MATHHX B
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1.1 Hovedregning
Det er vigtigt, at man kan regne i hovedet, da det er en forudsætning for at lære mere avanceret matematik. Det betyder ikke, at man skal kunne regne f.eks. svære gangestykker i hovedet, men det er vigtigt, at man mestrer de
forskellige regneoperationer og regneregler.
Simpel regning
Man skal gange og dividere, inden man lægger til og trækker fra. Vi skriver altid division med en brøkstreg, så vil vi f.eks. skrive \(20\) divideret med \(5\), skriver vi \(\frac {20}{5}\).
Øvelse 1.1.1
Regn:
-
a) \(4+5\cdot 2\)
-
b) \(7-2+3\cdot 2-5\)
-
c) \(4-\frac {6}{3}+2\)
1.1.1
-
a) \(14\)
-
b) \(6\)
-
c) \(4\)
I udtryk med brøker regnes først tæller og nævner, derefter udføres divisionen, og til sidst regnes resten af udtrykket.
1.1.2
-
a) \(14\)
-
b) \(9\)
-
c) \(1\)
Visse simple divisionsstykker kan forvirre.
-
Eksempel 1.1.3
Dividerer man noget med sig selv, giver det altid \(1\). Har man f.eks. \(3\) personer, der skal dele \(3\) bananer, får de \(1\) hver:
\[\frac {3}{3} = 1\]
Ved division med kommatal kan du tænke på det som et spørgsmål om, hvor mange gange nævneren går op i tælleren. Skal man f.eks. dividere \(1\) med \(0{,}5\), giver det \(2\), da \(0{,}5\) går \(2\) gange op i \(1\):
\[\frac {1}{0{,}5}=2\]
Dividerer man \(0\) med noget, får man \(0\). Hvis man f.eks. skal dele \(0\) bananer ud til \(5\) personer, får de ingen bananer:
\[\frac {0}{5} = 0\]
Man kan ikke dividere med \(0\). Skal man f.eks. dele \(5\) bananer ud til \(0\) personer, så kan man ikke komme af med bananerne:
\[\frac {5}{0}\ \text { kan ikke regnes}\]
Læg mærke til, at det er muligt at dividere nul med noget, men man kan ikke dividere noget med nul.
Øvelse 1.1.3
Regn, hvis det er muligt:
Regning med negative tal
Minus gange minus giver plus. Plus gange minus giver minus. Plus gange plus giver selvfølgelig plus. Tilsvarende gælder for division.
Læg mærke til, at \(-6\) står i parentes i de to nederste regnestykker. Det er, fordi det er forbudt at skrive ”\(\cdot -\)”, altså det er forbudt at skrive et gangetegn og et minustegn lige efter hinanden.
Hvad så med \(-4-3\)? Giver det mon så også plus? Der er jo to minustegn! Svaret er: NEJ! Vi regner:
\[-4-3=-7\]
Så det er kun ved gange eller division, at to minustegn giver plus.
1.1.4
-
a) \(-6\)
-
b) \(-14\)
-
c) \(15\)
-
d) \(-2\)
-
e) \(16\)
Øvelse 1.1.5
Regn, hvis det er muligt:
Potenser
En potens er et tal som f.eks. \(3^2\) (læses ”\(3\) i anden”) eller \(5^3\) (læses ”\(5\) i tredje”). Tallet \(3^2\) regnes ved:
\[3^2=3\cdot 3=9\]
og \(5^3\) regnes ved:
\[5^3=5\cdot 5\cdot 5=125\quad \]
Man regner potenser, før man ganger, dividerer, lægger sammen og trækker fra.
Øvelse 1.1.6
Regn:
-
a) \(4^2\)
-
b) \(3^3\)
-
c) \(7^1\)
-
d) \(5-2\cdot 5^2\)
1.1.6
-
a) \(16\)
-
b) \(27\)
-
c) \(7\)
-
d) \(-45\)
Rødder
En rod er ”det omvendte af en potens”. Du kender sikkert kvadratroden fra folkeskolen. Kvadratroden af et tal er et tal, som sat i anden giver det oprindelige tal. F.eks. er
\(\sqrt {16}=4\) fordi \(4^2=16\). Godt nok er \((-4)^2\) også \(16\), men kvadratroden er altid den positive mulighed.
Man kan ikke tage kvadratroden af negative tal, da noget i anden aldrig kan give et negativt tal.
Øvelse 1.1.7
Regn, hvis det er muligt:
-
a) \(\sqrt {9}\)
-
b) \(\sqrt {4}\)
-
c) \(\sqrt {1}\)
-
d) \(\sqrt {-4}\)
-
e) \(\sqrt {0}\)
1.1.7
-
a) \(3\)
-
b) \(2\)
-
c) \(1\)
-
d) Kan ikke regnes.
-
e) \(0\)
Der findes andre rødder end kvadratrødder. Det er nemmest at forklare med et eksempel.
-
Eksempel 1.1.6
Vi vil bestemme \(\sqrt [3]{8}\). Det betyder, at vi skal finde et tal, som giver \(8\), når man sætter det i tredje. Da \(2^3=8\), er \(\sqrt [3]{8}=2\).
Tallet \(\sqrt [3]{8}\) læses ”den tredje rod af otte” eller ”kubikroden af otte”.
Øvelse 1.1.8
Bestem følgende rødder:
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a) \(\sqrt [2]{25}\)
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b) \(\sqrt [3]{0}\)
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c) \(\sqrt [3]{27}\)
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d) \(\sqrt [4]{16}\)
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e) \(\sqrt [100]{1}\)
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f) \(\sqrt [1]{13}\)
1.1.8
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a) \(5\)
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b) \(0\)
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c) \(3\)
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d) \(2\)
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e) \(1\)
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f) \(13\)
Parenteser
En parentes betyder, at man skal regne det, der står inde i parentesen, først. Står der et tal foran eller bagved en parentes, betyder det, at tallet skal ganges med parentesen. Står der f.eks. \(2(5+3)\),
betyder det altså \(2\cdot (5+3)\).
Øvelse 1.1.10
Regn:
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a) \(2(3-1)\)
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b) \((5+2)3\)
Reglerne for rækkefølgen af de forskellige regneoperationer kaldes regningsarternes hierarki.
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Eksempel 1.1.8
Udtrykket \(2+\frac {2(3+1)^2}{6+2}-7 \) regnes på følgende måde:
\(\seteqnumber{0}{1.}{0}\)
\begin{align*}
2+\frac {2(3+1)^2}{6+2}-7 & = 2+\frac {2\cdot 4^2}{6+2}-7 && \text {(parentes regnet)} \\[10pt] & = 2+\frac {2\cdot 16}{6+2}-7 && \text {(potens regnet)} \\[10pt] &
=2+\frac {32}{8}-7&& \text {(tæller og nævner regnet)} \\[10pt] & =2+4-7 && \text {(division regnet)} \\[10pt] & =-1 && \text {(plus og minus regnet)}
\end{align*}
1.1.11
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a) \(14\)
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b) \(-\frac {1}{2}\).
Læg godt mærke til næste eksempel.
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Eksempel 1.1.9
Udtrykket \(-3^2\) betyder \(3\cdot 3\) med et minus foran:
\[ -3^2=-\mspace {-12mu} \underbrace {3 \cdot 3}_{\text {regn først}}=-9 \]
Udtrykket \((-3)^2\) betyder\(-3\) gange med \(-3\):
\[ (-3)^2=(-3)\cdot (-3) =9\]
Det er crazy så mange elever, der bliver ved med at forveksle \(-3^2\) og \((-3)^2\). Det er som om, en særlig forbandelse har ramt den danske ungdom. Læs eksemplet igen, og husk det.
Øvelse 1.1.12
Regn:
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a) \(-2^2\)
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b) \((-2)^2\)