12/24: OpenAI announces o3, its highest-ever performing model 1/25: Alibaba unveils Qwen2.5-Max, which surpasses the performance of other leading models (GPT- 4o, Claude 3.5) on some AI overviews to augment its search functions 9/24: Alibaba releases 100 open-source Qwen 2.5 models, with performance in line with Western competitors 1/25: DeepSeek releases its 2021-2025, per NVIDIA 2.5MM 6MM 0 3 6 Number of Developers (MM) 7K 27K 0 15 30 2021 2025 Number of AI Startups (K) Number of Applications Using GPUs (K) 1.7K 4K 0 2.5 5 +2.4x +3.9x +2

julia> 2.5e-4 0.00025 The above results are all Float64 values. Literal Float32 values can be entered by writing an f in place of e: julia> x = 0.5f0 0.5f0 julia> typeof(x) Float32 julia> 2.5f-4 2 + 0im julia> (-3 + 2im) - (5 - 1im) -8 + 3im julia> (-1 + 2im)^2 -3 - 4im julia> (-1 + 2im)^2.5 2.729624464784009 - 6.9606644595719im 38CHAPTER 6. COMPLEX AND RATIONAL NUMBERS 39 julia> (-1 + julia> (2 + 3im) - 1 1 + 3im julia> (1 + 2im) + 0.5 1.5 + 2.0im julia> (2 + 3im) - 0.5im 2.0 + 2.5im julia> 0.75(1 + 2im) 0.75 + 1.5im julia> (2 + 3im) / 2 1.0 + 1.5im julia> (1 - 3im) / (2 + 2im)

julia> 2.5e-4 0.00025 The above results are all Float64 values. Literal Float32 values can be entered by writing an f in place of e: julia> x = 0.5f0 0.5f0 julia> typeof(x) Float32 julia> 2.5f-4 2 + 0im julia> (-3 + 2im) - (5 - 1im) -8 + 3im julia> (-1 + 2im)^2 -3 - 4im julia> (-1 + 2im)^2.5 2.729624464784009 - 6.9606644595719im 38CHAPTER 6. COMPLEX AND RATIONAL NUMBERS 39 julia> (-1 + julia> (2 + 3im) - 1 1 + 3im julia> (1 + 2im) + 0.5 1.5 + 2.0im julia> (2 + 3im) - 0.5im 2.0 + 2.5im julia> 0.75(1 + 2im) 0.75 + 1.5im julia> (2 + 3im) / 2 1.0 + 1.5im julia> (1 - 3im) / (2 + 2im)

julia> 2.5e-4 0.00025 The above results are all Float64 values. Literal Float32 values can be entered by writing an f in place of e: julia> x = 0.5f0 0.5f0 julia> typeof(x) Float32 julia> 2.5f-4 2 + 0im julia> (-3 + 2im) - (5 - 1im) -8 + 3im julia> (-1 + 2im)^2 -3 - 4im julia> (-1 + 2im)^2.5 2.729624464784009 - 6.9606644595719im 42CHAPTER 7. COMPLEX AND RATIONAL NUMBERS 43 julia> (-1 + julia> (2 + 3im) - 1 1 + 3im julia> (1 + 2im) + 0.5 1.5 + 2.0im julia> (2 + 3im) - 0.5im 2.0 + 2.5im julia> 0.75(1 + 2im) 0.75 + 1.5im julia> (2 + 3im) / 2 1.0 + 1.5im julia> (1 - 3im) / (2 + 2im)

julia> 2.5e-4 0.00025 The above results are all Float64 values. Literal Float32 values can be entered by writing an f in place of e: julia> x = 0.5f0 0.5f0 julia> typeof(x) Float32 julia> 2.5f-4 2 + 0im julia> (-3 + 2im) - (5 - 1im) -8 + 3im julia> (-1 + 2im)^2 -3 - 4im julia> (-1 + 2im)^2.5 2.729624464784009 - 6.9606644595719im 42CHAPTER 7. COMPLEX AND RATIONAL NUMBERS 43 julia> (-1 + julia> (2 + 3im) - 1 1 + 3im julia> (1 + 2im) + 0.5 1.5 + 2.0im julia> (2 + 3im) - 0.5im 2.0 + 2.5im julia> 0.75(1 + 2im) 0.75 + 1.5im julia> (2 + 3im) / 2 1.0 + 1.5im julia> (1 - 3im) / (2 + 2im)

julia> 2.5e-4 0.00025 The above results are all Float64 values. Literal Float32 values can be entered by writing an f in place of e: julia> x = 0.5f0 0.5f0 julia> typeof(x) Float32 julia> 2.5f-4 2 + 0im julia> (-3 + 2im) - (5 - 1im) -8 + 3im julia> (-1 + 2im)^2 -3 - 4im julia> (-1 + 2im)^2.5 2.729624464784009 - 6.9606644595719im 42CHAPTER 7. COMPLEX AND RATIONAL NUMBERS 43 julia> (-1 + julia> (2 + 3im) - 1 1 + 3im julia> (1 + 2im) + 0.5 1.5 + 2.0im julia> (2 + 3im) - 0.5im 2.0 + 2.5im julia> 0.75(1 + 2im) 0.75 + 1.5im julia> (2 + 3im) / 2 1.0 + 1.5im julia> (1 - 3im) / (2 + 2im)

julia> 2.5e-4 0.00025 The above results are all Float64 values. Literal Float32 values can be entered by writing an f in place of e: julia> x = 0.5f0 0.5f0 julia> typeof(x) Float32 julia> 2.5f-4 2 + 0im julia> (-3 + 2im) - (5 - 1im) -8 + 3im julia> (-1 + 2im)^2 -3 - 4im julia> (-1 + 2im)^2.5 2.729624464784009 - 6.9606644595719im 42CHAPTER 7. COMPLEX AND RATIONAL NUMBERS 43 julia> (-1 + julia> (2 + 3im) - 1 1 + 3im julia> (1 + 2im) + 0.5 1.5 + 2.0im julia> (2 + 3im) - 0.5im 2.0 + 2.5im julia> 0.75(1 + 2im) 0.75 + 1.5im julia> (2 + 3im) / 2 1.0 + 1.5im julia> (1 - 3im) / (2 + 2im)

julia> 2.5e-4 0.00025 The above results are all Float64 values. Literal Float32 values can be entered by writing an f in place of e: julia> x = 0.5f0 0.5f0 julia> typeof(x) Float32 julia> 2.5f-4 2 + 0im julia> (-3 + 2im) - (5 - 1im) -8 + 3im julia> (-1 + 2im)^2 -3 - 4im julia> (-1 + 2im)^2.5 2.729624464784009 - 6.9606644595719im 42CHAPTER 7. COMPLEX AND RATIONAL NUMBERS 43 julia> (-1 + julia> (2 + 3im) - 1 1 + 3im julia> (1 + 2im) + 0.5 1.5 + 2.0im julia> (2 + 3im) - 0.5im 2.0 + 2.5im julia> 0.75(1 + 2im) 0.75 + 1.5im julia> (2 + 3im) / 2 1.0 + 1.5im julia> (1 - 3im) / (2 + 2im)

julia> 2.5e-4 0.00025 The above results are all Float64 values. Literal Float32 values can be entered by writing an f in place of e: julia> x = 0.5f0 0.5f0 julia> typeof(x) Float32 julia> 2.5f-4 2 + 0im julia> (-3 + 2im) - (5 - 1im) -8 + 3im julia> (-1 + 2im)^2 -3 - 4im julia> (-1 + 2im)^2.5 2.729624464784009 - 6.9606644595719im 42CHAPTER 7. COMPLEX AND RATIONAL NUMBERS 43 julia> (-1 + julia> (2 + 3im) - 1 1 + 3im julia> (1 + 2im) + 0.5 1.5 + 2.0im julia> (2 + 3im) - 0.5im 2.0 + 2.5im julia> 0.75(1 + 2im) 0.75 + 1.5im julia> (2 + 3im) / 2 1.0 + 1.5im julia> (1 - 3im) / (2 + 2im)

julia> 2.5e-4 0.00025 The above results are all Float64 values. Literal Float32 values can be entered by writing an f in place of e: julia> x = 0.5f0 0.5f0 julia> typeof(x) Float32 julia> 2.5f-4 2 + 0im julia> (-3 + 2im) - (5 - 1im) -8 + 3im julia> (-1 + 2im)^2 -3 - 4im julia> (-1 + 2im)^2.5 2.729624464784009 - 6.9606644595719im 42CHAPTER 7. COMPLEX AND RATIONAL NUMBERS 43 julia> (-1 + julia> (2 + 3im) - 1 1 + 3im julia> (1 + 2im) + 0.5 1.5 + 2.0im julia> (2 + 3im) - 0.5im 2.0 + 2.5im julia> 0.75(1 + 2im) 0.75 + 1.5im julia> (2 + 3im) / 2 1.0 + 1.5im julia> (1 - 3im) / (2 + 2im)