integer

Examples of integer in the following topics:

• Integer Coefficients and the Rational Zeros Theorem

  • Each solution to a polynomial, expressed as $x= \frac {p}{q}$, must satisfy that $p$ and $q$ are integer factors of $a_0$ and $a_n$, respectively.
  • If $a_0$ and $a_n$ are nonzero, then each rational solution $x= \frac {p}{q}$, where $p$ and $q$ are coprime integers (i.e. their greatest common divisor is $1$), satisfies:
  • Since any integer has only a finite number of divisors, the rational root theorem provides us with a finite number of candidates for rational roots.
  • When given a polynomial with integer coefficients, we can plug in all of these candidates and see whether they are a zero of the given polynomial.
  • Since every polynomial with rational coefficients can be multiplied with an integer to become a polynomial with integer coefficients and the same zeros, the Rational Root Test can also be applied for polynomials with rational coefficients.

• Zeroes of Polynomial Functions With Rational Coefficients

  • In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.
  • Since q may be equal to 1, every integer is a rational number.
  • These statements hold true not just for base 10, but also for binary, hexadecimal, or any other integer base.
  • Zero divided by any other integer equals zero.
  • The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers.

• Simplifying Exponential Expressions

  • Previously, we have applied these rules only to expressions involving integers.
  • However, they also apply to expressions involving a combination of both integers and variables.
  • In terms of conducting operations, exponential expressions that contain variables are treated just as though they are composed of integers.

• Exponential Growth and Decay

  • The formula for exponential growth of a variable $x$ at the (positive or negative) growth rate $r$, as time $t$ goes on in discrete intervals (that is, at integer times 0, 1, 2, 3, ...), is:
  • For example, with a growth rate of $r = 5 \% = 0.05$, going from any integer value of time to the next integer causes $x$ at the second time to be $1.05$ times (i.e., $5\%$ larger than) what it was at the previous time.

• Simplifying Radical Expressions

  • Radical expressions containing variables can be simplified to a basic expression in a similar way to those involving only integers.
  • Radical expressions that contain variables are treated just as though they are integers when simplifying the expression.
  • This follows the same logic that we used above, when simplifying the radical expression with integers: $\sqrt{32} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}$.

• Introduction to Factoring Polynomials

  • Factoring is the decomposition of an algebraic object, for example an integer or a polynomial, into a product of other objects, or factors, which when multiplied together give the original.
  • As an example, the integer $15$ factors as $3 \cdot 5$, and the polynomial $x^3 + 2x^2$ factors as $x^2(x+2)$.
  • In all cases, a product of simpler objects than the original (smaller integers, or polynomials of smaller degree) is obtained.
  • The aim of factoring is to reduce objects to "basic building blocks", such as integers to prime numbers, or polynomials to irreducible polynomials.

• Transposition

  • This is how I arrived at the T4 arrow label in the musical example above, by "subtracting" the pitch class integers of m. 1 from the pitch-class integers in m. 18.

• The Pauli Exclusion Principle

  • The Pauli exclusion principle governs the behavior of all fermions (particles with half-integer spin), while bosons (particles with integer spin) are not subject to it.
  • Half-integer spin means the intrinsic angular momentum value of fermions is $\hbar =\frac { h }{ 2\pi }$ (reduced Planck's constant) times a half-integer (1/2, 3/2, 5/2, etc.).
  • In contrast, particles with integer spin (bosons) have symmetric wave functions; unlike fermions, bosons may share the same quantum states.

• Additional Properties of the Binomial Distribution

  • If $np$ is an integer, then the mean, median, and mode coincide and equal $np$.
  • However, when $(n + 1)p$ is an integer and p is neither 0 nor 1, then the distribution has two modes: $(n + 1)p$ and $(n + 1)p 1$.

• Bases Other than e and their Applications

  • Thus, $\log_{10}(x)$ is related to the number of decimal digits of a positive integer $x$: the number of digits is the smallest integer strictly bigger than $\log_{10}(x)$.
  • The next integer is $4$ , which is the number of digits of $1430$ .