Explain number-system structure

The imaginary part is zero. Which conclusion follows?

Nested systems and their structure

The standard systems extend one another. Rational numbers include all integers; real numbers also include irrationals; complex numbers also include non-real values. Ordinary arithmetic properties such as associativity, commutativity and distributivity continue to hold in the complex system

1. 1Use the stated natural-number convention Here the natural numbers start at one. Integers also include zero and negative whole numbers. The ± notation includes both signs. Other sources may include zero in the natural numbers, so check the definition \(\displaystyle \begin{aligned}\mathbb N&=\{1,2,3,\ldots\}\\\mathbb Z&=\{0,\pm1,\pm2,\ldots\}\end{aligned}\)
2. 2Rational versus irrational A rational number is a ratio of integers with a nonzero denominator. Real numbers not expressible that way are irrational. Finite and recurring decimals are rational \(\displaystyle \mathbb Q=\left\{\frac ab:a,b\in\mathbb Z,\ b\ne0\right\}\)
3. 3Extend to the complex numbers A complex number has real and imaginary components. Every real number is complex with imaginary part zero. Non-real means the imaginary part is nonzero \(\displaystyle \begin{gathered}z=a+bi\\a,b\in\mathbb R,\quad i^2=-1\end{gathered}\)
4. 4Closure is a universal claim A set is closed under an operation when every permitted pair of members gives another member. One valid counterexample refutes closure; several successful examples do not prove it \(\displaystyle a,b\in\mathbb Q\implies a+b\in\mathbb Q\)
5. 5Identity and inverse depend on the operation Zero is the additive identity, one the multiplicative identity. The additive inverse of a is −a. A multiplicative inverse exists only for nonzero elements of a field such as the rationals, reals or complex numbers \(\displaystyle \begin{aligned}a+0&=a\\a+(-a)&=0\\a\cdot1&=a\\a\cdot\frac1a&=1\quad(a\ne0)\end{aligned}\)

Which conclusion follows after evaluating the expression?

A student claims integers are closed under division by a nonzero integer. Which pair disproves that claim?

Enter the additive inverse as a fraction

Enter the multiplicative inverse of the nonzero value as a fraction

Written practice, ungraded

On paper, prove that the sum of any two rational numbers is rational. State arbitrary numerator and denominator variables, their domains and the nonzero conditions. The next screen gives a model argument

Model argument

Let the displayed fractions be arbitrary rational numbers, with integer numerators and nonzero integer denominators. The displayed sum has an integer numerator and nonzero integer denominator, so the sum is rational. Because the pair was arbitrary, this proves closure under addition

Order is not preserved by the complex extension

The real ordered-field rules cannot extend to all complex numbers. In an ordered field every nonzero square is positive, but the square of i is −1. You can sort complex numbers using a chosen display convention, but that is not the usual real-number order with all of its arithmetic properties