Bring Learning to Life
Lesson 1: Number System
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.
1.1 The number system
1.2 What are natural numbers?
1.3 Zero and Brahmagupta
1.4 Whole numbers in everyday life
1.5 What are integers?
1.6 Fractions
1.7 Recurring decimals
1.8 Irrational numbers