Arithmetic & Algebra

Basic arithmetic is critical in most aspects of life, and algebra has nearly the same usefulness.

What You Probably Know – Arithmetic

As early as kindergarten education, we start math with what we observe:

  1. Descriptions and demarcations for the things we observe, which become the beginnings of geometry.
  2. Groupings of similar things we observe, which become the basics of arithmetic through identifying quantities (e.g., 3 apples).
  3. Incrementing more of those quantities as an interval of 1 (e.g., 1, 2, 3, 4, 5, 6…).
  4. Addition by combining those groups together (e.g., 2 + 3).
  5. Subtraction by removing groups from other groups (e.g., 7 – 2).
  6. Using language to define the concepts as expressions (e.g., 5 + 1).
  7. Composing numbers in different ways to come to the same answer (e.g., fact families).
  8. Exploring applied mathematics by giving story-based problems.
  9. Using a base-10 number system to familiarize everyone with it as a mathematical convention.

From there, most of the effort is applied toward repetition and scaling upward:

  • Counting increasingly higher values (i.e., 100, then 1,000).
  • Adding and subtracting more proficiently and with higher numbers.
  • Giving more story problems to sufficiently expand on how it can be useful.
  • Further applied mathematics with things like time and money.

A bit later, we expand on the ideas:

  1. Defining the concept of equality using equal groups of various types (e.g., 3 boxes with 2 things each in them).
  2. Learning about commutative properties within mathematics (e.g., 4 + 3 = 3 + 4).
  3. Introducing multiplication as compounding addition (e.g., 5 + 5 + 5 = 5 x 3 = 15).
  4. Reversing the grouping of multiplication to create division (e.g., 15 / 3 is 3 groups, which are 5 each).
  5. Demonstrating fractions as partial integers that represent unresolved division (e.g., 1/3).
  6. Partial base-12 and base-60 calculations with time.
  7. Factoring values into smaller or larger components, which form the beginnings of algebra.
  8. The standard addition and subtraction algorithm:
    • 8 5 4 3
      - 1 3 2 2
      7 2 2 1
  9. Indicating place value relationship (e.g., 0.001 is 10 times more than 0.0001).
  10. Comparing, ordering, and rounding numbers (e.g., 513 rounded to the nearest 100 is 500).
  11. Multiplicative comparison of values, which is the beginnings of mathematical analysis (e.g., A is twice as large as B, 1,000 is 100 times more than 10).
  12. Multiplying and dividing multi-digit numbers by factoring the numbers.

Around 4th and 5th grade, there’s a bit of a divergence into partial numbers:

  1. Indicating how fractions can be equivalent (e.g., 1/3 = 2/4).
  2. Showing how fractions can decompose (e.g., 4/3 can become 1 1/3)
  3. Adding and subtracting fractions with the same denominator.
  4. Multiplying fractions together (e.g., 4 x 2/3 = 4 sets of 2/3 combined = 8/3 or 2 2/3).
  5. Expressing decimals to the tenth and hundredth (e.g., 0.1, 0.001).
  6. Indicating how fractions are both quotients and whole numbers.
  7. Multiplying fractions together.
  8. Adding, subtracting, multiplying, and dividing decimals.
  9. Adding and subtracting fractions with different denominators by factoring to a common denominator (e.g., 1/2 + 1/3 = 3/6 + 2/6 = 5/6).
  10. Understanding the concept of scale with fractions relative to each other.

At this point, many people hit a mental wall, usually because they have a hard time imagining the point of representing 8/4 and 2 at the same time, or why 1/2 and 0.5 are both valid numbers. Most mathematicians are not aware that math represents a perspective upon reality, and their obsession with certainty and abstraction blinds them from the fact that math itself is expressing different points of view. Since this all heavily compounds, algebra becomes impossible to explain later.

However, whether we understand it or not, the cumulative education continues:

  1. Introducing exponents and square roots.
  2. Introducing the concept of ratios (e.g., 1:2).
  3. Adding to fractions with percentages.
  4. Dividing fractions and decimals, which effectively multiplies the numbers.

Arithmetic Tricks

There are several tricks that can make arithmetic much easier to calculate.

You can do any arithmetic you want, as long as you do the same thing to the other side to keep it equal.

While using the same operator on the opposite side of the formula creates an equivalent result, using the opposite operator on the same side does it as well:

  1. x = 50 * 6
  2. x = 25 * 12
  3. x = 5 * 60
  4. x = 300

Anything multiplied or divided by a multiple of 10 simply requires moving a decimal point or adding a zero:

  • 65.1 * 10 = 651
  • 8749 / 10 = 874.9
  • Generally, removing decimals is less complicated than adding them.

Multiplying by 9 is simply the number times 10, minus one instance:

  • 11 * 9 = 110 – 11 = 99
  • 95 * 9 = 950 – 95 = 855
  • 856 * 9 = 8,560 – 856 = 7,704

Multiplying 2-digit numbers by 11 involves adding the 2 digits together, then placing the sum in between those 2 digits:

  • 12 * 11, add 1 + 2 to make 3, convert 12 to 132
  • 72 * 11, add 7 + 2 to make 9, convert 72 to 792
  • 57 * 11, add 5 + 7 to make 12, convert 57 to 627 (because the 1 carries leftward)

Multiplying and dividing decimals is easy:

  1. Move the decimals all the way to the right and count the number of places, which is effectively multiplying the entire operation by increments of 10 (e.g., 5.8 * 7.21 becomes 58 * 721 with 3 decimal places).
  2. Perform standard multiplication or division (e.g., 58 * 721 = 41,818).
  3. Put the decimal places back in, which is effectively dividing the entire operation by the same increments of 10 (e.g., 41,818 becomes 41.818).

Fractions, like equations, can be multiplied and divided by arithmetic as long as the numerator and denominator are treated the same:

  • 7/10 = 14/20
  • 8.25/11 = 17/22
  • 360/400 = 9/10

Fractions can be easily divided if you flip the numerator and denominator and multiply them:

  • (4/5) / (8/3) = 4/5 * 3/8 = 12/40 = 3/10

Percentages can be converted to to fractions by making it the percentage, divided by 100:

  • 59% = 59/100
  • 63.8% = 63.8/100 = 638/1000

A number can be divided in no particular order, so you can factor a large number step-by-step by dividing it piece-by-piece:

  • It can be divided by 2 if the last digit is even.
  • It can be divided by 3 if the digits added together are divisible by 3 (e.g., 732).
  • It can be divided by 4 if the last 2 digits together are divisible by 4.
  • It can be divided by 5 if the last number is 0 or 5.
  • It can be divided by 6 if the rules for 2 and 3 above apply.
  • It can be divided by 9 if the sum of the digits can be divided by 9 (e.g., 360).
  • It can be divided by 10 if the number ends in a 0.
  • It can be divided by 12 if the rules for 3 and 4 apply.

What You Might Know – Algebra

The line between algebra and arithmetic is most clearly distinguished by the existence of variables. For highly practical minds, it’s easy to get lost in algebra because it requires abstractly thinking with a type of certainty about an unknown thing (e.g., let x be a not-yet-established number).

With the presence of variables, the equations have many options for factoring:

  • 5 + x = 12 can become x = 7 or x – 7 = 0
  • x2 + 5 = 2 can be x2 = -3 or x = √-3 or x2 + 3 = 0

Most equations require “solve for □” to give an absolute concept, but the vagueness of algebra comes from its relativity, which is probably the largest failure in mathematics pedagogy:

  • Everything in maths requires understanding the precise purpose you wish to accomplish.
  • Rote memorization literally gives the opposite effect by provoking people to calculate by habit without any awareness of the grander picture of why the calculation should be done.
  • The result of this is that most students can be easily thrown off by the relative thinking necessary to perform algebra.

One of the most significant aspects of algebra is how to work with the distributive property [a (b + c) = ab + ac]. It allows more rearranging of values, which becomes very useful once multiple variables are introduced.

Another issue with algebra for the uneducated comes through a few issues with math symbols:

  • A set of symbols may have a specific meaning in one equation (e.g., f(x) meaning “function of x”) but then feeds later into another symbolic representation (e.g., f(x) becomes y).
  • It’s not uncommon for a parenthesis to represent either an order of operations or the constraint of an idea, context-depending:
    • f(3x) = x + (2y * 7)
  • Some symbols (such as π for circles or σ in statistics) are shorthand for very specific meanings and purposes.
  • All of this is compounded by mathematicians not spelling out each tiny detail regarding why an idea exists or its specific rules.

Irrespective, the cumulative algebra education continues:

  1. Indicating how proportional correlations in tables can represent as formulas:
    • Given how a cost of 4 correlates to 8 products, 8 correlates to 16, and 50 correlates to 100…
    • …the same expression could be written as c = 2p.
    • At the same time, proportion can only represent with multiplication and division, not addition and subtraction alone.

Logarithms

While most people are familiar with exponents, many elements of computing (such as crypto mining) use logarithms, which are the opposite of exponents. While an exponent becomes dramatically higher with each iteration (e.g., 3, 9, 27, 81, etc.) logarithms become decreased with each iteration as it progresses infinitely toward that number (e.g., 0.6, 0.8, 0.9, 0.95, etc.).

Variables

Linear Algebra

Linear algebra is, quite simply, using formulas as shorthand for tracking where a line is. It establishes an x-axis and y-axis (and sometimes a z-axis), then sets an arbitrary point as (0,0) or (0,0,0). In a mathematical sense, you can set points forward and backward along any of those axes into infinity, but in a computer sense there’s a memory limit to how far you can go.

The math operation F represents a line, and often tracks how much that line rises (positively or negatively) relative to how far it runs (positively or negatively). The function of x becomes F(x), and often represents as:

  • F(x) = x + 5

In a straight line, any segment of that line will have the exact same properties:

  • A 4-foot run with a 1-foot rise means that a 10x-sized run (i.e., 40 feet) will have a 10-foot rise.
  • If you add 8 feet to a 4-foot run with a 1-foot rise, then it would have a proportional rise (i.e. 1/4) of 2 feet.

However, the only “linear” algebra is technically multiplication and division, since exponents/squares create curves and addition/subtraction simply translates the line elsewhere while keeping it the same.