Basic mathematics (basic arithmetic) are about manipulating numbers, so they govern the behavior of numbers. Mathematics cannot exist without numbers (ability to count) or our ability to compare and our ability to just know when two or more objects are similar or different. In this serial, we mark with a tally when we visualize quantity.
We use mathematics everyday when we want to know how much “time” we have left or counting the total number of fruits that you have harvested from your backyard. The semantics of the word “time” will be discussed in Serial 11 and 12.
Mathematics is also best written, so we have to use a suitable number writing system and mathematical symbols as shorthand.
The words we have learnt from elementary school for manipulating numbers are add, subtract, multiply and divide. We also learnt how to use the symbols for those words along with the Hindu-Arabic number symbols. Without a knowledge of place value and the best number display system for our vocabulary, mathematics will no make sense to us.
As we have seen in the previous serial, the proper sequence of numerals which form the continuum of numbers made measuring devices useful. While we are measuring something we might add or subtract any of our measurements.
We will go back to using sticks or tally marks to demonstrate addition and subtraction in the diagram below.
It means combining the value of two or more numbers. It also means adding more beads to the lower part of the abacus, thought it won’t be illustrated here. As we read aloud the sequence of number, we are adding by one. We have created the symbol (+) for addition. It is read aloud as “plus”.
The equals (=) symbol is the result we get after adding, subtracting, multiplying or dividing a number.
Practical Example: A neighbor has given to you some of her apples to taste and you also have your own apples from your backyard in the cupboard.
First, before your put her apple in your cupboard, you count your own apples, one tally or stick on your record book, one apple.
Now you say it numerals: “six” apples. Then you count with a tally the number of apples your neighbor gave you on your record book:
You counted three. You add it to the number of apples in your cupboard. We immediately visualize things growing when we add something.
You read aloud the numerals after six in the defined sequence. You say: I now have “nine” apples. You remember you have seen apples like that (typical apple). Anything that has a similar pattern can be added (included) to another group (categorizing), You know they are not apples when they are soft. In mathematical notation:
It is read aloud as “six (6) plus (+) three (3) equals (=) nine (9)”.
Subtraction means removing or taking away a number from a number. It also means removing beads from the lower part of the abacus. As we do a countdown, the number symbols read aloud in reverse means we are subtracting by one. We have created the symbol (-) for subtraction (-). It is read aloud as “minus”.
Practical Example: You count the number of apples when you wake up in the morning and count the number of apples your family has eaten throughout the day.
When you wake up the next day, you look again in your cupboard, you count “nine” apples. You know that you apple tree still has not produced anything ripe.
As the day progresses, you grab some for snacks at different moments. You mark the snack you enjoyed with a tally on your writing paper:
You say you have four moments you took a snack. You write these tally marks on your record book.
Now you say one snack, one apple, so how many apples I have left in my cupboard, so you cross out the counting you did before:
When something is subtracted things feel smaller, you know something had disappeared in your cupboard. You do a countdown based on the tallies you have crossed out.
You now say you have five apples. You can say it number and mathematical symbols:
It is read aloud as “nine (9) minus (-) four (4) equals (=) five (5)”.
Multiplication means duplicating a number by some number. It can be treated as a larger form of addition, because we are combining groups of numbers. We have created the symbol (×) for multiplication. To help remember the number when multiplied, we were asked to memorize the times tables (1-12). The numbers that are being multiplied by each other are called factors.
Practical Example: After fitting the number of apples in a single container, you want to create more containers to collect the fruits in your backyard. You ask, how many apples can now fit in seven (7) containers?
You design and build a container (basket) out of dried leaves that can fit six apples. You did this because this is more convenient than bringing an apple one by one from your tree to the cupboard.
You then think: “If I go back and forth seven times to my backyard carry apples for my simple physical exercise routine, how many apples will I store in my cupboard?”. This is one example of simple creative visualization.
To make it easier to imagine apples, you represent one apple as 1 tally mark. You cluster six tally marks into one rectangular box. The box also represents the basket or the container.
Then you imagine seven of those boxes:
Now as you point at each tally mark with your pen you read aloud the sequence of numerals. You reach forty-two. In Hindu-Arabic, it is “42”. You now say “I need to store 42 apples in my cupboard”. In mathematical symbols it is written as:
It is read aloud as “six (6) times (×) seven (4) equals (=) forty-two (42)”.
So that you don’t have to count each tally mark they created the multiplication tables (times table) for you to memorize as an elementary school student.
Example: “What numeral will I reach when I cluster four tally marks in one square box and visualize eight triangular boxes?”
Oh, the times table is very convenient, I don’t need to point at each tally mark with my pen I read aloud the sequence of numerals. I remember seeing 32 when I looked at my times table.
Below are some examples of multiplication tables for the square box and the rectangular box.
Position can be used to keep track or what we are multiplying. This is the precursor to algebra:
Division means reducing a number by some number, or grouping numbers evenly. It can be treated as a larger form of subtraction and a reverse of multiplication, meaning we are separating numbers. In the times table, any number at the right of the equals symbol (=) can be reversed with any number at the left of the equals symbol to give back one of the values. So the times table is also a tool for division.
Practical example: You have counted all of the ripe fruits you have harvested from your backyard. How many containers will you need for all you have harvested?
At harvest time for your apples, you leave them on the sundial in your backyard as you harvest them one by one. You mark on your record book tally marks to represent one apple. You found that you have harvested 57 apples.
Now you ask how many trips from my table to the cupboard I need, to store all my apples, now that I still using a basket that can store 6 apples?
How you look for any number close to 57 in the times table:
Based on the sequence, they are 54 and 60. Next you focus first on the number of tally marks on each box, which is six (6). You say you need ten (10) trips from your backyard sundial to your cupboard.
You know that if take only nine (9) trips there will three (3) apples left in the sundial plate after subtracting 54 from 57 [ 57 – 54 = 3 ]. So the three (3) need another trip, this is called the remainder in division.
The remainder can be expressed as a fraction, as will discussed below. The denominator is 6 (number of apple that can fit in to the box). The remainder (3) can be expressed as a numerator. The remainder can now be expressed in fraction form: 3/6. Simplifying: ½
Also, You know one trip is worth one box, because as you fill the cupboard, you need to empty the box, and the number of apples in the cupboard will increase, as you make each trip.
It is read aloud as “fifty-seven (57) divided by (÷) six (6) equals (=) nine (9) with remainder of three sixths”.
You were also taught that the number sequence continuum: 1, 2, 3, 4, 5 … are whole numbers. What they mean by whole is that they are not split.
But as you know from experience, one whole cake can be split into several slices to share with your party-mates. We use ordinal numerals to describe fractions below.
We can split the number one into two, forming a half. We can split the number 1 into three, forming a third. You can split the number 1 into four, forming a fourth or a quarter. We can split the number 1 into five forming fifths. We can split the number 1 into eight, forming eights. We commonly split a cake into eight smaller slices for our party mates. The graphics below will visualize this.
We can also simplify fractions: As shown in the graphics below, if a cake is cut into eights and there are two remaining slices, we say there are two eights left. It can be simplified into: a quarter is left, because two eights looks like a quarter in your eyes.
But we have a special case with fractions too: If we split one whole into ten to form tens, it is called a decimal fraction, or simply a decimal. Note the semantic difference between decimal fractions and the decimal display system as previously discussed. Due to our decimal English vocabulary, decimal fractions have become the most convenient way to write fractions, we simply put a decimal fraction dot as a shorthand.
How fractions, multiplication and division are related
But the number one is a universal whole and the numbers 2, 3, 4, 5, 6, 7, 8 and so on are only multipliers of 1 because they are defined to be greater than 1 and they are used for indicating individual and not combined wholes. When we see the same object such as an apple, we focus on an object one at a time, treating it as a separate and individual whole as we point and count.
We can only visualize individual and combined objects in our mind. We can visualize a cluster of individual objects as one object. Example: a box of apples has individual apples inside. We can also treat a cluster of objects (boxes) as an individual object. This was demonstrated in the multiplication graphics above.
The number 1 can be split into smaller parts called fractions. The split objects are smaller than the previously uncut whole, just as you split a cake for your party mates. The split objects look like individual wholes now and you can use the sequence of numbers.
When you see more individual whole cakes, you count the number of cakes with whole numbers. Let’s say you can put/combine 3 individual cakes into 1 whole box. The one box of cake is larger than that individual cake, according to your sense of sight. So you see that the box is much bigger than the individual cake. You can take the cakes out of the box to get back the individual 3 cakes. This is also what we did with the apples above.
This is what you also see in cooking, you combined all the separate measured ingredients to make one whole food recipe (multiplication). You cook the recipe in a single cooking vessel, only when the food is cooked, you share with your party mates by dividing it evenly to their desired amounts. You split the food with your knowledge of fractions (division). They are demonstrated above.
This is why fractions represent division (create individual wholes by splitting or separating) and combining represents multiplication.