- Convert 148 pounds to kilograms Round your answer to the nearest tenth. Kg
- In Z3i), a. Write the additive inverse of 2 + i. b. Write the multiplicative inverse of (21) (1 + i)2 in the form a + bi.
the correct answer and explanation is:
1. Convert 148 pounds to kilograms:
To convert pounds to kilograms, use the conversion factor 1 pound=0.453592 kilograms1 \, \text{pound} = 0.453592 \, \text{kilograms}. 148 pounds×0.453592 kg/pound=67.131024 kg148 \, \text{pounds} \times 0.453592 \, \text{kg/pound} = 67.131024 \, \text{kg}
Rounded to the nearest tenth: 67.1 kg67.1 \, \text{kg}
Answer: 67.1 kg67.1 \, \text{kg}
2. Z3i)
a. Write the additive inverse of 2+i2 + i:
The additive inverse of a complex number a+bia + bi is −a−bi-a – bi. For 2+i2 + i: Additive inverse=−2−i\text{Additive inverse} = -2 – i
Answer: −2−i-2 – i
b. Write the multiplicative inverse of (2+i)2(2 + i)^2 in the form a+bia + bi:
- Expand (2+i)2(2 + i)^2:
(2+i)2=(2+i)(2+i)=4+4i+i2(2 + i)^2 = (2 + i)(2 + i) = 4 + 4i + i^2
Since i2=−1i^2 = -1: (2+i)2=4+4i−1=3+4i(2 + i)^2 = 4 + 4i – 1 = 3 + 4i
- Find the multiplicative inverse of 3+4i3 + 4i: The multiplicative inverse of a complex number z=a+biz = a + bi is given by:
1z=conjugate of z∣z∣2\frac{1}{z} = \frac{\text{conjugate of } z}{\lvert z \rvert^2}
For z=3+4iz = 3 + 4i, the conjugate is 3−4i3 – 4i, and the magnitude squared is: ∣z∣2=32+42=9+16=25\lvert z \rvert^2 = 3^2 + 4^2 = 9 + 16 = 25
Thus: 13+4i=3−4i25=325−425i\frac{1}{3 + 4i} = \frac{3 – 4i}{25} = \frac{3}{25} – \frac{4}{25}i Multiplicative inverse=0.12−0.16i\text{Multiplicative inverse} = 0.12 – 0.16i
Answer: 0.12−0.16i0.12 – 0.16i
Explanation:
Complex numbers combine a real part and an imaginary part in the form a+bia + bi. To solve the problems in Z3i, understanding key properties like the additive and multiplicative inverses is crucial.
The additive inverse of a complex number is the number that, when added to the original number, gives zero. For example, the additive inverse of 2+i2 + i is −2−i-2 – i because: (2+i)+(−2−i)=0(2 + i) + (-2 – i) = 0
The multiplicative inverse involves finding a number that, when multiplied with the original, gives 11. For z=3+4iz = 3 + 4i, its multiplicative inverse is: 1z=conjugate of z∣z∣2\frac{1}{z} = \frac{\text{conjugate of } z}{\lvert z \rvert^2}
Here, ∣z∣2=25\lvert z \rvert^2 = 25 is derived from the Pythagorean theorem applied to a=3a = 3 and b=4b = 4. The conjugate, 3−4i3 – 4i, ensures the imaginary part cancels when dividing. Simplifying yields: 325−425i\frac{3}{25} – \frac{4}{25}i
Understanding these inverses helps in fields like signal processing or solving differential equations, where complex numbers play a key role.