Math101 · Independent Ontario learning library
Math101learn.math101.caWhole Numbers
Whole numbers describe zero and positive counting quantities through place value, operations, factors, and estimation.
Place value gives every digit meaning, and that structure powers every whole-number calculation.
What whole numbers are
The whole numbers are
They include zero and all positive counting numbers, with no fractions, decimals, or negative values. Whole numbers model counts such as books, people, or completed items.
Place value
In base ten, each place is ten times the value of the place to its right. For example,
The zero is a placeholder showing there are no tens. Removing it would change the number.
Reading, writing, and comparing
Read a number by grouping digits into periods of three: ones, thousands, millions, and so on. Compare whole numbers first by digit count, then from the greatest place value toward the right.
On a number line, the number farther right is greater. Symbols $<$ and $>$ point toward the smaller number.
Addition and subtraction
Align digits by place value. Addition combines quantities; subtraction finds a difference, a missing part, or how much remains.
Inverse operations provide a built-in check.
Multiplication
Multiplication represents equal groups, arrays, area, or repeated scaling. The distributive property can break a calculation into friendly pieces:
Partial products and the standard algorithm describe the same place-value reasoning.
Division and remainders
Division can mean sharing equally or finding how many groups fit. For
the check is
Interpret a remainder in context: leftover people, a partial group, rounding up buses, or a fraction of a unit may all require different reporting.
Order of operations
Use grouping symbols first, then exponents, multiplication/division from left to right, and addition/subtraction from left to right.
For example,
The convention ensures everyone interprets the same expression consistently.
Factors and multiples
A factor divides a number exactly. A multiple is produced by multiplying the number by a whole number. For $24$, factors include $1,2,3,4,6,8,12,24$, while multiples include $0,24,48,72,\ldots$.
Prime numbers have exactly two positive factors; composite numbers have more than two.
Estimation and reasonableness
Round to a useful place value or use compatible numbers. Since $37\cdot24$ is near $40\cdot20=800$, the exact result $888$ is plausible; $8\,880$ would signal a place-value error.
Estimate before calculating when the size of the answer matters.
Common mistakes
Ignoring zeros inside a number. They hold place value.
Aligning digits rather than places. Ones must align with ones.
Treating a remainder the same in every context. Interpret the situation.
Performing addition before multiplication without grouping. Follow operation order.
Accepting an answer without estimating. A quick benchmark catches many errors.
Quick self-check
- Can I write the number in expanded place-value form?
- Are comparisons based on the first differing place?
- Are operations aligned and regrouped by place value?
- Can the inverse operation verify the result?
- Are factors, multiples, and remainders interpreted correctly?
- Is the answer close to a reasonable estimate?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Calculate 37 × 24.
- 37 × 20 = 740.
- 37 × 4 = 148.
- 740 + 148 = 888.
End of lesson
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