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I make the seemingly fantastic claim in my two pages on
basic, single-digit addition and
basic, single-digit multiplication,
that all of math, at all levels,
involves nothing more than those simple operations.
Ok, that's a slight overstatement. The presumption is that you know what numbers are, and have mastered counting. If you ask, "But what about subtraction and division?", the answer comes barreling back:
Subtraction is bound up in addition; subtraction is addition just going the other way, stepping backwards instead of forwards; subtracting a number is the same as adding its negative; subtraction is actually performed in the mind using addition skills. You don't keep a spot in your brain for 11-7, and you'd better not be counting back! You think, "7 plus what gets me up to 11? Oh yeah..."
Likewise, division is bound up in multiplication; division is multiplication just going the other way, jumping backwards instead of forwards; dividing by a number is the same as multiplying by its inverse; division is actually performed in the mind using multiplication skills. You don't keep a spot in your brain for 42÷7, and you'd better not be jumping back to 0 by 7s! You think, "7 times what gives me 42? Oh yeah..."
Multiplication is the next operation up in power from addition. If you challenge me on exponentiation, the next operation up in power from multiplication, I say that exponentiation is just repeated multiplication, and is, in fact performed with multiplication and addition skills. Although some of us may know a few of the teensy-tiniest exponentiation facts, such as 33 and 34, no one memorizes exponentiation tables.
Yes, you have to learn some definitions, such as perimeter, angle, average, radius, tangent, derivative, standard deviation, etc., but working with those things will never involve more than the application of addition and/or multiplication.
Mr. Morabito is a math instructor at the college where I work. He knows my claim that "all of math is just addition and multiplication"; that mathematics is the only profession on earth in which you only need to know two things! I mean, can you imagine a gardener, plumber, auto mechanic, or ballerina only knowing two things??? Ok, I'm being a little facetious there, but not completely. Mathematics would do itself a big favor by emphasizing how little there is to it, really, rather than by intimidating everyone with its supposed enormity. You can find math practice sites which trumpet, "Practice in 265 sixth-grade math skills!", for example. Right, 265 ways to diddle with addition and multiplication...
Anyhow, one day I came up to Mr. Morabito while he was helping a student with a problem. He turned to me and said, "There. Solve that using just addition and multiplication."
The problem was,
Solve for x: Log x + Log(x-21) = 2 (1)
As if logarithms are going to scare me off. A logarithm is just an exponent, and I've already said that exponentiation is just repeated multiplication.
On with the show...
If a = b, then 10 times itself a times must equal 10 times itself b times. That is, 10a = 10b. Or, just apply the basic logic, "whatever you do to one side of the equation, you do to the other," slapping the same base (the thing that gets repeatedly multiplied) under each side. So (1) becomes,
10Log x + Log(x-21) = 102 (2)
Now, 10 times itself (a+b) times equals 10 times itself
a times times 10 times itself b times. That is,
10(a+b) = 10a · 10b. So (2) becomes,
10Log x · 10Log(x-21) = 100 (3)
The Log of a means the number of times 10 has to multiply itself to reach a. That is, 10Log a = a. So (3) becomes,
x · (x - 21) = 100
x2 - 21x - 100 = 0
(x - 25) · (x + 4) = 0
x = 25, -4 (Faint-hearts may throw out the -4.)
There. No mathematical activity whatever besides addition and
multiplication.
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