Indices and Bases
The decimal system doesn’t just refer to the numbers after a period – it refers to the base-10 system that uses 10 digits (0-9) to represent numbers.
If you input 1 + 1 into a calculator, you’d expect the result it gives you to be 2. Which it is… most of the time. See, there’s a little set of buttons on some scientific calculators that allows you to toggle the numeral system it uses – and if you do that, you just might find that basic arithmetic as we know it seems to go out the window
The above calculation was made using a binary or base-2 system – hence the letters BIN on the screen. The term “base” in numeral systems refers to how many digits are available per place value – an alternative term for it would be the word radix.
In math, there is another concept that makes use of the word base – indices. For example, if 2 was multiplied by itself 4 times to make 16 (2×2×2×2=16), we could simplify the notation by writing it in the form of indices – 24 = 16. In this scenario, 2 is our base, and 4 is the index (the singular form of indices).
While these two bases fall in distinct and separate concepts, numeral systems can actually involve the use of indices and bases.
To understand how, let’s take a look at the base-2 system – the basis on which many computer systems today operate.
A computer’s Central Processing Unit (CPU) – the “control center” of a computer – consists of many transistors that you may think of as on-and-off switches. When you run a program, the computer translates it into binary to be interpreted by these switches. 1 means the switch is closed or switched on, and 0 means the switch is open or switched off.
So, how does a base-2 numeral system work? Well, let’s start counting integers from zero with ones and zeros.
0000, 0001
And then we’re stuck. How can we represent the number two with ones and zeros? Well, we can think of it as replacing the tens place with a “twos” place, like as follows:
0000, 0001, 0010, 0011
Now, we have sufficiently counted to three. From here on, every time we’re “stuck”, we start to make use of an additional place value.
0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111
Congratulations! You’ve successfully counted to 15 using binary. In reality, computers use more than four digits to represent far greater numbers. Most computers today use a 64bit system, which means they store a string of ones and zeros 64 digits long. In the past, incidents such as Psy’s Gangnam Style surpassing 2,147,483,647 views on Youtube led to a phenomenon known as integer overflow, which “broke” Youtube’s then-32bit system.
However, it would be really tedious to count from zero every time programmers need to convert between base-2 and base-10. Is there a mathematical approach?
Well, fortunately for all programmers, there is. Since each place value can store two digits, we can utilise our understanding of the secondary school math topic of indices that we covered earlier to break down a long string of ones and zeros. Let’s take a 4-bit string of binary:
Remember how we thought of the tens place as a “twos” place? Let’s extend that logic to the other digits as well. In this scenario, we will then have an “eights” place, a “fours” place, a “twos” place and a “ones” place. Notice a pattern? Each additional place value corresponds to a sequentially larger integer index assigned to base 2 – 8 is 23, 4 is 22, 2 is 21, and 1 is 20. Hence, we can write the number as follows:
In this manner, 2 is both the “base” in the base-2 numeral system, and also the base in every instance of use of indices to convert between base-2 and base-10.
Standard Form refers to the format whereby a number is expressed as a number between 1 and 10, multiplied by 10 to the power of an integer.
The ten Arabic numerals – that’s your digits 0-9 – are used to represent nearly every other number in mathematics, aside from imaginary numbers and a few irrational numbers like π and e. When we wish to represent ten, the number after 9, we make use of an additional place value – the tens place, and then add one to it.
Similar to the base-2 system, each additional place value corresponds to a sequentially larger integer index assigned to base 10.
In this way, we are referring to 1733 as the sum of the numbers 1000, 700, 30, and 3 – individually expressed in standard form. To obtain a number between 1 and 10, we hence use the largest index – 3 – to be assigned to base 10.
At TSV Math Tuition Centre, we believe in making learning math fun and accessible for everyone. It’s important when studying math to not study in a vacuum – math has a wide range of interesting applications, both big and small, that are there for us to appreciate.
While we’ve covered indices and bases in both base-2 and base-10 systems, the truth is there are far more numeral systems beyond these two. In fact, while not as extensive as the base-2 and base-10 systems we just covered, much of our day-to-day life incorporates some part of other numeral systems.
For example, learning to convert between hours, minutes and seconds is a key component of the primary school math curriculum. Just like the base-2 and base-10 systems we’ve covered, when expressing time in a combination of hours, minutes and seconds, the unique digits for minutes and seconds would be limited to the 60 unique integers from 0-59, and hence can be said to be a form of use of a base-60 number counting system.
If we would like to express a certain time as a certain number of seconds from the most recent midnight before it, we could utilise the same method we’ve been using to do so:
Why would we ever want to store time as seconds? Well, interestingly enough, for technical reasons, the most common way that time is stored on a computer is using a piece of code called the Unix Time Stamp, which stores it as the number of seconds that have elapsed since January 1st, 1970 at Coordinated Universal Time (UTC).
Nonetheless, counting hours, minutes and seconds is a core part of the primary school math curriculum that, even if unknowingly, familiarises children with a base-60 numeral system.
While the depth and breadth of the math syllabus for O Level and even H2 math may seem daunting at first, getting a good grasp of the foundations is the first step to mastery. In truth, much like how base-60 numeral systems are quietly learnt by children all over the world, many of these math topics in the syllabus quietly build upon previously learnt foundations in other concepts and topics. It is for this very reason that it is so important to have the proper support available for learning maths.
At TheSequoiaVision, we understand there is no one-size-fits-all approach to learning. With our diagnostic class, our teachers pinpoint students’ weaker topics to strengthen their foundation, even before they start any classes. After building a strong foundation, they then go on to deepen their mastery in math, by being exposed to a larger variety of questions and question types.
Whether you’re in Primary School, Secondary School, or preparing for your JC H2 math exams, you’ll find that TheSequoiaVision has the perfect support system for you – find out more about our values, plentiful resources, expert consultations, and flexible class schedules on the TSV math tuition website here.
At TheSequoiaVision, we’re passionate about making math fun and accessible for everyone! Whether you’re in Primary School, Secondary School,, or preparing for your JC H2 math, we’ve got the perfect support system for you. With teachers who are math experts with strong teaching experience, be prepared to excel with us through our proven track record!
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