In November 2023, the Financial Accounting Standards Board, or FASB, issued a new accounting standard requiring disclosures of significant expenses in operating segments. We adopted this standard in our fiscal year 2025 annual report. Refer to Note 16 of the Notes to the Consolidated Financial Statements in Part IV, Item 15 of this Annual Report on Form 10-K for further information.
Recent Accounting Pronouncements Not Yet Adopted
In December 2023, the FASB issued a new accounting standard which includes new and updated income tax disclosures, including disaggregation of information in the rate reconciliation and income taxes paid. We expect to adopt this standard in our fiscal year 2026 annual report. We do not expect the adoption of this standard to have a material impact on our Consolidated Financial Statements other than additional disclosures.
In November 2024, the FASB issued a new accounting standard requiring disclosures of certain additional expense information on an annual and interim basis, including, among other items, the amounts of purchases of inventory, employee compensation, depreciation and intangible asset amortization included within each income statement expense caption, as applicable. We expect to adopt this standard in our fiscal year 2028 annual report. We do not expect the adoption of this standard to have a material impact on our Consolidated Financial Statements other than additional disclosures.
Note 2 - Business Combination
Termination of the Arm Share Purchase Agreement
In February 2022, NVIDIA and SoftBank Group Corp, or SoftBank, announced the termination of the Share Purchase Agreement whereby NVIDIA would have acquired Arm from SoftBank. The parties agreed to terminate it due to significant regulatory challenges preventing the completion of the transaction. We recorded an acquisition termination cost of $1.4 billion in fiscal year 2023 reflecting the write-off of the prepayment provided at signing.
Note 3 - Stock-Based Compensation
Stock-based compensation expense is associated with RSUs, PSUs, market-based PSUs, and our ESPP.
Consolidated Statements of Income include stock-based compensation expense, net of amounts capitalized into inventory and subsequently recognized to cost of revenue, as follows:
In November 2023, the Financial Accounting Standards Board, or FASB, issued a new accounting standard requiring disclosures of significant expenses in operating segments. We adopted this standard in our fiscal year 2025 annual report. Refer to Note 16 of the Notes to the Consolidated Financial Statements in Part IV, Item 15 of this Annual Report on Form 10-K for further information.
Recent Accounting Pronouncements Not Yet Adopted
In December 2023, the FASB issued a new accounting standard which includes new and updated income tax disclosures, including disaggregation of information in the rate reconciliation and income taxes paid. We expect to adopt this standard in our fiscal year 2026 annual report. We do not expect the adoption of this standard to have a material impact on our Consolidated Financial Statements other than additional disclosures.
In November 2024, the FASB issued a new accounting standard requiring disclosures of certain additional expense information on an annual and interim basis, including, among other items, the amounts of purchases of inventory, employee compensation, depreciation and intangible asset amortization included within each income statement expense caption, as applicable. We expect to adopt this standard in our fiscal year 2028 annual report. We do not expect the adoption of this standard to have a material impact on our Consolidated Financial Statements other than additional disclosures.
Note 2 - Business Combination
Termination of the Arm Share Purchase Agreement
In February 2022, NVIDIA and SoftBank Group Corp, or SoftBank, announced the termination of the Share Purchase Agreement whereby NVIDIA would have acquired Arm from SoftBank. The parties agreed to terminate it due to significant regulatory challenges preventing the completion of the transaction. We recorded an acquisition termination cost of $1.4 billion in fiscal year 2023 reflecting the write-off of the prepayment provided at signing.
Note 3 - Stock-Based Compensation
Stock-based compensation expense is associated with RSUs, PSUs, market-based PSUs, and our ESPP.
Consolidated Statements of Income include stock-based compensation expense, net of amounts capitalized into inventory and subsequently recognized to cost of revenue, as follows:
In November 2023, the Financial Accounting Standards Board, or FASB, issued a new accounting standard requiring disclosures of significant expenses in operating segments. We adopted this standard in our fiscal year 2025 annual report. Refer to Note 16 of the Notes to the Consolidated Financial Statements in Part IV, Item 15 of this Annual Report on Form 10-K for further information.
Recent Accounting Pronouncements Not Yet Adopted
In December 2023, the FASB issued a new accounting standard which includes new and updated income tax disclosures, including disaggregation of information in the rate reconciliation and income taxes paid. We expect to adopt this standard in our fiscal year 2026 annual report. We do not expect the adoption of this standard to have a material impact on our Consolidated Financial Statements other than additional disclosures.
In November 2024, the FASB issued a new accounting standard requiring disclosures of certain additional expense information on an annual and interim basis, including, among other items, the amounts of purchases of inventory, employee compensation, depreciation and intangible asset amortization included within each income statement expense caption, as applicable. We expect to adopt this standard in our fiscal year 2028 annual report. We do not expect the adoption of this standard to have a material impact on our Consolidated Financial Statements other than additional disclosures.
Note 2 - Business Combination
Termination of the Arm Share Purchase Agreement
In February 2022, NVIDIA and SoftBank Group Corp, or SoftBank, announced the termination of the Share Purchase Agreement whereby NVIDIA would have acquired Arm from SoftBank. The parties agreed to terminate it due to significant regulatory challenges preventing the completion of the transaction. We recorded an acquisition termination cost of $1.4 billion in fiscal year 2023 reflecting the write-off of the prepayment provided at signing.
Note 3 - Stock-Based Compensation
Stock-based compensation expense is associated with RSUs, PSUs, market-based PSUs, and our ESPP.
Consolidated Statements of Income include stock-based compensation expense, net of amounts capitalized into inventory and subsequently recognized to cost of revenue, as follows:
Major General Hitchcock, Commissioner of Exchanges, is authorized and directed to offer Brigadier General Trimble, now a prisoner of war in Fort McHenry, in exchange for Major White, who is held as a prisoner at Richmond. He is also directed to send forward the offer of exchange by Henry M. Warfield, Esq. of Baltimore, under a flag of truce, and give him a pass to City Point.
Major General Hitchcock, Commissioner of Exchanges, is authorized and directed to offer Brigadier General Trimble, now a prisoner of war in Fort McHenry, in exchange for Major White, who is held as a prisoner at Richmond. He is also directed to send forward the offer of exchange by Henry M. Warfield, Esq. of Baltimore, under a flag of truce, and give him a pass to City Point.
Major General Hitchcock, Commissioner of Exchanges, is authorized and directed to offer Brigadier General Trimble, now a prisoner of war in Fort McHenry, in exchange for Major White, who is held as a prisoner at Richmond. He is also directed to send forward the offer of exchange by Henry M. Warfield, Esq. of Baltimore, under a flag of truce, and give him a pass to City Point.
Major General Hitchcock, Commissioner of Exchanges, is authorized and directed to offer Brigadier General Trimble, now a prisoner of war in Fort McHenry, in exchange for Major White, who is held as a prisoner at Richmond. He is also directed to send forward the offer of exchange by Henry M. Warfield, Esq. of Baltimore, under a flag of truce, and give him a pass to City Point.
Major General Hitchcock, Commissioner of Exchanges, is authorized and directed to offer Brigadier General Trimble, now a prisoner of war in Fort McHenry, in exchange for Major White, who is held as a prisoner at Richmond. He is also directed to send forward the offer of exchange by Henry M. Warfield, Esq. of Baltimore, under a flag of truce, and give him a pass to City Point.
Proof. Let $S$ be the generating set associated with $D$ as described in Proposition 2.5. By the circulant diagonalization theorem, the spectrum of $G_R(D) = \Gamma(R, S)$ is the multiset \(\{\lambda_g \}_{g \in R} \) where
We remark that by Corollary 2.7, if $s \in R$ such that $Rs = I_i = Rx_i$ then $s$ has a unique representation of the form $s = ux_i$ where $u \in (R/\text{Ann}_R(x_i))^\times$ and $\hat{u}$ is a fixed lift of $u$ to $R^\times$. With this presentation, we can write
Here we recall that $\psi_{x_i}$ is the induced linear functional on $R/\text{Ann}_R(x_i)$. We conclude that $\lambda_g = \sum_{i=1}^k c(g, R/\text{Ann}_R(x_i))$. \(\square \)
The following corollary is simple yet important for our future work on perfect state transfers on gcd-graphs.
**Corollary 4.17.** Suppose that $g' = ug$ for some $u \in R^\times$. Then $\lambda_g = \lambda_{g'}$.
**ACKNOWLEDGEMENTS**
We thank the Department of Mathematics and Computer Science at Lake Forest College for their generous financial support through an Overleaf subscription. We also thank Ján Mináč for his constant encouragement and support.
**REFERENCES**
1. Reza Akhtar, Megan Boggess, Tiffany Jackson-Henderson, Isidora Jiménez, Rachel Karpman, Amanda Kinzel, and Dan Pritikin, *On the unitary Cayley graph of a finite ring*, Electron. J. Combin. **16** (2009), no. 1, Research Paper 117, 13 pages.
2. Milan Bašić, Aleksandar Ilić, and Aleksandar Stamenković, *Maximal diameter of integral circulant graphs*, Information and Computation **301** (2024), 105208.
3. Maria Chudnovsky, Michal Cizek, Logan Crew, Ján Mináč, Tung T. Nguyen, Sophie Spirkl, and Nguyễn Duy Tấn, *On prime Cayley graphs*, arXiv:2401.06062, to appear in Journal of Combinatorics (2024).
4. Thomas Honold, *Characterization of finite frobenius rings*, Archiv der Mathematik **76** (2001), no. 6, 406–415.
5. Irving Kaplansky, *Elementary divisors and modules*, Transactions of the American Mathematical Society **66** (1949), no. 2, 464–491.
6. Walter Klotz and Torsten Sander, *Some properties of unitary Cayley graphs*, The Electronic Journal of Combinatorics **14** (2007), no. 1, R45, 12 pages.
7. Erich Lamprecht, *Allgemeine theorie der Gaußschen Summen in endlichen kommutativen Ringen*, Mathematische Nachrichten **9** (1953), no. 3, 149–196.
8. Ján Mináč, Tung T Nguyen, and Nguyen Duy Tấn, *Isomorphic gcd-graphs over polynomial rings*, arXiv preprint arXiv:2411.01768 (2024).
9. ______, *On the gcd graphs over polynomial rings*, arXiv preprint arXiv:2409.01929 (2024).
Proof. Let $S$ be the generating set associated with $D$ as described in Proposition 2.5. By the circulant diagonalization theorem, the spectrum of $G_R(D) = \Gamma(R, S)$ is the multiset \(\{\lambda_g \}_{g \in R} \) where
We remark that by Corollary 2.7, if $s \in R$ such that $Rs = I_i = Rx_i$ then $s$ has a unique representation of the form $s = ux_i$ where $u \in (R/\text{Ann}_R(x_i))^\times$ and $\hat{u}$ is a fixed lift of $u$ to $R^\times$. With this presentation, we can write
Here we recall that $\psi_xi$ is the induced linear functional on $R/\text{Ann}_R(x_i)$. We conclude that $\lambda_g = \sum_{i=1}^k c(g, R/\text{Ann}_R(x_i))$. \qed
The following corollary is simple yet important for our future work on perfect state transfers on gcd-graphs.
**Corollary 4.17.** Suppose that $g' = ug$ for some $u \in R^\times$. Then $\lambda_g = \lambda_{g'}$.
**ACKNOWLEDGEMENTS**
We thank the Department of Mathematics and Computer Science at Lake Forest College for their generous financial support through an Overleaf subscription. We also thank Ján Mináč for his constant encouragement and support.
**REFERENCES**
1. Reza Akhtar, Megan Boggess, Tiffany Jackson-Henderson, Isidora Jiménez, Rachel Karpman, Amanda Kinzel, and Dan Pritikin, *On the unitary Cayley graph of a finite ring*, Electron. J. Combin. **16** (2009), no. 1, Research Paper 117, 13 pages.
2. Milan Bašić, Aleksandar Ilić, and Aleksandar Stamenković, *Maximal diameter of integral circulant graphs*, Information and Computation **301** (2024), 105208.
3. Maria Chudnovsky, Michal Cizek, Logan Crew, Ján Mináč, Tung T. Nguyen, Sophie Spirkl, and Nguyễn Duy Tấn, *On prime Cayley graphs*, arXiv:2401.06062, to appear in Journal of Combinatorics (2024).
4. Thomas Honold, *Characterization of finite frobenius rings*, Archiv der Mathematik **76** (2001), no. 6, 406–415.
5. Irving Kaplansky, *Elementary divisors and modules*, Transactions of the American Mathematical Society **66** (1949), no. 2, 464–491.
6. Walter Klotz and Torsten Sander, *Some properties of unitary Cayley graphs*, The Electronic Journal of Combinatorics **14** (2007), no. 1, R45, 12 pages.
7. Erich Lamprecht, *Allgemeine theorie der Gaußschen Summen in endlichen kommutativen Ringen*, Mathematische Nachrichten **9** (1953), no. 3, 149–196.
8. Ján Mináč, Tung T Nguyen, and Nguyen Duy Tấn, *Isomorphic gcd-graphs over polynomial rings*, arXiv preprint arXiv:2411.01768 (2024).
9. ______, *On the gcd graphs over polynomial rings*, arXiv preprint arXiv:2409.01929 (2024).
Proof. Let $S$ be the generating set associated with $D$ as described in Proposition 2.5. By the circulant diagonalization theorem, the spectrum of $G_R(D) = \Gamma(R, S)$ is the multiset \(\{\lambda_g \}_{g \in R} \) where
We remark that by Corollary 2.7, if $s \in R$ such that $Rs = I_i = Rx_i$ then $s$ has a unique representation of the form $s = \hat{u}x_i$ where $u \in (R/\text{Ann}_R(x_i))^\times$ and $\hat{u}$ is a fixed lift of $u$ to $R^\times$. With this presentation, we can write
Here we recall that $\psi_{x_i}$ is the induced linear functional on $R/\text{Ann}_R(x_i)$. We conclude that $\lambda_g = \sum_{i=1}^{k} c(g, R/\text{Ann}_R(x_i))$. \(\square \)
The following corollary is simple yet important for our future work on perfect state transfers on gcd-graphs.
**Corollary 4.17.** Suppose that $g' = ug$ for some $u \in R^\times$. Then $\lambda_g = \lambda_{g'}$.
**ACKNOWLEDGEMENTS**
We thank the Department of Mathematics and Computer Science at Lake Forest College for their generous financial support through an Overleaf subscription. We also thank Ján Mináč for his constant encouragement and support.
**REFERENCES**
1. Reza Akhtar, Megan Boggess, Tiffany Jackson-Henderson, Isidora Jiménez, Rachel Karpman, Amanda Kinzel, and Dan Pritikin, *On the unitary Cayley graph of a finite ring*, Electron. J. Combin. **16** (2009), no. 1, Research Paper 117, 13 pages.
2. Milan Bašić, Aleksandar Ilić, and Aleksandar Stamenković, *Maximal diameter of integral circulant graphs*, Information and Computation **301** (2024), 105208.
3. Maria Chudnovsky, Michal Cizek, Logan Crew, Ján Mináč, Tung T. Nguyen, Sophie Spirkl, and Nguyễn Duy Tấn, *On prime Cayley graphs*, arXiv:2401.06062, to appear in Journal of Combinatorics (2024).
4. Thomas Honold, *Characterization of finite frobenius rings*, Archiv der Mathematik **76** (2001), no. 6, 406–415.
5. Irving Kaplansky, *Elementary divisors and modules*, Transactions of the American Mathematical Society **66** (1949), no. 2, 464–491.
6. Walter Klotz and Torsten Sander, *Some properties of unitary Cayley graphs*, The Electronic Journal of Combinatorics **14** (2007), no. 1, R45, 12 pages.
7. Erich Lamprecht, *Allgemeine theorie der Gaußschen Summen in endlichen kommutativen Ringen*, Mathematische Nachrichten **9** (1953), no. 3, 149–196.
8. Ján Mináč, Tung T Nguyen, and Nguyen Duy Tấn, *Isomorphic gcd-graphs over polynomial rings*, arXiv preprint arXiv:2411.01768 (2024).
9. ______, *On the gcd graphs over polynomial rings*, arXiv preprint arXiv:2409.01929 (2024).
Proof. Let $S$ be the generating set associated with $D$ as described in Proposition 2.5. By the circulant diagonalization theorem, the spectrum of $G_R(D) = \Gamma(R, S)$ is the multiset \(\{\lambda_g \}_{g \in R} \) where
We remark that by Corollary 2.7, if $s \in R$ such that $Rs = I_i = Rx_i$ then $s$ has a unique representation of the form $s = \hat{u}x_i$ where $u \in (R/\text{Ann}_R(x_i))^\times$ and $\hat{u}$ is a fixed lift of $u$ to $R^\times$. With this presentation, we can write
Here we recall that $\psi_{x_i}$ is the induced linear functional on $R/\text{Ann}_R(x_i)$. We conclude that $\lambda_g = \sum_{i=1}^k c(g, R/\text{Ann}_R(x_i))$. \(\square \)
The following corollary is simple yet important for our future work on perfect state transfers on gcd-graphs.
**Corollary 4.17.** Suppose that $g' = ug$ for some $u \in R^\times$. Then $\lambda_g = \lambda_{g'}$.
**ACKNOWLEDGEMENTS**
We thank the Department of Mathematics and Computer Science at Lake Forest College for their generous financial support through an Overleaf subscription. We also thank Ján Mináč for his constant encouragement and support.
**REFERENCES**
1. Reza Akhtar, Megan Boggess, Tiffany Jackson-Henderson, Isidora Jiménez, Rachel Karpman, Amanda Kinzel, and Dan Pritikin, *On the unitary Cayley graph of a finite ring*, Electron. J. Combin. **16** (2009), no. 1, Research Paper 117, 13 pages.
2. Milan Bašić, Aleksandar Ilić, and Aleksandar Stamenković, *Maximal diameter of integral circulant graphs*, Information and Computation **301** (2024), 105208.
3. Maria Chudnovsky, Michal Cizek, Logan Crew, Ján Mináč, Tung T. Nguyen, Sophie Spirkl, and Nguyễn Duy Tấn, *On prime Cayley graphs*, arXiv:2401.06062, to appear in Journal of Combinatorics (2024).
4. Thomas Honold, *Characterization of finite frobenius rings*, Archiv der Mathematik **76** (2001), no. 6, 406–415.
5. Irving Kaplansky, *Elementary divisors and modules*, Transactions of the American Mathematical Society **66** (1949), no. 2, 464–491.
6. Walter Klotz and Torsten Sander, *Some properties of unitary Cayley graphs*, The Electronic Journal of Combinatorics **14** (2007), no. 1, R45, 12 pages.
7. Erich Lamprecht, *Allgemeine theorie der Gaußschen Summen in endlichen kommutativen Ringen*, Mathematische Nachrichten **9** (1953), no. 3, 149–196.
8. Ján Mináč, Tung T Nguyen, and Nguyen Duy Tấn, *Isomorphic gcd-graphs over polynomial rings*, arXiv preprint arXiv:2411.01768 (2024).
9. ______, *On the gcd graphs over polynomial rings*, arXiv preprint arXiv:2409.01929 (2024).
Proof. Let $S$ be the generating set associated with $D$ as described in Proposition 2.5. By the circulant diagonalization theorem, the spectrum of $G_R(D) = \Gamma(R, S)$ is the multiset \(\{\lambda_g \}_{g \in R} \) where
We remark that by Corollary 2.7, if $s \in R$ such that $Rs = I_i = Rx_i$ then $s$ has a unique representation of the form $s = \hat{u}x_i$ where $u \in (R/\text{Ann}_R(x_i))^\times$ and $\hat{u}$ is a fixed lift of $u$ to $R^\times$. With this presentation, we can write
Here we recall that $\psi_{x_i}$ is the induced linear functional on $R/\text{Ann}_R(x_i)$. We conclude that $\lambda_g = \sum_{i=1}^{k} c(g, R/\text{Ann}_R(x_i))$. \(\square \)
The following corollary is simple yet important for our future work on perfect state transfers on gcd-graphs.
**Corollary 4.17.** Suppose that $g' = ug$ for some $u \in R^\times$. Then $\lambda_g = \lambda_{g'}$.
**ACKNOWLEDGEMENTS**
We thank the Department of Mathematics and Computer Science at Lake Forest College for their generous financial support through an Overleaf subscription. We also thank Ján Mináč for his constant encouragement and support.
**REFERENCES**
1. Reza Akhtar, Megan Boggess, Tiffany Jackson-Henderson, Isidora Jiménez, Rachel Karpman, Amanda Kinzel, and Dan Pritikin, *On the unitary Cayley graph of a finite ring*, Electron. J. Combin. **16** (2009), no. 1, Research Paper 117, 13 pages.
2. Milan Bašić, Aleksandar Ilić, and Aleksandar Stamenković, *Maximal diameter of integral circulant graphs*, Information and Computation **301** (2024), 105208.
3. Maria Chudnovsky, Michal Cizek, Logan Crew, Ján Mináč, Tung T. Nguyen, Sophie Spirkl, and Nguyễn Duy Tấn, *On prime Cayley graphs*, arXiv:2401.06062, to appear in Journal of Combinatorics (2024).
4. Thomas Honold, *Characterization of finite frobenius rings*, Archiv der Mathematik **76** (2001), no. 6, 406–415.
5. Irving Kaplansky, *Elementary divisors and modules*, Transactions of the American Mathematical Society **66** (1949), no. 2, 464–491.
6. Walter Klotz and Torsten Sander, *Some properties of unitary Cayley graphs*, The Electronic Journal of Combinatorics **14** (2007), no. 1, R45, 12 pages.
7. Erich Lamprecht, *Allgemeine theorie der Gaußschen Summen in endlichen kommutativen Ringen*, Mathematische Nachrichten **9** (1953), no. 3, 149–196.
8. Ján Mináč, Tung T Nguyen, and Nguyen Duy Tấn, *Isomorphic gcd-graphs over polynomial rings*, arXiv preprint arXiv:2411.01768 (2024).
9. ______, *On the gcd graphs over polynomial rings*, arXiv preprint arXiv:2409.01929 (2024).
A journey through the most elegant and influential formulas in mathematics
1. Euler’s Identity
\[ e^{i\pi} + 1 = 0 \]
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
2. Pythagorean Theorem
\[ a^2 + b^2 = c^2 \]
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.
3. The Fundamental Theorem of Calculus
\[\int_a^b f(x) \, dx = F(b) - F(a) \]
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving.
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
A journey through the most elegant and influential formulas in mathematics
1. Euler’s Identity
\[ e^{i\pi} + 1 = 0 \]
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
2. Pythagorean Theorem
\[ a^2 + b^2 = c^2 \]
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.
3. The Fundamental Theorem of Calculus
\[\int_a^b f(x) \, dx = F(b) - F(a) \]
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving.
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
A journey through the most elegant and influential formulas in mathematics
1. Euler’s Identity
\[ e^{i\pi} + 1 = 0 \]
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
2. Pythagorean Theorem
\[ a^2 + b^2 = c^2 \]
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.
3. The Fundamental Theorem of Calculus
\[\int_a^b f(x) \, dx = F(b) - F(a) \]
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving.
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
A journey through the most elegant and influential formulas in mathematics
1. Euler’s Identity
\[ e^{i\pi} + 1 = 0 \]
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
2. Pythagorean Theorem
\[ a^2 + b^2 = c^2 \]
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.
3. The Fundamental Theorem of Calculus
\[\int_a^b f(x) \, dx = F(b) - F(a) \]
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving.
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
A journey through the most elegant and influential formulas in mathematics
1. Euler’s Identity
\[ e^{i\pi} + 1 = 0 \]
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
2. Pythagorean Theorem
\[ a^2 + b^2 = c^2 \]
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.
3. The Fundamental Theorem of Calculus
\[\int_a^b f(x) \, dx = F(b) - F(a) \]
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving.
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
A journey through the most elegant and influential formulas in mathematics
1. Euler's Identity
\[ e^{i\pi} + 1 = 0 \]
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
2. Pythagorean Theorem
\[ a^2 + b^2 = c^2 \]
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.
3. The Fundamental Theorem of Calculus
\[\int_{a}^{b} f(x) \, dx = F(b) - F(a) \]
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving.
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
A journey through the most elegant and influential formulas in mathematics
1. Euler's Identity
\[ e^{i\pi} + 1 = 0 \]
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
2. Pythagorean Theorem
\[ a^2 + b^2 = c^2 \]
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.
3. The Fundamental Theorem of Calculus
\[\int_{a}^{b} f(x) \, dx = F(b) - F(a) \]
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving.
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
