Number Bases
Representing numbers with place values that are powers of a base b.
Key idea
Representing numbers with place values that are powers of a base b.
In depth
A base b system has b digits and place values powers of b. To convert from base b to base 10, multiply each digit by its place value. To convert from base 10, divide repeatedly by b and read remainders backwards.
Worked example
45 in base 10 to base 6: 45 = 7×6 + 3, 7 = 1×6 + 1, 1 = 0×6 + 1, so 113₆.
Formula
Real-world uses
- Converting between decimal and computer bases
- Reading clock arithmetic and data encoding
Common mistake
Using digits equal to or larger than the base.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Number Bases?
Representing numbers with place values that are powers of a base b. It is a topic at the advanced level in Arithmetic & Number Sense.
What is the formula for Number Bases?
The key formula is Σ dᵢ bⁱ.
Where is Number Bases used in real life?
Real-world examples: Converting between decimal and computer bases; Reading clock arithmetic and data encoding.
What should I learn before Number Bases?
Make sure you are comfortable with Binary Numbers, the previous topic in Arithmetic & Number Sense.
What is a common mistake with Number Bases?
Using digits equal to or larger than the base.
How can I practise Number Bases?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Number Bases?
Move on to Hexadecimal and Octal next.
Related advanced topics in Arithmetic & Number Sense
- Repeating Decimals – Decimals with a repeating block are rational and convertible to fractions.
- Compound Interest – Interest earned on principal and past interest.
- Inverse Proportion – When one quantity doubles, the other halves.
- Rational Numbers – Numbers expressible as p/q with integers p, q, q≠0.
- Irrational Numbers – Real numbers that cannot be written as a fraction, such as √2 and π.
- Real Numbers – All rational and irrational numbers; the continuous number line.