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実数のまとめ | 夢を叶える塾
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実数のまとめ

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数学に関する投稿の第3回目です。
こちらは単元のポイントなどをまとめた記事になります。

実数の特徴

すごい単元名ですよね。
深く語ろうとすると、すごく長くなってしまうと思います。
ここでは、
・実数の大きさ
・平方根の扱い
をマスターしておくことが重要です。

何がうれしいのか

数の扱いが正しく行えるようになること、でしょうか。
式の変形で、
「√(a2+4a+4)=2」
等は、両辺を二乗して、
「a2+4a+4=4」
「a(a+4)=0」
「a=0,-4」
として求めることができますが、
「√(a2+4a+4)=-2」
を両辺を二乗してaの値を求めることはNGです。
「√(a2+4a+4)≧0」ですから。

そういった数の扱いに対しての意識が高まると思います。

実数

以下のように分けられます。

分類
実数有理数整数・・・,-3,-2,-1,0,1,2,3,・・・
有限小数1/2=0.5など
無限小数1/3=0.333…など
無理数循環しない無限小数√2,e(自然対数の底),π(円周率)など

注意点として、0は自然数に含まれない(0は数えることができない)ことを理解しておくと良いです。

絶対値

絶対値は次単元の一次不等式など、不等式と問題に出しやすいという意味で相性が良いです。
絶対値を外すときに符号が変わることを意識し、場合分けを行いましょう。

平方根

有理化、二重根号の外し方をマスターしておきましょう。

有理化では、分母の「√(a)±√(b)」等に対し、分母分子に「√(a)∓√(b)」等を掛けます。これは、
「(a+b)(a-b)=a2-b2
の「和(a+b)」と「差(a-b)」の「積(a+b)(a-b)」が「平方」になる乗法公式をうまく利用しています。
二数の和でない場合は、「A±√(a)」等と見て、分母分子に「A∓√(a)」等を掛けます。
この計算で少なくとも一つの平方根は外れますので、一つずつ外していきましょう。
また、分母分子に同じ数を掛けることは1を掛けることで等式となることもおさらいしておきましょう。

二重根号の外し方

二重根号の外し方は、外の根号の中で平方を作ります。
「a2±2ab+b2=(a±b)2
の乗法公式を利用します。
「√(m±2√(n))=√(a+b±2√(ab))=√((√(a)±√(b))2)」
を目指します。※見にくくてすみません。
「√(m±2√(n))」に対し、nを素因数分解し、「n=ab」となるa,bの組み合わせを考えるところから取り掛かるとよいでしょう。

まとめ

複雑な問題を解くうえで、正しく数字を扱えることが必要になります。
整式の演算や因数分解同様、数学のベースになるでしょう。
演習問題では正答率9割以上を目指したいところです。
単純な計算ミスを防止できるよう、数字の大小関係の感覚を身に着けておくとよいでしょう。

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