The real number system describes every point on the ordinary number line. SAT questions often hide a familiar value inside a fraction, radical, or decimal, so simplify first and classify second.
The real number line
Nested number sets
| Set | Meaning | Examples | Important boundary |
|---|---|---|---|
| Natural | Positive counting numbers | \(1,2,3,\ldots\) | This course uses the convention that \(0\) is not natural. |
| Whole | Natural numbers and zero | \(0,1,2,3,\ldots\) | No negatives or fractions |
| Integers | Whole numbers and their negatives | \(\ldots,-2,-1,0,1,2,\ldots\) | No fractional part |
| Rational | \(a/b\), with integers \(a,b\) and \(b\ne0\) | \(-3,2/7,0.125,0.\overline{6}\) | Decimal terminates or eventually repeats |
| Irrational | Real but not rational | \(\sqrt2,\pi\) | Decimal neither terminates nor repeats |
| Real | All rational and irrational values | Every point on this number line | Does not include \(\sqrt{-1}\) |
Terminating, repeating, and nonrepeating decimals
Every terminating decimal is rational because place value writes it over a power of \(10\): \(0.375=375/1000=3/8\). Every repeating decimal is also rational. A decimal such as \(0.101001000100001\ldots\) has a pattern but no fixed repeating block, so it is irrational.
Locate an irrational number
Between which consecutive integers does \(-\sqrt{20}\) lie?
- Interpret
Bracket the positive root using perfect squares: \(4^2=16<20<25=5^2\).
- Set up
Therefore \(4<\sqrt{20}<5\).
- Work
Negating reverses the order: \(-5<-\sqrt{20}<-4\).
- Check
The number line confirms the negative root lies between \(-5\) and \(-4\).
Rounding with place-value precision
- Locate
Identify the requested place exactly.
- Inspect
Look one digit to its right.
- Round
If that digit is \(5\) or greater, increase the target digit by \(1\); otherwise leave it.
- Report precision
Retain trailing zeros when they communicate the requested decimal places, such as \(4.00\) to the nearest hundredth.
Check your understanding
Which is irrational?
- \(-\sqrt{36}\)
- \(0.272727\ldots\)
- \(\sqrt{12}/2\)
- \(7/20\)
Show answer and explanation
Answer: \(\sqrt{12}/2=\sqrt3\)
The other values simplify to \(-6\), a repeating rational decimal, and a terminating rational fraction. \(\sqrt3\) is irrational.
Key takeaways
What to remember
- Natural, whole, integer, and rational sets are nested inside the real numbers.
- Rational decimals terminate or eventually repeat; irrational decimals do neither.
- Simplify fractions and radicals before classifying.
- Use exact positions or sufficient approximations to compare, and retain requested rounding precision.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.