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A unit fraction is a common fraction with a numerator of 1 (e.g., 1 / 7 ). Unit fractions can also be expressed using negative exponents, as in 2 −1, which represents 1/2, and 2 −2, which represents 1/(2 2) or 1/4. A dyadic fraction is a common fraction in which the denominator is a power of two, e.g. 1 / 8 = 1 / 2 3 .
For example, the sequence (1; 1.4; 1.41; 1.414; 1.4142; 1.41421; ...), where each term adds a digit of the decimal expansion of the positive square root of 2, is Cauchy but it does not converge to a rational number (in the real numbers, in contrast, it converges to the positive square root of 2).
Let a and b be positive integers such that 1< a / b < 3/2 (as 1<2< 9/4 satisfies these bounds). Now 2b 2 and a 2 cannot be equal, since the first has an odd number of factors 2 whereas the second has an even number of factors 2.
For a real number x and an integer n ≥ 0, let [x] n denote the (finite) decimal expansion of the greatest number that is not greater than x that has exactly n digits after the decimal mark. Let d i denote the last digit of [x] i. It is straightforward to see that [x] n may be obtained by appending d n to the right of [x] n−1. This way one has
The number π (/ p aɪ / ⓘ; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter.It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.
Five and ten: Sums of the form 5 + x and 10 + x are usually memorized early and can be used for deriving other facts. For example, 6 + 7 = 13 can be derived from 5 + 7 = 12 by adding one more. [36] Making ten: An advanced strategy uses 10 as an intermediate for sums involving 8 or 9; for example, 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14. [36]
For 2011 and 2012, the employee's contribution was temporarily reduced to 4.2%, while the employer's portion remained at 6.2%. [ 124 ] [ 125 ] In 2024, the Social Security Wage Base (the upper limit on earnings taxed) increased to $168,600.