Complex numbers #1 - Q&A

Question 1 Which operation could we perform in order to find the number of milliseconds in a year? 60 · 60 · 24 · 7 · 365 1000 · 60 · 24 · 365 24 · 60 · 100 · 7 · 52 1000 · 60 · 24 · 7 · 52? Answer 1 We need to determine the number of milliseconds in a year. First of all, "milli" means thousandth and thus one second contains 1000 milliseconds. 1   s = 1000   ms 1s=1000ms Next, there are 60 seconds in one minute and thus the number of milliseconds in a minute is then obtained by multiplying the number of milliseconds in a second by the number of seconds in a minute, 1   min = 60   s = 1000 ⋅ 60   ms 1min=60s=1000⋅60ms Next, there are 60 minutes in one hour and thus the number of milliseconds in an hour is then obtained by multiplying the number of milliseconds in a minute by the number of minutes in an hour. 1   h = 60   min = 1000 ⋅ 60 ⋅ 60   ms 1h=60min=1000⋅60⋅60ms Next, there are 24 hours in a day and thus the number of milliseconds in a day is then obtained by multiplying the number of milliseconds in an hour by the number of hours in a day. 1   day = 24   h = 1000 ⋅ 60 ⋅ 60 ⋅ 24   ms 1day=24h=1000⋅60⋅60⋅24ms Finally, we assume that there are 365 days in a year (assuming no leap years). The number of milliseconds in a year is then obtained by multiplying the number of milliseconds in a day by the number of days in a year. 1   year = 365   days = 1000 ⋅ 60 ⋅ 60 ⋅ 24 ⋅ 365   ms 1year=365days=1000⋅60⋅60⋅24⋅365ms We then note that the correct operation is: 1000 ⋅ 60 ⋅ 60 ⋅ 24 ⋅ 365 1000⋅60⋅60⋅24⋅365. Answer 2 To find the number of milliseconds in a year, we need to multiply the number of milliseconds in a day by the number of days in a year. There are different ways to express the number of milliseconds in a day. One common way is to use the fact that there are 24 hours in a day, 60 minutes in an hour, 60 seconds in a minute, and 1000 milliseconds in a second. Therefore, we have: milliseconds in a day = 24 × 60 × 60 × 1000 = 86 , 400 , 000 =24×60×60×1000=86,400,000 Alternatively, we could use the fact that there are 1440 minutes in a day, 60 seconds in a minute, and 1000 milliseconds in a second. Therefore, we have: milliseconds in a day = 1440 × 60 × 1000 = 86 , 400 , 000 =1440×60×1000=86,400,000 Either way, the number of milliseconds in a day is 86,400,000. To find the number of milliseconds in a year, we can multiply the number of milliseconds in a day by the number of days in a year. There are different ways to express the number of days in a year, depending on whether we consider a year to have 365 or 365.25 days. For simplicity, we will assume that a year has 365 days. Therefore, we have: milliseconds in a year = 86 , 400 , 000 × 365 = 31 , 536 , 000 , 000 =86,400,000×365=31,536,000,000 Therefore, the correct operation to perform in order to find the number of milliseconds in a year is: milliseconds in a year = 86 , 400 , 000 × 365 = 31 , 536 , 000 , 000 =86,400,000×365=31,536,000,000 Answer 3 Answer: = 31 , 536 , 000 , 000 =31,536,000,000 Explanation: We can start by finding the number of milliseconds in one day, which is given by: milliseconds in a day = 1000 × 60 × 60 × 24 = 86 , 400 , 000 =1000×60×60×24=86,400,000 Next, we can find the number of milliseconds in one year by multiplying the number of milliseconds in one day by the number of days in one year. Since there are 365 days in a non-leap year, we have: milliseconds in a year = milliseconds in a day × days in a year =milliseconds in a day×days in a year = 86 , 400 , 000 × 365 =86,400,000×365 = 31 , 536 , 000 , 000 =31,536,000,000 Therefore, the number of milliseconds in one year is 31,536,000,000. Answer 4 Solution: We know that a year has 365 days and each day has 24 hours, 60 minutes, 60 seconds, and 1000 milliseconds. Therefore, we can find the number of milliseconds in a year by first finding the number of milliseconds in one hour, one minute, and one second, and then multiplying these values by the number of hours, minutes, and seconds in a year, respectively. One hour has 60 minutes, and each minute has 60 seconds, so there are: 60 × 60 = 3600 60×60=3600 seconds in an hour. Since each second has 1000 milliseconds, there are: 3600 × 1000 = 3 , 600 , 000 3600×1000=3,600,000 milliseconds in one hour. Similarly, one minute has 60 seconds, so there are: 60 × 1000 = 60 , 000 60×1000=60,000 milliseconds in one minute. Finally, we can find the number of milliseconds in a year by multiplying the number of hours, minutes, and seconds in a year by the number of milliseconds in one hour, one minute, and one second, respectively: milliseconds in a year = 365 × 24 × 60 × 60 × 1000 =365×24×60×60×1000 = 365 × 24 × 3 , 600 , 000 =365×24×3,600,000 = 31 , 536 , 000 , 000 =31,536,000,000 Therefore, the number of milliseconds in a year is 31,536,000,000. Question 2 Complex number in rectangular form What is ( 1 + 2 𝑗 ) + ( 1 + 3 𝑗 ) (1+2j)+(1+3j)? Your answer should contain three significant figures. Answer 1 Sum of complex numbers ( 𝑎 1 + 𝑏 1 𝑖 ) + ( 𝑎 2 + 𝑏 2 𝑖 ) = ( 𝑎 1 + 𝑎 2 ) + ( 𝑏 1 + 𝑏 2 ) 𝑖 = 𝑎 + 𝑏 𝑖 (a 1 ​ +b 1 ​ i)+(a 2 ​ +b 2 ​ i)=(a 1 ​ +a 2 ​ )+(b 1 ​ +b 2 ​ )i=a+bi ( 1 + 2 𝑖 ) + ( 1 + 3 𝑖 ) = ( 1 + 1 ) + ( 2 + 3 ) 𝑖 = 2 + 5 𝑖 (1+2i)+(1+3i)=(1+1)+(2+3)i=2+5i this is the binomial form Then find 𝑟 r and 𝜃 θ 𝑟 = 𝑎 2 + 𝑏 2 = 2 2 + 5 2 = 29 r= a 2 +b 2 ​ = 2 2 +5 2 ​ = 29 ​ if 𝑎 a and 𝑏 b are positive then 𝜃 = tan ⁡ − 1 ( 𝑏 𝑎 ) = tan ⁡ − 1 ( 5 2 ) = 68. 2 ∘ θ=tan −1 ( a b ​ )=tan −1 ( 2 5 ​ )=68.2 ∘ Rectangle form 𝑧 = 𝑟 ( cos ⁡ ( 𝜃 ) + sin ⁡ ( 𝜃 ) 𝑖 ) z=r(cos(θ)+sin(θ)i) then 𝑧 = 29 ( cos ⁡ ( 68.2 ) + sin ⁡ ( 68.2 ) 𝑖 ) z= 29 ​ (cos(68.2)+sin(68.2)i) Answer 2 Answer: ( 1 + 2 𝑗 ) + ( 1 + 3 𝑗 ) = 2 + 5 𝑗 (1+2j)+(1+3j)=2+5j Explanation: Given: ( 1 + 2 𝑗 ) + ( 1 + 3 𝑗 ) (1+2j)+(1+3j) We can group the real and imaginary parts: ( 1 + 1 ) + ( 2 𝑗 + 3 𝑗 ) (1+1)+(2j+3j) Simplifying the real part: 1 + 1 = 2 1+1=2 Simplifying the imaginary part: 2 𝑗 + 3 𝑗 = 5 𝑗 2j+3j=5j Combining the real and imaginary parts, we get the result: 2 + 5 𝑗 2+5j Therefore, ( 1 + 2 𝑗 ) + ( 1 + 3 𝑗 ) = 2 + 5 𝑗 (1+2j)+(1+3j)=2+5j. Answer 3 Let's start by adding the real parts together: ( 1 + 2 𝑗 ) + ( 1 + 3 𝑗 ) = 1 + 1 + 2 𝑗 + 3 𝑗 (1+2j)+(1+3j)=1+1+2j+3j Next, we add the imaginary parts together: = 2 + 5 𝑗 =2+5j Therefore, the sum of ( 1 + 2 𝑗 ) + ( 1 + 3 𝑗 ) (1+2j)+(1+3j) is 2 + 5 𝑗 2+5j. In rectangular form, the complex number 2 + 5 𝑗 2+5j can be represented as 2 + 5 𝑖 2+5i, where 𝑖 i is also used to denote the imaginary unit. Answer 4 To solve the expression ( 1 + 2 𝑗 ) + ( 1 + 3 𝑗 ) (1+2j)+(1+3j) in rectangular form, we simply add the real parts and the imaginary parts separately. The real part is obtained by adding the real parts of the complex numbers, and the imaginary part is obtained by adding the imaginary parts. Let's break down the calculation: Real part: 1 + 1 = 2 1+1=2 Imaginary part: 2 𝑗 + 3 𝑗 = 5 𝑗 2j+3j=5j Putting it all together, the sum of ( 1 + 2 𝑗 ) + ( 1 + 3 𝑗 ) (1+2j)+(1+3j) is 2 + 5 𝑗 2+5j. Hence, the result in rectangular form is 2 + 5 𝑗 2+5j. Question 3 Write each quotient as a complex number. 4 − 3 𝑖 − 1 − 4 𝑖 −1−4i 4−3i ​ Answer 1 Each quotient should be written as a complex number. 4 − 3 𝑖 − 1 − 4 𝑖 −1−4i 4−3i