being approximated. Numbers like π have occasional large terms in their continued fraction, which makes them easy to approximate with rational numbers. Other numbers like e have only small terms early in their continued fraction, which makes them more difficult to approximate rationally. The golden ratio Φ has terms equal to 1 everywhere—the smallest values possible—which makes Φ the most difficult number to approximate rationally. In this sense, therefore, it is the "most irrational" of all irrational numbers. Even-numbered convergents are smaller than the original number, while odd-numbered ones are larger. For a continued fraction , the first four convergents (numbered 0 through 3) are . The numerator of the third convergent is formed by multiplying the numerator of the second convergent by the third coefficient, and adding the numerator of the first convergent. The denominators are formed similarly. Therefore, each convergent can be expressed explicitly in terms of the continued fraction as the ratio of certain multivariate polynomials called continuants. If successive convergents are found, with numerators , , ... and denominators , , ... then the relevant recursive relation is that of Gaussian brackets: , . The successive convergents are given by the formula . Thus to incorporate a new term into a rational approximation, only the two previous convergents are necessary. The initial "convergents" (required for the first two terms) are 0⁄1 and 1⁄0. For example, here are the convergents for [0;1,5,2,2]. {| class="wikitable" |- align="right" ! | −2|| −1|| 0 || 1 || 2 || 3 || 4 |- align="right" ! | || || 0 || 1 || 5 || 2 || 2 |- align="right" ! | 0 || 1 || 0 || 1 || 5 || 11 || 27 |- align="right" ! | 1 || 0 || 1 || 1 || 6 || 13 || 32 |} When using the Babylonian method to generate successive approximations to the square root of an integer, if one starts with the lowest integer as first approximant, the rationals generated all appear in the list of convergents for the continued fraction. Specifically, the approximants will appear on the convergents list in positions 0, 1, 3, 7, 15, ... , , ... For example, the continued fraction expansion for is [1;1,2,1,2,1,2,1,2,...]. Comparing the convergents with the approximants derived from the Babylonian method: {| class="wikitable" |- align="right" ! | −2|| −1|| 0 || 1 || 2 || 3 || 4 ||