphere.
What are the 4 elements of climate change? ›Changing temperature and precipitation patterns. Increases in ocean temperatures, sea level, and acidity. Melting of glaciers and sea ice. Changes in the frequency, intensity, and duration of extreme weather events.
What are some factors that affect climate? ›The temperature characteristics of a region are influenced by natural factors such as latitude, elevation and the presence of ocean currents. The precipitation characteristics of a region are influenced by factors such as proximity to mountain ranges and prevailing winds.
What was the first climate model? ›In the late 1960s, NOAA's Geophysical Fluid Dynamics Laboratory in Princeton, New Jersey, developed the first-of-its-kind general circulation climate model that combined both oceanic and atmospheric processes.
Will 2023 be hotter than 2022? ›What's next. Heading into 2023, models generally anticipate that the ongoing La Niña event will subside, leading to neutral conditions in the Pacific Ocean and possibly an El Niño in the latter part of the year. That suggests that 2023 will be warmer than 2022 and among the hottest years on record.
How hot will it be in 2030? ›The study, published Jan. 30 in Proceedings of the National Academy of Sciences, provides new evidence that global warming is on track to reach 1.5 degrees Celsius (2.7 Fahrenheit) above pre-industrial averages in the early 2030s, regardless of how much greenhouse gas emissions rise or fall in the coming decade.
How hot will the US be in 2050? ›Swaths of the country will see at least one day per year with a heat index of 125 degrees Fahrenheit. This summer is already marked by megadroughts and extreme heat waves across the globe.
How accurate are the global warming predictions? ›
They found that, depending on the metric used, 63-83 percent of the global warming projections reported by Exxon scientists were consistent with actual temperatures over time.
What part of the climate system is the most difficult to model? ›Cumulus are difficult to represent in models because each cloud covers only a small part of Earth's surface, but when taken together, cumulus clouds have a large influence on global circulation.
What is a tipping point in climate change? ›For the climate system, the term refers to a critical threshold at which global or regional climate changes from one stable state to another stable state.". In ecosystems and in social systems, a tipping point can trigger a regime shift, a major systems reorganisation into a new stable state.
What are the 3 major climates of USA? ›The five main groups are A (tropical), B (dry), C (temperate), D (continental), and E (polar). All climates except for those in the E group are assigned a seasonal precipitation sub-group (second letter). Climate classifications are identified by hovering your mouse over the legend.
What climate zone is Florida in? ›Most of the state is classified as a humid subtropical climate that experiences extremely long, warm, typically humid summers and mild, cool winters making Florida planting zones some of the higher ranges. Southern Florida has a tropical climate.
Which country has the most climate zones? ›As the proportionally longest and narrowest country in the world – stretching over 33 degrees in latitude – or 4,200km – from north to south, Chile has its fair share of different climate zones.
Which is more predictable weather or climate? ›Climate is also a complex system, but it's an average of decades of weather patterns in a region, and it changes much more slowly than weather. Because of this, it's easier to predict climate than weather.
How can scientists predict what the climate will be like in 100 years? ›The Short Answer:
To predict future climate, scientists use computer programs called climate models to understand how our planet is changing. Climate models work like a laboratory in a computer. They allow scientists to study how different factors interact to influence a region's climate.
The greater Earth's axial tilt angle, the more extreme our seasons are, as each hemisphere receives more solar radiation during its summer, when the hemisphere is tilted toward the Sun, and less during winter, when it is tilted away.
What two 2 factors are used to describe climate? ›The two most important factors in the climate of an area are temperature and precipitation. The yearly average temperature of the area is obviously important, but the yearly range in temperature is also important.
How does the change of temperature affect precipitation? ›
As average temperatures at the Earth's surface rise, more evaporation occurs, which, in turn, increases overall precipitation. Therefore, a warming climate is expected to increase precipitation in many areas.
How many types are there in climate? ›one of five classifications of the Earth's climates: tropical, dry, mild, continental, and polar.
What are the most common climate models? ›When creating climate models, scientists use one of three common types of simple climate models: energy balance models, intermediate complexity models, and general circulation models.
How do we test climate models? ›How are Climate Models Tested? Once a climate model is set up, it can be tested via a process known as “hind-casting.” This process runs the model from the present time backwards into the past. The model results are then compared with observed climate and weather conditions to see how well they match.
What are the three main purposes of climate modeling? ›Climate models are important tools for improving our understanding and predictability of climate behavior on seasonal, annual, decadal, and centennial time scales. Models investigate the degree to which observed climate changes may be due to natural variability, human activity, or a combination of both.
What is the simplest climate model? ›The Very Simple Climate Model is, as the name implies, very simple. In this model, average global temperature is determined entirely by the atmospheric carbon dioxide concentration via greenhouse warming of the atmosphere and uptake of carbon dioxide by the ocean and biosphere, which are kept constant through time.
What are the best cmip6 climate models? ›The top three models are EC-Earth3-Veg, EC-Earth3, and HadGEM3-GC31-MM. Although created by simply averaging all the models, ENS_MEAN can compete with the best GCMs and ranks 4th overall. In the same model family, the higher-resolution experiment exhibits a higher ranking than the lower-resolution one.
Who has the best climate policy? ›Denmark, Sweden, and Chile are leading countries in climate action. Here's why and what other countries can learn from their climate policy strategies.
What are the most widely used climate models? ›When creating climate models, scientists use one of three common types of simple climate models: energy balance models, intermediate complexity models, and general circulation models.
What are 2 limitations of climate models? ›- an incomplete understanding of the climate system,
- an imperfect ability to transform our knowledge into accurate mathematical equations,
- the limited power of computers,
- the models' inability to reproduce important atmospheric phenomena, and.
What are the 3 main types of climates? ›
According to the three cell convection model of each hemisphere the Earth neatly separates itself into three distinct climate zones; the polar, temperate, and the tropical zones.
What does SSP1 1.9 mean? ›SSP1-1.9 : Very ambitious scenario to comply with the 1.5°C objective of the Paris Agreement. This is the most optimistic scenario. Global CO2 emissions fall to zero by 2050. Societies adopt more environmentally friendly practices, with the focus shifting from economic growth to general well-being.
Are climate models running hot? ›It's evident that, even in the past, many models run too hot, with only a small number coming close to actual measurements. Current projections of future warming represent an average of an ensemble of typically 55 different models.
What is the largest source of unpredictability in climate models? ›The uncertainty seen in climate modelling stems from two main sources: inter-model variability and intra-model variability.
What country has the worst climate change? ›- CHAD. It is considered the most vulnerable country on the planet according to the University of Notre Dame's country vulnerability studies, ...
- SOMALIA. ...
- DEMOCRATIC REPUBLIC OF CONGO. ...
- AFGHANISTAN.
- Somalia. Climate change has had a devastating impact on Somalia, worsening challenges of drought and extreme food insecurity. ...
- The Democratic Republic of the Congo. ...
- Yemen. ...
- Chad. ...
- South Sudan. ...
- Nigeria.
Climate change is the result of the total concentration of greenhouse gases in the atmosphere. And carbon dioxide, the primary greenhouse gas driving climate change, stays in the atmosphere for hundreds of years.