Bring Learning to Life
Lesson 25: Statistics
UNIT 1 - Number System & Number Sense
This lesson covers:
- Real numbers and their representations on the number line
- Operations on Real Numbers
- Rationalisation of real numbers
- Laws of exponents with integral powers
Number System & Number Sense (Unit1)
- Real numbers and their representations on a number line (LM01 and LM03)
- Rational Numbers:
- Definition: Rational numbers (Q) are numbers that can be written as simple fractions: p/q, where p and q are integers, and q ≠ 0.
- Examples:
- 1/2, 0, -14
- All natural numbers, whole numbers, and integers are rational.
- Terminating and recurring decimals:
- Terminating decimals end after a finite number of steps (e.g., 1/8 = 0.125).
- Recurring decimals repeat a digit or block of digits infinitely (e.g., 1/3 = 0.333...).
Decimal Expansions
To write a decimal expansion for a fraction, use long division. If the division ends, the number is terminating; otherwise, it's recurring.
Worked Examples
- 1/8: This is a terminating decimal.
- 1/3: This is a non-terminating, recurring decimal.
Challenge Question
What is the decimal expansion of 2/7? Is it terminating or recurring?
Representing Numbers on the Number Line
Rational numbers like 1/2 can be located using successive magnification, while irrational numbers like √2 can use geometric construction.
Operations on Real Numbers
- Rules:
- The sum of two rationals is rational.
- The product of a non-zero rational and an irrational is irrational.
Rationalisation
To rationalise a denominator, multiply numerator and denominator by the conjugate if necessary.
- Example:
1/√3 → (1/√3) × (√3/√3) = √3/3
Laws of Exponents
- a^m · a^n = a^(m+n)
- (a^m)^n = a^(m·n)
- a^(-1) = 1/a
DEFINITIONS - Number System & Number Sense
Integers (Z):
Integers (Z) are all the positive and negative whole numbers, as well as the number 0.
Irrational Numbers:
Irrational Numbers are numbers that cannot be written as simple fractions. They include all positive and negative numbers which cannot be written in the form p/q, where p and q are integers, and q ≠ 0.
Natural Numbers (N):
Natural Numbers (N) are whole counting numbers like 1, 2 and 3. The smallest natural number is 1, and the numbers continue increasing infinitely from 1.
Non-terminating decimal:
A non-terminating decimal is a number with a decimal expansion that goes on forever – you never reach the final digit.
nth root:
The nth root of a real number is a number that can be multiplied by itself n times, to get to a given number. The n is a placeholder that represents any positive real number.
Rational Numbers (Q):
Rational Numbers (Q) are numbers that can be written as simple fractions. They include all positive and negative numbers that can be written in the form p/q, where p and q are integers, and q ≠ 0.
Rationalize the denominator:
To rationalize the denominator of a fraction with an irrational denominator means to find an equivalent fraction that has a rational denominator.
Real Numbers (R):
Real Numbers (R) are the collection of all rational and irrational numbers. Each real number can be represented by a unique point on the number line.
Recurring decimal:
A recurring decimal is a non-terminating decimal that has a repeating digit or block of digits after the decimal point. The repeated digit or digits will continue repeating forever.
Successive Magnification:
Successive Magnification is a method we can use to locate rational decimal numbers on the number line.
Terminating decimal:
A terminating decimal is a number with a decimal expansion that ends after a finite number of steps.
Whole Numbers (N0):
Whole Numbers (N0) include all the natural numbers and the number 0.
10.2 Statistics