Ordering, Comparison and Place Value Core Skills
Number • Structure and calculation
Flashcards
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Key concepts
What you'll likely be quizzed about
Place value fundamentals
Place value assigns value to each digit depending on its position relative to the decimal point. Digits left of the decimal represent units, tens, hundreds and higher powers of ten; digits right of the decimal represent tenths, hundredths and smaller powers of ten. Clear identification of these positions allows direct comparison of magnitude because higher place values always outweigh lower ones.
Comparing whole numbers and integers
Comparison of whole numbers uses place value first: the number with more digits is larger. When digits count equals, comparison proceeds left to right until a differing digit determines order. Negative integers reverse the order because smaller absolute value corresponds to a larger negative number; therefore -2 > -5 since -2 is closer to zero. Careful sign checking prevents ordering errors.
Comparing decimals
Comparison of decimals aligns digits by the decimal point and compares place by place, starting at the highest place value. Adding trailing zeros to equalise decimal places does not change value but assists direct comparison. For example 3.05 < 3.1 because 0.05 < 0.10 after alignment. Decimal comparison follows the same left-to-right rule as whole numbers once alignment is complete.
Comparing fractions
Fraction comparison uses common denominators or conversion to decimals. Converting to equivalent fractions with the same denominator makes direct numerator comparison valid; converting to decimals allows place-value comparison. Cross-multiplication provides a quick inequality test: a/b < c/d if ad < bc for positive denominators. Sign handling and reduction to simplest form prevent incorrect judgments.
Symbols for equality and inequality
The symbols =, ≠, <, >, ≤ and ≥ state exact equality or order relationships. The equals sign (=) indicates identical values; not equal (≠) indicates different values. Less than (<) and greater than (>) indicate strict order, while ≤ and ≥ allow equality as well as order. Correct use of symbols requires prior determination of magnitude by comparison methods.
Standard form explained
Standard form represents numbers as A × 10^n where 1 ≤ A < 10 and n is an integer. Shifting the decimal point in A changes n: shifting left increases n, shifting right decreases n. Standard form simplifies representation of very large or very small numbers and clarifies magnitude because the exponent n indicates scale directly.
Converting to and from standard form
Conversion to standard form moves the decimal point in the number until one non-zero digit appears before the decimal; the number of moves sets n. For numbers greater than or equal to 10, move the decimal left and take n positive. For numbers less than 1, move the decimal right and take n negative. Conversion back multiplies A by 10^n to restore the original magnitude.
Calculating with standard form
Multiplication and division in standard form separate coefficients and powers of ten: (A × 10^n) × (B × 10^m) = (A×B) × 10^(n+m). After calculation, adjust the coefficient to satisfy 1 ≤ A < 10 by shifting the decimal and changing the exponent. Addition and subtraction require equal powers of ten first, so convert terms to a common exponent before combining coefficients.
Key notes
Important points to keep in mind