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Of the 4 basic mathematical operations, this is the one which needs some explanation.
As a reminder, an integer is a whole number, be it positive or negative, or zero.
Here I have used the slash symbol, / rather than ÷. The latter is termed an obelus. In Scandinavia it means "minus", such as a sign outside a shop saying "Fiskboller ÷20%", 20% off fish balls; and in parts of southern Europe, a range of values "220 ÷ 240 volt", meaning a product can use 220 to 240 volts.
The symbol ≈ means "approximately equals", and is more correct than ≑, also used in this way at times.
There are several variations on division.
The simplest is to say you have a pack of 6 eggs, and eat two for each breakfast - how many breakfasts can you get from the box? 6 / 2 = 3
On the left is the Numerator, the number of objects we have. On the right is the Denominator or Divisor.
A couple have 4 eggs between them, and a box of 12 (1 dozen) how many breakfasts can the have? 12 / 4 = 3
What about a European holiday for three, so 6 eggs a day, but only 10 in a box? 10 / 6 = ???
In this case you get 1 full breakfast, and have 10 - 6 = 4 left over. Maybe the second day is scrambled eggs, or maybe they stop at a farm shop which will sell any number they want.
In this format you get an integer (whole number), and a remainder. This method is introduced first at school.
Lets say you you have 11 Tim-Tams (chocolate coated biscuits) in a pack, and you have 3 people at a meeting: 11 / 3 = 3, with a remainder of 2. Spreadsheet software, and various programming languages allow this kind of calculation.
Sometimes we need to know that there are 3 biscuits for each person; but other times the remainder is what we need: Suppose we want to know the direction something (or someone) is pointing after it (or they) has been rotating, and we know the number of degrees turned. 1080 / 360 is 3, with no (0) remainder, so the dancer is facing the same way as when she started spinning. If they turn 1620 degrees, 1620 / 360 is 4 times, with a remainder of 180, meaning they are facing the opposite direction to when they started.
Numerator / Denominator gives a Quotient and a Remainder. This method is called Euclidean division. In computing remainder is often called the "mod", or modulo operation.
A common question asks whether one number is divisible by another, such as "Is 10 divisible by 3?" As there is a remainder the answer is "No". Is 4 divisible by 2? Yes, as there is no remainder, or the answer is a whole number, without a decimal portion.
Not even infinity is big enough to provide an answer if you attempt to divide a value by zero. If you type 5 [÷] 0 [=] into a calculator you will get a message such as Error 0. Given V = P / I, no voltage, no matter how large can cause even 1 microwatt to flow with zero current.
If you just divide one number by another using a calculator, you get a single number as the result.
This might be a whole number, if you divide 2400 by 10, to get 240; a tidy decimal, like 25 / 2 = 12.5; or a repeating decimal. 10 / 3 is 3.333333, with the 3 going on forever. That number can however be expressed as 3 1/3 or 3⅓. Divide 10 by 7, and you get 1.42857142857142657... which repeats every 6 digits. .
Perhaps used more in the past, a value of 1.5 volts can be expressed as 1½ volts. Early camera flashes used a single use bulb containing fine magnesium wire, triggered by a 22.5 volt battery, termed as 22½ volts.
Fractions were used in the past as a way to approximate important values, such as those termed "constants". The most famous is π (named for the Greek character, pi). This is used to calculate the circumference of a circle from the diameter, by multiplication, or vis-a-versa, by division. It is also important in certain parts of electronics, especially radio. Before calculators, especially those which memorised π, most often 22/7 were used to approximate it.
22 / 7 ≈ 3.142857142857...
Pi (π) ≈ 3.141592653589793
You will notice that 22/7 repeats. Pi is a sequence of digits which give he appearance of being random; it has been defined to trillions of decimal places, and has not repeated. 22/7 would only be accurate to about 1 millimetre in 6 metres, although that is probably near enough or many practical uses. In school tests they usually presented a questions based around multiples of 7, or maybe 22. If a locomotive wheel has a radius if 3' 6", how far does it travel after one rotation? The diameter is 7 smelly feet, so it travels 7/1 × 22/7 = 7/1 × 22/7 = 22/1 = 22 feet. The strike-through with a pencil was used to show one value that the other cancels out.
Fractions were (or are, in one case) used heavily with inches. Halves, quarters, eighths, and sixteenths, and thirty-seconds, and so on, are common. Occasionally 12ths or 10ths are used.
Say you measure a screw with basic calipers, and find that it is 9.5 mm in diameter. It is too far from 10 mm to be metric, and too big to be BA, so it is likely fractional inch. You type 9.5 [÷] 25.4 [=] into a calculator, and get 0.37401574803. Maybe you got 9.6 mm, so did 9.6 [÷] 25.4 [=] and got 0.3779527559. You can consult a fraction chart or drill size listing, or something similar, and find that it is close to the value listed, 3/8", being 0.375. You call also try multiplying by 8, getting 2.99212598424 or 3.0236220472, both very close to 3, meaning 3/8". If you got an answer close to say 3.5 then the size would be 7/16".
Dividing one decimal by another certainly can provide an integer. If you find a brass item to be 76.2 mm in diameter you may wonder if it was made in inches, and dividing thus: 76.2 [÷] 25.4 [=] gives 3. While the machinist may have worked in inches, if they may have done 3 [Enter] 25.4 [×] on their RPN calculator, and got 76.2 mm.
You will see more mm is bigger, and more inches are bigger, meaning that there is a proportional growth by increasing either. Another example is: The greater the voltage across a resister, the more current flows through it; the more current flowing through a resistor the greater the voltage across it. The latter can be used to measure current, a small known resistance means a proportional increase in the voltage. E = IR so is 10 ampls flows through a 0.01 ohm resistor. 10 × 0.01 = 0.1 volts, or 100 millivolts. This can be read using a low cost voltmeter module set to the 200 mV range.
Some calculators with output answers in fractions, depending on the mode used.
Division on 10, 100, 1000, or 1,000,000 can be done easily, just by moving the decimal point. Say we have calculated an antenna 4150 mm of timber. If we move the decimal 3 places to the left we divide the number by 1000, to get 4.15 metres. 4.2 metres as a multiple of 0.3 metres is a commonly available size in Australia. For multiplication, if a club net is on 3.6085 MHz, this can be expressed as 3608.5 kHz by moving the decimal 3 points to the right, multiplying the number by 1000. This is done with little though by native metric users.
We can also mentally say that a 4-pack of 500 ml energy drinks will weigh (or rather, have a mass of) just over 2 kilograms, 500 × 4 = 2000, then divide this by 1000 by shifting the implied decimal 3 places to the left, to get to 2 litres, which for water-like liquids is approximately 2 kilograms, plus packaging.
As an aside, the building industry, including things like kitchen design, often uses minor and major units, being 100 and 300 mm in the vast majority of countries; and 101.6 and 304.8 mm, aka 4 and 12 inches in the other.
Light, and thus radio signals, travel at about 299,700,000 metres per second, although you will often see 300,000,000 m/s. Frequency is measured in Hertz*, so a radio might transmit on 144,100,000 Hz. 299700000 [÷] 144100000 [=] is a lot of keying. Thankfully we can chop 6 zeros off both the top and the bottom. 299.7 [÷] 144.1 [=] gives exactly the same answer, 2.07980569049271. We can halve or quarter this to make elements, or multiply by 1.002 to get the length of wire needed for a loop. Multiplying the former element lengths by 0.95 factors in "end effect" and/or the velocity factor of RF flowing in the element. For a quarterwave this woudl calculate as 0.493953851492019,which we woudl roand to 494 mm.
* Previously called "cycles per second", or cps or c/s; and thus kc, mc, or kmc; being kilocycles, megacycles, and kilo-megacycles, in place of gigacycles (GHz).
Division into a factor or constant is bidirectional! That is, 300 / x. Thus if you forget the frequency of a certain band, such as 15 metres, you can just do a quick 300 / 15, and get 20, and hopefully there is only one answer remotely near it, this being 21-odd MHz. Without a calculator you can reduce this to 100 / 5, then to 20 / 1, meaning 20.
As a further example many countries measure the ICE vehicles by saying how many litres of fuel necessary to travel 100 kilometres. In the past miles which can be travelled using one imperial gallon was used. In another place miles per one US gallon is still used. Simpler, a few countries use kilometres per litre, perhaps informally. In the last case just divide the known value into 100. Thus 20 km per litre is 100 / 20 = 5 litres per 100 km. An 4x4 might get 20 litres / 100 km, so the 5 indicates 5 km per litre.
Thus you could say that there is an inverse relationship between what we might call consumption (litres per 100 km), and efficiency (km per litre, or MPG).
The Numerators are 282.481 for Imperial gallons, and 235.215 for US.
Negative numbers are used in a wide range of ways. If you use Celsius temperature are relative to the freezing point of water, so on a chilly day (or night) how far below his point is expressed with a simple - sign in front of the value, say -4°C or -4℃ (yes, -4° does look like a guy on the potty). Clearly, the fact that this 269.15 K shows it is only relative. However, in other cases it means that you owe the bank an amount, or that current flows from a battery, rather than it charging the battery. Audio amplifiers often use both positive and negative rails, as speakers both pull and push.
Negative numbers generate a negative when divided by a positive number. If we have a -10 volt rail, and a voltage divider made from two identical resistors, the output will be: -10 / 2 = -5 volts
A positive over a negative is negative: 5 / -2 = -2.5
Division of a negative by a negative gets a positive. Examples may be odd, but say you paid $20 from an account, which shows as as -$20 on your bank statement / app, for 20 special bolts. The company finds they only have 10 in stock and no more are available, so refunds / reverses half the payment. -$20 / -2 = a credit of $10, so your statement shows +$10.
Sometimes the polarity, being plus or minus, does not matter. If you were walking around with a 3 metre antenna on an 11 metre (CB) or 10 metre band handheld or man-pack radio, and it touched a 1500 volt DC overhead rail power line, you and the radio will likely cease to function, no matter if it is positive or negative (it is positive). This is termed the absolute value, for example: |-5| = 5; or |4-5| = |-1| = 1. On a calculator it is the ABS key, or potentially |x|. Most programming languages express the function as ABS(), such as y = ABS(x); and spreadsheets follow the pattern =ABS(C3).
"Magnitude" is an alternative term.
Ambulances in Australia may have a T-pattern 2-pin outlet for heating pads in mobile baby cribs. These do now care which way the electrons flow, just that they do, and that it is around 12 volts. That is, only the absolute value matters. Thus if you need to establish a station during an emergency or exercise powered from an ambulance, be aware of this, as all the ICs and transistors in radios definitely do care.
When formulas are written out attention needs to be paid to the order in which calculations are performed. If you use base level social media such as FB, you will see people who spent too much time on external studies (looking out the window) during maths arguing that 2 + 3 × 4 is 20, instead of 14, being 3 × 4 = 12, then adding 2, so 2 + 12 = 14.
If a total current includes a constant current and 1 voltage / resistance calculation, it can be written as I = 0.1 + 12/240 means that first you must assess 12/240 = 1/20 amps, or 0.05 amps (50 mA). Add the 0.1 to that, and get 0.15 amps. If you just type this into a $2 calculator you likely get 0.050416666.
Say we have a 24 volt regulated supply and an LED with a forward drop of 2 volt, with 2 resistors limiting the current. The maths becomes: How much current flows in a 1200 plus a 1000 ohm resistor in series with 22 volts across them? We can use to use brackets (parentheses), to form I = 22 / (1200 + 1000) = 22 / 2200 = 1/100 = 0.01 amps = 10 mA. If you write the 22 over a long bar with the resistor values summed below groups that addition without the need for brackets.
See: Wikipedia: Order of operations
Apparently, if you want the square of negative 4, you need to use (-4)² for clarity, the answer being +16, otherwise it may be read as the negative of the square of 4, -(4²), being -16. All a bit strange.
Have you ever seen a calculator with no [=] key, but an [Enter] bar instead? Early calculator engineering was done by Hewlett Packard before the became ink slingers. Early scientific and financial calculators, such as the HP-12C (financial, still sold), HP-15C, and the highly sought-after HP-16C use a different order.
First you type the first value, then press [Enter}, then the second, and press the operation. 65 [Enter] 2 [×] gives 130. You do not need to use Enter for single value operations: 7 [x!] gives 5040.
You can do calculations progressively, such as for a circuit with 4700 ohms at 240 volts: 240 [Enter] 4700 [÷] displays 0.051063829787234, (0.51 amps), and continuing with 240 [×] gives 12.2553191489362, meaning 12.255 watts.
Note that these are programmable, meaning that they are not permitted for these exams. Very old HP-35, perhaps the improved HP-45, Sinclair, and perhaps some Soviet calculators are among the older RPM calculators which are not programmable, so permitted for these exams. There is also a modern hardware emulator of the Sinclair.
The most common meaning of "Reciprocal" in maths is the "Multiplicative inverse".
On a (scientific) calculator pressing this key with anything other than 0 on the display will cause it to display the inverse of it, expressed as 1/x. Pressing 5 [1/x] means the calculator displays 0.2. Entering 0.02 [1/x] displays 50. This is relationship is 20 milliseconds to 50 Hz in mains power. 60 [1/x] displays 0.0166666666666667 seconds, or 16.666667 milliseconds.
If you have a basic calculator you can type 1 [÷] 400 [=] and so get 0.0025
x cannot be 0, due to the rule against division by zero.
The practical use is then you have calculated the bottom part of a formula, then need to divide that by the top part. You can divide by the top number, then hit the [1/x] key; or you can hit that key, then multiply by the top part.
To find the reactance of a capacitor the following formula is used: Xc = 1/(2πfc). This is because a capacitor has less reactance the higher the frequency, meaning it passes more current. Note that Pi or π is a constant which is the ration of the diameter (radius × 2) of a circle to its circumference, but it also applies to electrical cycles. It is about 3.14159265358979. On some calculators it is the second or third function of a key, so you must press [f], [g], [2nd] or similar first to obtain it.
Say we have a 100 μF capacitor with 400 Hz across it.
Xc = 1 / (2 × π × 400 × 0.0001). Key: 2 [×] [π] [×] 400 [×] .0001 [=] and you get 0.251327412287183. Press [1/x] and you get 3.97887357729738, meaning about 3.98 ohms.
Say we have a 0.47 μF capacitor with 6 kHz across it.
If you use a RPN (Reverse Polish Notation) Calculator, such an physical HP one, an app, or this online one you can use this key sequence in the example below. You may have to manually enter 3.14159 in place of the [π] key below.
Xc = 1 / (2 × π × 6,000 × 0.00000047). Key: 2 [Enter] [π] [×] 6000 [×] .00000047 [×] and you get 0.0177185825662464. Press [1/x] and you get 56.4379230822324, meaning about 56.4 ohms.
The lowercase Greek latter mu, μ means micro, meaning 1 millionth, 1/1000000, or 0.000001. As, until super- or ultra-capacitors came along, a Farad was a very large value. Thus values are typically expressing in μF, Occasionally written uF, even for traditionally large values, such as 80,000 μF, one of the power supply capacitors in a power amplifier. Very small capacitors are marked or measured in pico farads, pF, occasionally uuF or μμF. This is 1 over 1 trillion, 1/1000,000,000,000. A capacitor marked "154" meaning 15 × 104, or 150,000 pF, which may also be marked in nano farads, as 150 nF.
Off topic, but placing an exclamation mark (!) after a number has a specific mathematical meaning, so it is best not to use it. "I bought my mates some beers, and it cost $100!" does not mean it cost a number of dollars significantly greater than the number of atoms in the universe. Maybe use (!).
! after an integer means you multiply it by each smaller integer, down to 1. Pronounced "Five Factorial" this is an example:
5! = 5 × 4 × 3 × 2 × 1 = 120. This escalates very quickly, with 15! exceeding 1 trillion.
That said x × 1/x = 1! is valid, because 1! = 1 × 1 = 1. 😁
You can read more at: Wikipedia: Factorial
Maybe a better way to write something surprising as: The voltage on the feedpoint of an end-fed halfwave (EFHW) antenna fed via a 1:49 unun at 1500 watts: V = √(1500 × (50 × 49)) = 1917.02895126808 (!). You'd REALLY feel that if you touched it! (While perhaps not in this case, note that the voltage at the end of some antennas, where the current is lower than at the feed-point, may thus be much higher).
For division you work left to right. Divide 690 by 3: 6/3 = 2; 9/3 = 3; 0/3 = 0. Put these together and you get 230.
Divide 240 by 5. 2/5 is no-go (or rather, 0, carry 2). 24/5 is 4, with 4 to carry. 5/40 is 8. Write 48. A shortcut for 5 is to divide by 10, and double: 24 × 2 is 48. These are the order of currents involved for the water heater at a small restaurant. A unit the same size as a domestic one might be used, but with extra elements to allow a larger flow.
Divide 240 by 20 (old Christmas lights had 20 lamps in series across this voltage): 2/2 = 1; 2/4 = 2; 0/2 = 0; giving 120, then move the decimal one left, to get 12 volts across each lamp.
Transformers which put out 12.6 volts are a hangover from the valve / tube era, being a common filament or heater voltage. They were often centre-tapped: 12.6 / 2 provides 6; and 6/2 gives 3. Clearly, the decimal goes between these digits, giving 6.3 volts, also a common filament voltage.
Unless you are fortunate with values, such as 480 / 12, where 48/12 = 4, and 0/12 is 0, giving 40 then you will need to use "Long Division". You can do a web search for this.
How many 1.5 volts cells do you need to get 6 volts. While the answer is perhaps obvious, you kind of have to carry the 6 to be 60, and divide by 15, making 4.
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Written by Julian Sortland, VK2YJS & AG6LE, March 2026.
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