In a paper by W. Parry, the β-expansions of real numbers are discussed. A. Rényi proved that for β > 1, every non-negative number x has a β-expansion. This expansion is defined by a sequence of digits ε₀(x), ε₁(x), ε₂(x), etc., where each εₙ(x) is the integer part of βⁿ(x). Rényi also showed that the transformation T(x) = βx is ergodic and that for almost all x, the average of g(Tᵏ(x)) over k converges to the integral of g(x) against a specific measure h(x). This measure is unique and equivalent to Lebesgue measure. For the special case where β satisfies β² = β + 1, the invariant measure can be represented with a specific function h(x) that has two different values on different intervals. β-numbers are those with recurrent tails in their β-expansions, while simple β-numbers have zero tails. In Section 1, the paper finds an explicit determination of the unique invariant measure for each β > 1, equivalent to Lebesgue measure. The derivative of this measure with respect to Lebesgue measure is a pure jump function, which becomes a step function with a finite number of steps for β-numbers. In Section 2, the paper determines the restrictions on the sequences of digits in β-expansions and shows that the conjugates of each β-number with respect to its characteristic equation must have absolute values less than 2. Simple β-numbers are everywhere dense in (1, ∞), as shown in Section 3. Although there is an explicit formula for the invariant measure corresponding to each β > 1, it does not generally yield normalized measures. In Section 4, the paper focuses on the properties of the normalizing factor as a function of β. It shows that this function is continuous from the right, and the set of points of discontinuity from the left coincides with the set of simple β-numbers. The normalizing function tends to infinity as β approaches 1 and tends to 1 as β approaches infinity. The paper thanks Yaël Naim Dowker and H. Halberstam for their helpful suggestions.In a paper by W. Parry, the β-expansions of real numbers are discussed. A. Rényi proved that for β > 1, every non-negative number x has a β-expansion. This expansion is defined by a sequence of digits ε₀(x), ε₁(x), ε₂(x), etc., where each εₙ(x) is the integer part of βⁿ(x). Rényi also showed that the transformation T(x) = βx is ergodic and that for almost all x, the average of g(Tᵏ(x)) over k converges to the integral of g(x) against a specific measure h(x). This measure is unique and equivalent to Lebesgue measure. For the special case where β satisfies β² = β + 1, the invariant measure can be represented with a specific function h(x) that has two different values on different intervals. β-numbers are those with recurrent tails in their β-expansions, while simple β-numbers have zero tails. In Section 1, the paper finds an explicit determination of the unique invariant measure for each β > 1, equivalent to Lebesgue measure. The derivative of this measure with respect to Lebesgue measure is a pure jump function, which becomes a step function with a finite number of steps for β-numbers. In Section 2, the paper determines the restrictions on the sequences of digits in β-expansions and shows that the conjugates of each β-number with respect to its characteristic equation must have absolute values less than 2. Simple β-numbers are everywhere dense in (1, ∞), as shown in Section 3. Although there is an explicit formula for the invariant measure corresponding to each β > 1, it does not generally yield normalized measures. In Section 4, the paper focuses on the properties of the normalizing factor as a function of β. It shows that this function is continuous from the right, and the set of points of discontinuity from the left coincides with the set of simple β-numbers. The normalizing function tends to infinity as β approaches 1 and tends to 1 as β approaches infinity. The paper thanks Yaël Naim Dowker and H. Halberstam for their helpful suggestions.