Lesson · Grade 11 · Pre Calculus
The Real Number System: From Natural Numbers to Irrationals
- Grade 11
- Pre Calculus
- 1 min 56 s
- British Columbia
- BC11-MA-63
Seven numbers (√16, 0, −3, 3/4, 0.333..., √2 and π) are placed one by one into nested boxes: natural inside whole inside integers inside rational, with irrational beside rational and both inside the real numbers. Each placement is checked in a working panel (√16 = 4, −3 = −3/1, 3/4 = 0.75, 0.333... = 1/3), then two common mistakes are corrected: not every square root is irrational, and 22/7 differs from π in the third decimal place, so it is a rational number that is only close to π.
Anyone can watch two lessons on the website without an account. Signing in is free, no payment and no card, and opens three a day; Plus has no limit.
Loading
In one sentence
The question this lesson answers
Where do these numbers belong: the square root of 16, 0, negative 3, three quarters, 0.333 repeating, the square root of 2, and pi?
Every number on the number line is a real number.
Transcript
What the lesson says
Where do these numbers belong: the square root of 16, 0, negative 3, three quarters, 0.333 repeating, the square root of 2, and pi?
Every number on the number line is a real number.
Natural numbers are the counting numbers: 1, 2, 3 and so on.
The square root of 16 is 4, since 4 times 4 is 16, so it is natural.
Whole numbers are the natural numbers and zero.
Integers are the whole numbers and their negatives, like negative 3.
A rational number can be written as a fraction of two integers, with a denominator that is not zero.
Every integer is rational too: negative 3 is negative 3 over 1.
Three quarters is rational, and its decimal, 0.75, ends.
0.333 repeating is one third, so a repeating decimal is rational too.
An irrational number cannot be written as a fraction of integers: its decimals never end and never repeat.
Both the square root of 2 and pi are irrational.
A common mistake is calling every square root irrational, but the square root of 16 is exactly 4.
Another mistake is thinking 22 over 7 equals pi. Compare their decimals.
They differ in the third decimal place, so 22 over 7 is rational and only close to pi.
Every real number is either rational, a fraction of integers, or irrational, with decimals that never end or repeat.
Curriculum
Where this fits
- British Columbia, Grade 11
Pre Calculus 11, Real numbers and powers: real number systemBC11-MA-63
The outcomes and strands follow the British Columbia curriculum.
Keep going
1 min 59 sRational exponents: a fraction power is a root
Grade 11 · 1 min 59 s
2 min 23 sSimplifying, Adding and Multiplying Radicals
Grade 11 · 2 min 23 s
2 min 7 sSolving a Radical Equation and Checking for Extraneous Roots
Grade 11 · 2 min 7 s
2 min 11 sAdding and Subtracting Rational Expressions
Grade 11 · 2 min 11 s
2 min 3 sAngles in Standard Position and Reference Angles
Grade 11 · 2 min 3 s
1 min 58 sAngles made by parallel lines and a transversal
Grade 11 · 1 min 58 s
2 min 12 sAngles of Elevation and Depression: Measuring What You Cannot Reach
Grade 10 · 2 min 12 s
1 min 58 sChoosing the best average: mean, median or mode
Grade 8 · 1 min 58 s
- All Grade 11 Pre Calculus lessons
- Create a free account to watch three lessons a day, no payment and no card.