MathChapter 1: The Language and Tools of Algebra
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Card 1 of 20. Answer hidden.
Prompt
What are natural numbers here?
💡This course does not include zero in this set.
Answer
The positive counting numbers \(1,2,3,\ldots\).
Prompt
What are whole numbers?
💡No negatives or fractional parts.
Answer
Zero and the natural numbers.
Prompt
What are integers?
💡They have no fractional part.
Answer
Whole numbers and their negatives.
Example\(-4,0,9\) are integers.
Prompt
Define a rational number.
💡The denominator cannot be zero.
Answer
A number expressible as \(a/b\), where \(a,b\) are integers and \(b\ne0\).
Prompt
Define an irrational number.
💡Its decimal neither terminates nor repeats.
Answer
A real number that cannot be written as a ratio of integers.
Example\(\sqrt2\) and \(\pi\).
Prompt
What are real numbers?
💡This set excludes \(\sqrt{-1}\).
Answer
All rational and irrational numbers—the values on the ordinary number line.
ExampleEvery integer is real.
Prompt
What is the origin?
💡It separates negative and positive values.
Answer
The point \(0\) on the number line.
ExampleZero is neither positive nor negative.
Prompt
What are opposites?
💡Their sum is zero.
Answer
Numbers with equal distance from zero and opposite signs.
Example\(-3.2\) and \(3.2\).
Prompt
What is the distance between \(a\) and \(b\)?
💡Distance is nonnegative.
Answer
\(|a-b|\).
ExampleBetween \(-2\) and \(5\): \(|-2-5|=7\).
Prompt
What decimal forms can a rational number have?
💡A repeating block may begin after initial digits.
Answer
It terminates or eventually repeats.
Example\(1/8=0.125\); \(1/6=0.1666\ldots\).
Prompt
What decimal form identifies an irrational number?
💡A pattern is not necessarily a fixed repeating block.
Answer
It neither terminates nor eventually repeats.
Example\(0.1010010001\ldots\).
Prompt
When does a reduced fraction terminate?
💡Factor the denominator after reducing.
Answer
When its denominator's only prime factors are \(2\) and \(5\).
Example\(7/40\) terminates because \(40=2^3\cdot5\).
Prompt
When is \(\sqrt n\) rational for a nonnegative integer \(n\)?
💡Simplify before classifying.
Answer
When \(n\) is a perfect square.
Example\(\sqrt{49}=7\), but \(\sqrt{11}\) is irrational.
Prompt
How can you locate \(\sqrt n\) without a calculator?
💡Use nearby squares.
Answer
Bracket \(n\) between consecutive perfect squares, then take roots.
Example\(25<30<36\) gives \(5<\sqrt{30}<6\).
Prompt
How are negative values ordered?
💡Reflect the positive ordering across zero.
Answer
The value with greater magnitude is farther left and therefore smaller.
Example\(-\pi<-3<-\sqrt8\).
Prompt
How do you round to a requested place?
💡Keep requested trailing zeros.
Answer
Inspect the digit one place to its right; \(5\) or more rounds up.
Example\(3.996\) to hundredths is \(4.00\).
Prompt
Why can an irrational expression have a rational result?
💡Simplify first.
Answer
Irrational parts may cancel or multiply to a rational value.
Example\(\sqrt3\cdot\sqrt3=3\).
Prompt
Is \(0.999\ldots\) less than \(1\)?
💡Use the repeating-decimal algebra proof.
Answer
No. It equals \(1\).
ExampleIf \(x=0.999\ldots\), then \(9x=9\).
Prompt
What is the smallest set containing \(-\sqrt{49}\)?
💡Simplify before classifying.
Answer
The integers, because \(-\sqrt{49}=-7\).
ExampleIt is also rational and real, but integer is more restrictive.
Prompt
SAT check: how much should an irrational be approximated?
💡Keep exact form until approximation is useful.
Answer
Only enough digits to decide the comparison or satisfy the requested precision.
ExampleUse \(\sqrt2\approx1.414\) to compare with \(1.4\) and \(1.5\).
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