Intuition The ONE core ideaA decimal is just a fraction whose bottom number (denominator) is a power of ten — like 100 235 2.35 see that hidden fraction, all four operations (+ , − , × , ÷ keeping track of where the dot lands .
Before we can operate on decimals, we must earn every symbol the parent page throws at you. Nothing below assumes you have seen it before. We build from a single idea: a digit's value depends on which column it sits in.
A digit is one of the ten single symbols 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 value comes entirely from where you place it .
Look at the number 30 3 column .
Intuition Why columns existWe only have ten digit-shapes but we need to write every number. The trick: when we run out of shapes, we don't invent a new one — we shift left into a new column that is worth ten times more . That single rule is the whole reason numbers work.
The columns don't grow randomly; each is worth ten times the one to its right. We need a compact way to write "ten times ten times ten…", and that is what a power does.
Definition Power / exponentThe little raised number is the exponent . It counts how many times you multiply the base by itself .
1 0 2 = 10 × 10 = 100 , 1 0 1 = 10 , 1 0 0 = 1 1 0 2 repeat-counter , not a multiplier.
Keep dividing by ten past 1 0 0 negative :
Definition Negative exponent = "one over"A negative exponent means a reciprocal (one divided by that power):
1 0 − 1 = 10 1 = 0.1 , 1 0 − 2 = 100 1 = 0.01 Why negative? Because to the right of the point each column is worth ten times less , not ten times more. Going left, exponents rise; going right, they fall below zero.
Recall What is1 0 − 3 1 0 − 3 = 1000 1 = 0.001
You will need this idea again in Place Value and Powers of Ten and Scientific Notation , where "exponents add when you multiply" is the star rule.
The decimal point (the dot ".") is a fence . Everything on its left is a whole amount (1 0 0 piece of one (1 0 − 1 in a column — it sits between the 1 0 0 1 0 − 1
The word "decimal" comes from Latin decem = ten, because every column is a power of ten.
Place value means: the value of a digit = × expanded form :
347.28 = 3 ⋅ 1 0 2 + 4 ⋅ 1 0 1 + 7 ⋅ 1 0 0 + 2 ⋅ 1 0 − 1 + 8 ⋅ 1 0 − 2
Worked example Reading each digit's weightIn 347.28
the 3 300 1 0 2
the 7 7 1 0 0
the 2 10 2 = 0.2 1 0 − 1
the 8 100 8 = 0.08 1 0 − 2 300 + 40 + 7 + 0.2 + 0.08 = 347.28
Everything above lets us see the single most important sentence of the whole topic.
A fraction B A A B A numerator (how many parts you have); the bottom B denominator (how many parts make one whole).
Intuition Why this is the master keyOnce a decimal is a fraction, the four operations are no longer new rules to memorise — they are the fraction rules you meet in Fractions to Decimals conversion . Adding needs a common denominator (same power of ten → "line up the point"). Multiplying multiplies denominators (1 0 m ⋅ 1 0 n = 1 0 m + n scales top and bottom equally → "shift both points". Every rule the parent page states is this in disguise.
Definition The four operation signs
a + b addition : combine two amounts into one total.a − b subtraction : how much is left after removing b a a × b multiplication : a b b a a ÷ b division : how many b a a b b a
Definition The equals sign= exactly the same value — a balance scale, not a "here comes the answer" arrow. So 2.35 = 100 235
× Why it feels right: with whole numbers, × The trap: 1.2 × 0.05 = 0.06 smaller than both inputs! Multiplying by a number below 1 shrinks . Division by a number below 1 ÷ 0.15 grows the result.
Fix: think of × scaling , not "making bigger". A scale factor below one shrinks.
Padding means writing extra 0 12.4 = 12.40 = 12.400 Why is this legal? Because 10 4 = 100 40 0 same denominator so their columns line up.
Recall Does padding change a number's value?No. Trailing zeros after the point are empty columns; 12.4 = 12.40
Columns worth ten times more
Negative powers 10^-1 10^-2
Place value expanded form
Decimal equals fraction over 10^n
Padding to a common denominator
Four operations on decimals
Every arrow says "you need the box behind it first". The topic page (Operations on decimals — all four operations ) sits at the bottom — it only works because every foundation above is in place.
Test yourself: cover the right side and answer out loud.
What does an exponent count? How many times you multiply the base by itself, e.g. 1 0 3 = 10 × 10 × 10
What is 1 0 0 1 1 0 1 = 10 → 1 0 0 = 1
What does a negative exponent like 1 0 − 2 A reciprocal: 1 0 2 1 = 100 1 = 0.01
What sits between the 1 0 0 1 0 − 1 The decimal point (the fence between whole and part).
Write 347.28 3 ⋅ 1 0 2 + 4 ⋅ 1 0 1 + 7 ⋅ 1 0 0 + 2 ⋅ 1 0 − 1 + 8 ⋅ 1 0 − 2
What single fact turns every decimal into a fraction? A decimal equals a whole number over a power of ten, e.g. 2.35 = 100 235
Why is 12.4 = 12.40 Trailing zeros are empty columns; 10 4 = 100 40
What does the = Both sides are the exact same value (a balance), not "the answer follows".
Does × No — scaling by a value below 1 × 0.05
