IB Maths Studies/Number and Algebra

The Number Sets of N, Z, Q and R

N - Natural Numbers. These are in the set (0, 1, 2, 3...)

We say "March has 31 days" or "There are 15 students in my math class" We count with the numbers 1, 2, 3, 4...

We also use these numbers for ordering. We say "This is the first year of my math studies course" or "Alison came second in the race"

There is actually rather a lot of debate within the math community about the inclusion of zero as a natural number. However, the IB Studies curriculum considers 0 to be a natural number.

Z - Integers. The set includes (...-3, -2, -1, 0, 1, 2, 3...)

Q - Rational Numbers. These are the fractions created by dividing any integer by any non-zero integer. This set can include decimals numbers as long as they can be transformed into a fraction (in other words, either a terminating decimal number or a repeating decimal number).

R - The Real Numbers. A real number is any number that can be used to measure a distance (positive or negative). These include all of the above. In addition, it includes the irrational numbers such as pi (3.14...) or e (2.71...). Irrationals are real decimal numbers that follow no pattern and are without end. Any decimal that ends or follows a pattern is a rational number.

Real Numbers can be understood best by comparing them with their "partners" - the imaginary numbers. Imaginary numbers are found when someone tries to take the square root of a negative number. Mathematicians use "i" to represent an imaginary number. Real numbers and Imaginary numbers work together as pairs to make up a large number system called "The Complex Numbers". Real numbers are called so because they measure real distances. Complex numbers measure more complex items and are used in a variety of scientific areas. Luckily, we do not use imaginary numbers in Math Studies.

Significant Figures

Unless otherwise written, answers in IB Math Studies are always to be given to three significant figures. So, what, you are dying to know, is a significant figure? Significant figures are like gremlins, they have to follow certain rules or things get out of control. So here are the rules for keeping Gizmo safe:

1. Always count nonzero digits

Example: 348 has three significant figures. 27 has two.

2. Never count leading zeros

Example: 021 and 0.021 both have two significant figures

3. Always count zeros which fall somewhere between two nonzero digits

Example: 20.8 has three significant figures, while 0.00104009 has six

4. Zeros after the last non-zero digit in whole numbers may not be significant (see NOTE)

Example: 498,000 has three or more significant figures depending on the context.

5. Zeros after the last non-zero digit in decimal numbers are significant, and are required to explicitly show the degree of accuracy

Example: 3.98 is accurate to three significant digits whereas 3.9800 is accurate to five significant digits.

NOTE: The ambiguity of accuracy in a number such as 5100 is one reason scientific notation is often preferred for general usage.

Example: 5.1 x 10^3 signifies two digit accuracy whereas 5.10 x 10^3 signifies three digit accuracy.