• The Mathematical Education
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• v.62 no.4
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• pp.551-567
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• 2023

Even and odd numbers are taught in elementary school mathematics, but the introductory activities, definitions, and properties of sum on even and odd numbers vary depending on which grade they are presented. The purpose of this study was to compare and analyze the activities related to even and odd numbers presented in Korean mathematics textbooks developed under the different curriculum revisions, and to further analyze the related activities in foreign textbooks to draw implications for the teaching of even and odd numbers. In Korean textbooks, from the time of the fourth mathematics curriculum until the 2007 revision, even and odd numbers were covered in the multiples and divisors unit of the fifth grade textbook, while since the 2009 revision, the first grade textbook has covered the topic along with teaching numbers up to 50 or 100. In addition, the definitions of even and odd numbers varied depending on the grade level and the nature of the unit being taught, and activities addressing the properties of sum were only presented in the mathematics textbook under the third curriculum along with a few mathematics workbooks. In foreign textbooks, even and odd numbers were introduced in Grades 1, 2, or 5, and their related activities varied accordingly. Based on these findings, this study discusses the implications for the teaching of even and odd numbers.

• Proceedings of the KIEE Conference
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• 1996.07c
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• pp.1453-1455
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• 1996

The odd-even effect of the alkyl chain length of rubbed polyimide Langmuir-Blodgett (LB) surfaces on the extrapolation length of 5CB has been successfully evaluated for the first time by measuring polar anchoring strength. The extrapolation length of 5CB for rubbed PI-LB surfaces with even-numbers is small compared with odd-numbers for alkyl chain lengths of greater than 7 carbons. The extrapolation length of 5CB on rubbed PI-LB surfaces with odd-numbers increases gradually as the temperature increases but tends to diverge near the clearing temperature (Tc=$35.3^{\circ}C$). The extrapolation length diverges because of rapidly decreasing surface order near $T_c$. We suggest that the polar anchoring strength on rubbed PI-LB surfaces with even-number is strong because of relatively high surface ordering caused by more crystalline surfaces. Finally, we conclude that the odd-even effects of the polar anchoring strength in NLCs are strongly related to the character of the polymer and observed clearly for long alkyl chain lengths.

• Journal of Elementary Mathematics Education in Korea
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• v.5 no.1
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• pp.1-18
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• 2001

This note includes that repeating patterns, knowledge of odd and even numbers, and the patterns in processing and learning addition facts. The potential to mathematical development of repeating patterns is Idly realized if the unit of repeat is recognized. Through the partition of numbers greater then 9 into two equal sets and into sets of 2s, It is necessary the teaching of children's knowledge of odd and even numbers. Being taught derivation strategies through patterns in numbers, we suggest that the teaching seguence to accelerate development of children's learning of additions facts.

• Journal of Astronomy and Space Sciences
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• v.38 no.1
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• pp.23-29
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• 2021

Utilizing a new version of the sunspot number and group sunspot number dataset available since 2015, we have statistically studied the relationship between solar activity parameters describing solar cycles and the slope of the linear relationship between the monthly sunspot numbers and the monthly number of active days in percentage (AD). As an effort of evaluating possibilities in use of the number of active days to predict solar activity, it is worthwhile to revisit and extend the analysis performed earlier. In calculating the Pearson's linear correlation coefficient r, the Spearman's rank-order correlation coefficient rs, and the Kendall's τ coefficient with the rejection probability, we have calculated the slope for a given solar cycle in three different ways, namely, by counting the spotless day that occurred during the ascending phase and the descending phase of the solar cycle separately, and during the period corresponding to solar minimum ± 2 years as well. We have found that the maximum solar sunspot number of a given solar cycle and the duration of the ascending phase are hardly correlated with the slope of a linear function of the monthly sunspot numbers and AD. On the other hand, the duration of a solar cycle is found to be marginally correlated with the slope with the rejection probabilities less than a couple of percent. We have also attempted to compare the relation of the monthly sunspot numbers with AD for the even and odd solar cycles. It is inconclusive, however, that the slopes of the linear relationship between the monthly group numbers and AD are subject to the even and odd solar cycles.

• Macromolecular Research
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• v.10 no.4
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• pp.230-235
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• 2002

Alternating polyols of oxymethylene and oxyalkylene were synthesized and used as precursors for thermoplastic poly(ether-ester) elastomers (TPEs). The polyols were synthesized by reacting diols having different methylene units with dichloromethane in the presence of a phase transfer catalyst. The number of methylene units in the alkylene oxides was varied from 3 to 6. TPEs were prepared using the polyols as soft segments and poly(butylene terephthalate) units as hard segments. The polyols and TPEs synthesized were characterized using FTIR, NMR, GPC, DSC, and polarized optical microscopy. The polyols showed a profound odd-even effect on the melting (T$_{m}$) and glass transition temperatures (T$_{g}$). Polyols with odd numbers of methylene groups in the alkylene units have higher transition temperatures than polyols with odd number of methylene groups. The tendency is still kept in TPEs, even though the T$_{g}$ of soft segment in TPEs are slightly higher than those of corresponding neat polyols. The T$_{m}$ and T$_{g}$ of soft segments are almost constant in the range of 20 to 60 wt % contents of soft segments. On the other hand, the normalized heat of fusion of hard segment decreased with increasing the content of loft segment.ent.t.ent.

• 전기의세계
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• v.24 no.6
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• pp.92-96
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• 1975

A generalized theroretical method is developed to eliminate a given number of harmonics in AC chopper output waves. The results show that halfwave symmetric and sinusoidal symmetric chopping are required to eliminate all even numbers of harmonics and , at least, M+1 times per half cycle chopping is required to eliminate any M odd number of harmonics in the given effective value of the output wave.

• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.583-590
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• 2008

In this article, we focuss on. sequences of polynomials with {$\pm1$} coefficients constructed by recursive argument that is known as Rudin-Shapiro polynomials. The asymptotic behavior of these polynomials defines as the ratio of their 2q-norm with 2-norm to be dominated by some number depending on q or "the best" by an absolute constant. In this work we first show the conjecture holds for some finite numbers of m and then introduce a technique that give the result for any positive odd integer m whenever it holds for all pervious even numbers.

• Journal of the Korean School Mathematics Society
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• v.18 no.1
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• pp.1-14
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• 2015

In this paper, in order to improve the teaching contents on even and odd number, composition and decomposition of numbers, and (1 digit)+(1 digit) with carrying, (10 and 1 digit)-(1 digit) with borrowing, the corresponding teaching contents in ${\ll}$Math 1-1${\gg}$, ${\ll}$Math 1-2${\gg}$ are critically reviewed. Implications obtained through this review can be summarized as follows. First, the current incomplete definition of even and odd numbers would need to be reconsidered, and the appropriateness of dealing with even and odd numbers in first grade would need to be reconsidered. Second, it is necessary to deal with composition and decomposition of numbers less than 20. That is, it need to be considered to compose (10 and 1 digit) with 10 and (1 digit) and to decompose (10 and 1 digit) into 10 and (1 digit) on the basis of the 10. And the sequence dealing with composition and decomposition of 10 before dealing with composition and decomposition of (10 and 1 digit) need to be considered. And it need to be considered that composing (10 and 1 digit) with (1 digit) and (1 digit) and decomposing (10 and 1 digit) into (1 digit) and (1 digit) are substantially useless. Third, it is necessary to eliminate the logical leap in the calculation process. That is, it need to be considered to use the composing (10 and 1 digit) with 10 and (1 digit) and decomposing (10 and 1 digit) into 10 and (1 digit) on the basis of the 10 to eliminate the leap which can be seen in the explanation of calculating (1 digit)+(1 digit) with carrying, (10 and 1 digit)-(1 digit) with borrowing. And it need to be considered to deal with the vertical format for calculation of (1 digit)+(1 digit) with carrying and (10 and 1 digit)-(1 digit) with borrowing in ${\ll}$Math 1-2${\gg}$, or it need to be considered not to deal with the vertical format for calculation of (1 digit)+(1 digit) with carrying and (10 and 1 digit)-(1 digit) with borrowing in ${\ll}$Math 1-2 workbook${\gg}$ for the consistency.

• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.835-840
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• 2008

The main result of this paper shows that every reciprocal Littlewood polynomial, one with {-1, 1} coefficients, of odd degree at least 7 has at least five unimodular roots, and every reciprocal Little-wood polynomial of even degree at least 14 has at least four unimodular roots, thus improving the result of Mukunda. We also give a sketch of alternative proof of the well-known theorem characterizing Pisot numbers whose minimal polynomials are in $$A_N=\{[{X^d+ \sum\limits^{d-1}_{k=0} a_k\;X^k{\in} \mathbb{Z}[X]\;:\;a_k={\pm}N,\;0{\leqslant}k{\leqslant}d-1}\}$$ for positive integer $N{\geqslant}2$.

• Journal of the Korean Society of Costume
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• v.50 no.7
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• pp.113-127
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• 2000

The purpose of this study was to examine the symbolic meaning of oriental numbers based on Yin-Yang(陰陽) theory. Based on the thought of Yin-Yang Wu-Hsing(陰陽五行), the number was divided the number of the heaven (positive number) 1, 3. 5, 7, 9 from the number cf the earth(negative number) 2. 4, 6, 8, 10. It was descrived very well in the dress and its ornaments and the folk customs. In the costume of the Court, there were 9, 7, 5, 3 patterns costume for the king and queen. Even though an even number, 12 patterns costume for the emperor symbolized 12 months and made it of the principal of the universe. Korean traditional costume Han-bock(韓服) was formed with the three dimentional principal of circle (圓.$\bigcirc$), square(方.$\square$) and triangularity(角.$\Delta$). In the middle of odd numbers, number 3 was regarded as a holy number of the heaven (天), the earth(地) and a man(人). Taken for a highest number. number 3 had the symbolism of wishes for good fortune. Number 10 and number 100, which meant the fullness and the long life, were used regularly. With Ten longevity patterns(十長生紋), the feast of a hundred-day-old baby, our race prayed for the healthy long life. As mentioned above. the symbolism of the number though the costume prefered the positive number to the negative one. Accommodating to the universal principal and the cycle. The deep meaning and the symbolism of the number has been implied the mental wishes.